Merge branch 'dev'
This commit is contained in:
commit
73c3b35631
34
BACKLOG.md
34
BACKLOG.md
|
|
@ -1,19 +1,40 @@
|
||||||
Backlog
|
Backlog
|
||||||
=======
|
=======
|
||||||
|
|
||||||
Prove postulates in `Cat.Wishlist`
|
Prove postulates in `Cat.Wishlist`:
|
||||||
`ntypeCommulative` might be there as well.
|
* `ntypeCommulative : n ≤ m → HasLevel ⟨ n ⟩₋₂ A → HasLevel ⟨ m ⟩₋₂ A`
|
||||||
|
|
||||||
Prove that the opposite category is a category.
|
Prove that these two formulations of univalence are equivalent:
|
||||||
|
|
||||||
|
∀ A B → isEquiv (A ≡ B) (A ≅ B) (id-to-iso A B)
|
||||||
|
∀ A → isContr (Σ[ X ∈ Object ] A ≅ X)
|
||||||
|
|
||||||
Prove univalence for the category of
|
Prove univalence for the category of
|
||||||
* sets
|
* the opposite category
|
||||||
* functors and natural transformations
|
* functors and natural transformations
|
||||||
|
|
||||||
Prove:
|
Prove:
|
||||||
* `isProp (Product ...)`
|
* `isProp (Product ...)`
|
||||||
* `isProp (HasProducts ...)`
|
* `isProp (HasProducts ...)`
|
||||||
|
|
||||||
|
Ideas for future work
|
||||||
|
---------------------
|
||||||
|
|
||||||
|
It would be nice if my formulation of monads is not so "stand-alone" as it is at
|
||||||
|
the moment.
|
||||||
|
|
||||||
|
We can built up the notion of monads and related concept in multiple ways as
|
||||||
|
demonstrated in the two equivalent formulations of monads (kleisli/monoidal):
|
||||||
|
There seems to be a category-theoretic approach and an approach more in the
|
||||||
|
style of functional programming as e.g. the related typeclasses in the
|
||||||
|
standard library of Haskell.
|
||||||
|
|
||||||
|
It would be nice to build up this hierarchy in two ways: The
|
||||||
|
"category-theoretic" way and the "functional programming" way.
|
||||||
|
|
||||||
|
Here is an overview of some of the concepts that need to be developed to acheive
|
||||||
|
this:
|
||||||
|
|
||||||
* Functor ✓
|
* Functor ✓
|
||||||
* Applicative Functor ✗
|
* Applicative Functor ✗
|
||||||
* Lax monoidal functor ✗
|
* Lax monoidal functor ✗
|
||||||
|
|
@ -24,7 +45,8 @@ Prove:
|
||||||
* Monad
|
* Monad
|
||||||
* Monoidal monad ✓
|
* Monoidal monad ✓
|
||||||
* Kleisli monad ✓
|
* Kleisli monad ✓
|
||||||
* Problem 2.3 in voe
|
* Kleisli ≃ Monoidal ✓
|
||||||
|
* Problem 2.3 in [voe] ✓
|
||||||
* 1st contruction ~ monoidal ✓
|
* 1st contruction ~ monoidal ✓
|
||||||
* 2nd contruction ~ klesli ✓
|
* 2nd contruction ~ klesli ✓
|
||||||
* 1st ≃ 2nd ✗
|
* 1st ≃ 2nd ✓
|
||||||
|
|
|
||||||
19
CHANGELOG.md
19
CHANGELOG.md
|
|
@ -1,6 +1,25 @@
|
||||||
Changelog
|
Changelog
|
||||||
=========
|
=========
|
||||||
|
|
||||||
|
Version 1.4.1
|
||||||
|
-------------
|
||||||
|
Defines a module to work with equivalence providing a way to go between
|
||||||
|
equivalences and quasi-inverses (in the parlance of HoTT).
|
||||||
|
|
||||||
|
Finishes the proof that the category of homotopy-sets are univalent.
|
||||||
|
|
||||||
|
Defines a custom "prelude" module that wraps the `cubical` library and provides
|
||||||
|
a few utilities.
|
||||||
|
|
||||||
|
Reorders Category.isIdentity such that the left projection is left identity.
|
||||||
|
|
||||||
|
Include some text for the half-time report.
|
||||||
|
|
||||||
|
Renames IsProduct.isProduct to IsProduct.ump to avoid ambiguity in some
|
||||||
|
circumstances.
|
||||||
|
|
||||||
|
[WIP]: Adds some stuff about propositionality for products.
|
||||||
|
|
||||||
Version 1.4.0
|
Version 1.4.0
|
||||||
-------------
|
-------------
|
||||||
Adds documentation to a number of modules.
|
Adds documentation to a number of modules.
|
||||||
|
|
|
||||||
3
Makefile
3
Makefile
|
|
@ -1,2 +1,5 @@
|
||||||
build: src/**.agda
|
build: src/**.agda
|
||||||
agda src/Cat.agda
|
agda src/Cat.agda
|
||||||
|
|
||||||
|
clean:
|
||||||
|
find src -name "*.agdai" -type f -delete
|
||||||
|
|
|
||||||
22
proposal/BACKLOG.md
Normal file
22
proposal/BACKLOG.md
Normal file
|
|
@ -0,0 +1,22 @@
|
||||||
|
Remove stuff about models of type theory
|
||||||
|
|
||||||
|
Add references to specific (noteable) implementaitons of category theory:
|
||||||
|
* Unimath
|
||||||
|
* cubicaltt
|
||||||
|
* https://github.com/pcapriotti/agda-categories
|
||||||
|
* https://github.com/copumpkin/categories
|
||||||
|
* ...
|
||||||
|
|
||||||
|
Talk about structure of library:
|
||||||
|
===
|
||||||
|
|
||||||
|
Propositional- and non-propositional stuff split up
|
||||||
|
Providing "equiality principles"
|
||||||
|
Provide overview of what has been proven.
|
||||||
|
|
||||||
|
What can I say about reusability?
|
||||||
|
|
||||||
|
Misc
|
||||||
|
====
|
||||||
|
|
||||||
|
Propositional content
|
||||||
177
proposal/halftime.tex
Normal file
177
proposal/halftime.tex
Normal file
|
|
@ -0,0 +1,177 @@
|
||||||
|
\section{Halftime report}
|
||||||
|
I've written this as an appendix because 1) the aim of the thesis changed
|
||||||
|
drastically from the planning report/proposal 2) partly I'm not sure how to
|
||||||
|
structure my thesis.
|
||||||
|
|
||||||
|
My work so far has very much focused on the formalization, i.e. coding. It's
|
||||||
|
unclear to me at this point what I should have in the final report. Here I will
|
||||||
|
describe what I have managed to formalize so far and what outstanding challenges
|
||||||
|
I'm facing.
|
||||||
|
|
||||||
|
\subsection{Implementation overview}
|
||||||
|
The overall structure of my project is as follows:
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item Core categorical concepts
|
||||||
|
\subitem Categories
|
||||||
|
\subitem Functors
|
||||||
|
\subitem Products
|
||||||
|
\subitem Exponentials
|
||||||
|
\subitem Cartesian closed categories
|
||||||
|
\subitem Natural transformations
|
||||||
|
\subitem Yoneda embedding
|
||||||
|
\subitem Monads
|
||||||
|
\subsubitem Monoidal monads
|
||||||
|
\subsubitem Kleisli monads
|
||||||
|
\subsubitem Voevodsky's construction
|
||||||
|
\item Category of \ldots
|
||||||
|
\subitem Homotopy sets
|
||||||
|
\subitem Categories
|
||||||
|
\subitem Relations
|
||||||
|
\subitem Functors
|
||||||
|
\subitem Free category
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
I also started work on the category with families as well as the cubical
|
||||||
|
category as per the original goal of the thesis. However I have not gotten so
|
||||||
|
far with this.
|
||||||
|
|
||||||
|
In the following I will give an overview of overall results in each of these
|
||||||
|
categories (no pun).
|
||||||
|
|
||||||
|
As an overall design-guideline I've defined concepts in a such a way that the
|
||||||
|
``data'' and the ``laws'' about that data is split up in seperate modules. As an
|
||||||
|
example a category is defined to have two members: `raw` which is a collection
|
||||||
|
of the data and `isCategory` which asserts some laws about that data.
|
||||||
|
|
||||||
|
This allows me to reason about things in a more mathematical way, where one can
|
||||||
|
reason about two categories by simply focusing on the data. This is acheived by
|
||||||
|
creating a function embodying the ``equality principle'' for a given record. In
|
||||||
|
the case of monads, to prove two categories propositionally equal it enough to
|
||||||
|
provide a proof that their data is equal.
|
||||||
|
|
||||||
|
\subsubsection{Categories}
|
||||||
|
Defines the basic notion of a category. This definition closely follows that of
|
||||||
|
[HoTT]: That is, the standard definition of a category (data; objects, arrows,
|
||||||
|
composition and identity, laws; preservation of identity and composition) plus
|
||||||
|
the extra condition that it is univalent - namely that you can get an equality
|
||||||
|
of two objects from an isomorphism.
|
||||||
|
|
||||||
|
I make no distinction between a pre-category and a real category (as in the
|
||||||
|
[HoTT]-sense). A pre-category in my implementation would be a category sans the
|
||||||
|
witness to univalence.
|
||||||
|
|
||||||
|
I also prove that being a category is a proposition. This gives rise to an
|
||||||
|
equality principle for monads that focuses on the data-part.
|
||||||
|
|
||||||
|
I also show that the opposite category is indeed a category. (\WIP{} I have not
|
||||||
|
shown that univalence holds for such a construction)
|
||||||
|
|
||||||
|
I also show that taking the opposite is an involution.
|
||||||
|
|
||||||
|
\subsubsection{Functors}
|
||||||
|
Defines the notion of a functor - also split up into data and laws.
|
||||||
|
|
||||||
|
Propositionality for being a functor.
|
||||||
|
|
||||||
|
Composition of functors and the identity functor.
|
||||||
|
|
||||||
|
\subsubsection{Products}
|
||||||
|
Definition of what it means for an object to be a product in a given category.
|
||||||
|
|
||||||
|
Definition of what it means for a category to have all products.
|
||||||
|
|
||||||
|
\WIP Prove propositionality for being a product and having products.
|
||||||
|
|
||||||
|
\subsubsection{Exponentials}
|
||||||
|
Definition of what it means to be an exponential object.
|
||||||
|
|
||||||
|
Definition of what it means for a category to have all exponential objects.
|
||||||
|
|
||||||
|
\subsubsection{Cartesian closed categories}
|
||||||
|
Definition of what it means for a category to be cartesian closed; namely that
|
||||||
|
it has all products and all exponentials.
|
||||||
|
|
||||||
|
\subsubsection{Natural transformations}
|
||||||
|
Definition of transformations\footnote{Maybe this is a name I made up for a
|
||||||
|
family of morphisms} and the naturality condition for these.
|
||||||
|
|
||||||
|
Proof that naturality is a mere proposition and the accompanying equality
|
||||||
|
principle. Proof that natural transformations are homotopic sets.
|
||||||
|
|
||||||
|
The identity natural transformation.
|
||||||
|
|
||||||
|
\subsubsection{Yoneda embedding}
|
||||||
|
|
||||||
|
The yoneda embedding is typically presented in terms of the category of
|
||||||
|
categories (cf. Awodey) \emph however this is not stricly needed - all we need
|
||||||
|
is what would be the exponential object in that category - this happens to be
|
||||||
|
functors and so this is how we define the yoneda embedding.
|
||||||
|
|
||||||
|
\subsubsection{Monads}
|
||||||
|
|
||||||
|
Defines an equivalence between these two formulations of a monad:
|
||||||
|
|
||||||
|
\subsubsubsection{Monoidal monads}
|
||||||
|
|
||||||
|
Defines the standard monoidal representation of a monad:
|
||||||
|
|
||||||
|
An endofunctor with two natural transformations (called ``pure'' and ``join'')
|
||||||
|
and some laws about these natural transformations.
|
||||||
|
|
||||||
|
Propositionality proofs and equality principle is provided.
|
||||||
|
|
||||||
|
\subsubsubsection{Kleisli monads}
|
||||||
|
|
||||||
|
A presentation of monads perhaps more familiar to a functional programer:
|
||||||
|
|
||||||
|
A map on objects and two maps on morphisms (called ``pure'' and ``bind'') and
|
||||||
|
some laws about these maps.
|
||||||
|
|
||||||
|
Propositionality proofs and equality principle is provided.
|
||||||
|
|
||||||
|
\subsubsubsection{Voevodsky's construction}
|
||||||
|
|
||||||
|
Provides construction 2.3 as presented in an unpublished paper by the late
|
||||||
|
Vladimir Voevodsky. This construction is similiar to the equivalence provided
|
||||||
|
for the two preceding formulations
|
||||||
|
\footnote{ TODO: I would like to include in the thesis some motivation for why
|
||||||
|
this construction is particularly interesting.}
|
||||||
|
|
||||||
|
\subsubsection{Homotopy sets}
|
||||||
|
The typical category of sets where the objects are modelled by an Agda set
|
||||||
|
(henceforth ``type'') at a given level is not a valid category in this cubical
|
||||||
|
settings, we need to restrict the types to be those that are homotopy sets. Thus the objects of this category are:
|
||||||
|
%
|
||||||
|
$$\Set_\ell \defeq \sum_{A \tp \MCU_\ell} \isSet\ A$$
|
||||||
|
%
|
||||||
|
\WIP{} I'm still missing a few details for the proof that this category is
|
||||||
|
univalent. Indeed this doesn't not follow immediately from
|
||||||
|
%
|
||||||
|
$$\mathit{univalence} \tp (A \cong B) \simeq (A \simeq B)$$
|
||||||
|
%
|
||||||
|
since $A$ and $B$ are of type $\MCU \neq \Set$.
|
||||||
|
\subsubsection{Categories}
|
||||||
|
Note that this category does in fact not exist. In stead I provide the
|
||||||
|
definition of the ``raw'' category as well as some of the laws.
|
||||||
|
|
||||||
|
Furthermore I provide some helpful lemmas about this raw category. For instance
|
||||||
|
I have shown what would be the exponential object in such a category.
|
||||||
|
|
||||||
|
These lemmas can be used to provide the actual exponential object in a context
|
||||||
|
where we have a witness to this being a category. This is useful if this library
|
||||||
|
is later extended to talk about higher categories.
|
||||||
|
|
||||||
|
\subsubsection{Functors}
|
||||||
|
The category of functors and natural transformations. An immediate corrolary is
|
||||||
|
the set of presheaf categories.
|
||||||
|
|
||||||
|
\WIP{} I have not shown that the category of functors is univalent.
|
||||||
|
|
||||||
|
\subsubsection{Relations}
|
||||||
|
The category of relations. \WIP I have not shown that this category is
|
||||||
|
univalent. Not sure I intend to do so either.
|
||||||
|
|
||||||
|
\subsubsection{Free category}
|
||||||
|
The free category of a category. \WIP I have not shown that this category is
|
||||||
|
univalent.
|
||||||
|
|
@ -18,3 +18,9 @@
|
||||||
\newcommand{\fmap}{\mathit{fmap}}
|
\newcommand{\fmap}{\mathit{fmap}}
|
||||||
\newcommand{\idFun}{\mathit{id}}
|
\newcommand{\idFun}{\mathit{id}}
|
||||||
\newcommand{\Sets}{\mathit{Sets}}
|
\newcommand{\Sets}{\mathit{Sets}}
|
||||||
|
\newcommand{\Set}{\mathit{Set}}
|
||||||
|
\newcommand{\MCU}{\UU}
|
||||||
|
\newcommand{\isSet}{\mathit{isSet}}
|
||||||
|
\newcommand{\tp}{\;\mathord{:}\;}
|
||||||
|
\newcommand{\subsubsubsection}[1]{\textbf{#1}}
|
||||||
|
\newcommand{\WIP}[1]{\textbf{[WIP]}}
|
||||||
|
|
|
||||||
|
|
@ -13,6 +13,8 @@
|
||||||
\usepackage{multicol}
|
\usepackage{multicol}
|
||||||
\usepackage{amsmath,amssymb}
|
\usepackage{amsmath,amssymb}
|
||||||
\usepackage[toc,page]{appendix}
|
\usepackage[toc,page]{appendix}
|
||||||
|
\usepackage{xspace}
|
||||||
|
|
||||||
% \setlength{\parskip}{10pt}
|
% \setlength{\parskip}{10pt}
|
||||||
|
|
||||||
% \usepackage{tikz}
|
% \usepackage{tikz}
|
||||||
|
|
@ -282,5 +284,6 @@ The thesis shall conclude with a discussion about the benefits of Cubical Agda.
|
||||||
\bibliography{refs}
|
\bibliography{refs}
|
||||||
\begin{appendices}
|
\begin{appendices}
|
||||||
\input{planning.tex}
|
\input{planning.tex}
|
||||||
|
\input{halftime.tex}
|
||||||
\end{appendices}
|
\end{appendices}
|
||||||
\end{document}
|
\end{document}
|
||||||
|
|
|
||||||
|
|
@ -8,7 +8,10 @@ open import Cat.Category.Exponential
|
||||||
open import Cat.Category.CartesianClosed
|
open import Cat.Category.CartesianClosed
|
||||||
open import Cat.Category.NaturalTransformation
|
open import Cat.Category.NaturalTransformation
|
||||||
open import Cat.Category.Yoneda
|
open import Cat.Category.Yoneda
|
||||||
|
open import Cat.Category.Monoid
|
||||||
open import Cat.Category.Monad
|
open import Cat.Category.Monad
|
||||||
|
open Cat.Category.Monad.Monoidal
|
||||||
|
open Cat.Category.Monad.Kleisli
|
||||||
open import Cat.Category.Monad.Voevodsky
|
open import Cat.Category.Monad.Voevodsky
|
||||||
|
|
||||||
open import Cat.Categories.Sets
|
open import Cat.Categories.Sets
|
||||||
|
|
|
||||||
|
|
@ -3,22 +3,14 @@
|
||||||
|
|
||||||
module Cat.Categories.Cat where
|
module Cat.Categories.Cat where
|
||||||
|
|
||||||
open import Agda.Primitive
|
open import Cat.Prelude renaming (proj₁ to fst ; proj₂ to snd)
|
||||||
open import Data.Product renaming (proj₁ to fst ; proj₂ to snd)
|
|
||||||
|
|
||||||
open import Cubical
|
|
||||||
open import Cubical.Sigma
|
|
||||||
|
|
||||||
open import Cat.Category
|
open import Cat.Category
|
||||||
open import Cat.Category.Functor
|
open import Cat.Category.Functor
|
||||||
open import Cat.Category.Product
|
open import Cat.Category.Product
|
||||||
open import Cat.Category.Exponential hiding (_×_ ; product)
|
open import Cat.Category.Exponential hiding (_×_ ; product)
|
||||||
open import Cat.Category.NaturalTransformation
|
open import Cat.Category.NaturalTransformation
|
||||||
|
open import Cat.Categories.Fun
|
||||||
open import Cat.Equality
|
|
||||||
open Equality.Data.Product
|
|
||||||
|
|
||||||
open Category using (Object ; 𝟙)
|
|
||||||
|
|
||||||
-- The category of categories
|
-- The category of categories
|
||||||
module _ (ℓ ℓ' : Level) where
|
module _ (ℓ ℓ' : Level) where
|
||||||
|
|
@ -35,19 +27,18 @@ module _ (ℓ ℓ' : Level) where
|
||||||
ident-l = Functor≡ refl
|
ident-l = Functor≡ refl
|
||||||
|
|
||||||
RawCat : RawCategory (lsuc (ℓ ⊔ ℓ')) (ℓ ⊔ ℓ')
|
RawCat : RawCategory (lsuc (ℓ ⊔ ℓ')) (ℓ ⊔ ℓ')
|
||||||
RawCat =
|
RawCategory.Object RawCat = Category ℓ ℓ'
|
||||||
record
|
RawCategory.Arrow RawCat = Functor
|
||||||
{ Object = Category ℓ ℓ'
|
RawCategory.𝟙 RawCat = identity
|
||||||
; Arrow = Functor
|
RawCategory._∘_ RawCat = F[_∘_]
|
||||||
; 𝟙 = identity
|
|
||||||
; _∘_ = F[_∘_]
|
|
||||||
}
|
|
||||||
private
|
private
|
||||||
open RawCategory RawCat
|
open RawCategory RawCat
|
||||||
isAssociative : IsAssociative
|
isAssociative : IsAssociative
|
||||||
isAssociative {f = F} {G} {H} = assc {F = F} {G = G} {H = H}
|
isAssociative {f = F} {G} {H} = assc {F = F} {G = G} {H = H}
|
||||||
ident : IsIdentity identity
|
|
||||||
ident = ident-r , ident-l
|
isIdentity : IsIdentity identity
|
||||||
|
isIdentity = ident-l , ident-r
|
||||||
|
|
||||||
-- NB! `ArrowsAreSets RawCat` is *not* provable. The type of functors,
|
-- NB! `ArrowsAreSets RawCat` is *not* provable. The type of functors,
|
||||||
-- however, form a groupoid! Therefore there is no (1-)category of
|
-- however, form a groupoid! Therefore there is no (1-)category of
|
||||||
|
|
@ -72,11 +63,13 @@ module CatProduct {ℓ ℓ' : Level} (ℂ 𝔻 : Category ℓ ℓ') where
|
||||||
module ℂ = Category ℂ
|
module ℂ = Category ℂ
|
||||||
module 𝔻 = Category 𝔻
|
module 𝔻 = Category 𝔻
|
||||||
|
|
||||||
Obj = Object ℂ × Object 𝔻
|
module _ where
|
||||||
|
private
|
||||||
|
Obj = ℂ.Object × 𝔻.Object
|
||||||
Arr : Obj → Obj → Set ℓ'
|
Arr : Obj → Obj → Set ℓ'
|
||||||
Arr (c , d) (c' , d') = ℂ [ c , c' ] × 𝔻 [ d , d' ]
|
Arr (c , d) (c' , d') = ℂ [ c , c' ] × 𝔻 [ d , d' ]
|
||||||
𝟙' : {o : Obj} → Arr o o
|
𝟙 : {o : Obj} → Arr o o
|
||||||
𝟙' = 𝟙 ℂ , 𝟙 𝔻
|
𝟙 = ℂ.𝟙 , 𝔻.𝟙
|
||||||
_∘_ :
|
_∘_ :
|
||||||
{a b c : Obj} →
|
{a b c : Obj} →
|
||||||
Arr b c →
|
Arr b c →
|
||||||
|
|
@ -87,17 +80,19 @@ module CatProduct {ℓ ℓ' : Level} (ℂ 𝔻 : Category ℓ ℓ') where
|
||||||
rawProduct : RawCategory ℓ ℓ'
|
rawProduct : RawCategory ℓ ℓ'
|
||||||
RawCategory.Object rawProduct = Obj
|
RawCategory.Object rawProduct = Obj
|
||||||
RawCategory.Arrow rawProduct = Arr
|
RawCategory.Arrow rawProduct = Arr
|
||||||
RawCategory.𝟙 rawProduct = 𝟙'
|
RawCategory.𝟙 rawProduct = 𝟙
|
||||||
RawCategory._∘_ rawProduct = _∘_
|
RawCategory._∘_ rawProduct = _∘_
|
||||||
|
|
||||||
open RawCategory rawProduct
|
open RawCategory rawProduct
|
||||||
|
|
||||||
arrowsAreSets : ArrowsAreSets
|
arrowsAreSets : ArrowsAreSets
|
||||||
arrowsAreSets = setSig {sA = ℂ.arrowsAreSets} {sB = λ x → 𝔻.arrowsAreSets}
|
arrowsAreSets = setSig {sA = ℂ.arrowsAreSets} {sB = λ x → 𝔻.arrowsAreSets}
|
||||||
isIdentity : IsIdentity 𝟙'
|
isIdentity : IsIdentity 𝟙
|
||||||
isIdentity
|
isIdentity
|
||||||
= Σ≡ (fst ℂ.isIdentity) (fst 𝔻.isIdentity)
|
= Σ≡ (fst ℂ.isIdentity) (fst 𝔻.isIdentity)
|
||||||
, Σ≡ (snd ℂ.isIdentity) (snd 𝔻.isIdentity)
|
, Σ≡ (snd ℂ.isIdentity) (snd 𝔻.isIdentity)
|
||||||
postulate univalent : Univalence.Univalent rawProduct isIdentity
|
|
||||||
|
postulate univalent : Univalence.Univalent isIdentity
|
||||||
instance
|
instance
|
||||||
isCategory : IsCategory rawProduct
|
isCategory : IsCategory rawProduct
|
||||||
IsCategory.isAssociative isCategory = Σ≡ ℂ.isAssociative 𝔻.isAssociative
|
IsCategory.isAssociative isCategory = Σ≡ ℂ.isAssociative 𝔻.isAssociative
|
||||||
|
|
@ -167,7 +162,7 @@ module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
|
||||||
RawProduct.proj₂ rawProduct = P.proj₂
|
RawProduct.proj₂ rawProduct = P.proj₂
|
||||||
|
|
||||||
isProduct : IsProduct Catℓ _ _ rawProduct
|
isProduct : IsProduct Catℓ _ _ rawProduct
|
||||||
IsProduct.isProduct isProduct = P.isProduct
|
IsProduct.ump isProduct = P.isProduct
|
||||||
|
|
||||||
product : Product Catℓ ℂ 𝔻
|
product : Product Catℓ ℂ 𝔻
|
||||||
Product.raw product = rawProduct
|
Product.raw product = rawProduct
|
||||||
|
|
@ -181,109 +176,70 @@ module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
|
||||||
-- it is therefory also cartesian closed.
|
-- it is therefory also cartesian closed.
|
||||||
module CatExponential {ℓ : Level} (ℂ 𝔻 : Category ℓ ℓ) where
|
module CatExponential {ℓ : Level} (ℂ 𝔻 : Category ℓ ℓ) where
|
||||||
private
|
private
|
||||||
open Data.Product
|
|
||||||
open import Cat.Categories.Fun
|
|
||||||
module ℂ = Category ℂ
|
module ℂ = Category ℂ
|
||||||
module 𝔻 = Category 𝔻
|
module 𝔻 = Category 𝔻
|
||||||
Categoryℓ = Category ℓ ℓ
|
Categoryℓ = Category ℓ ℓ
|
||||||
open Fun ℂ 𝔻 renaming (identity to idN)
|
open Fun ℂ 𝔻 renaming (identity to idN)
|
||||||
|
|
||||||
omap : Functor ℂ 𝔻 × Object ℂ → Object 𝔻
|
omap : Functor ℂ 𝔻 × ℂ.Object → 𝔻.Object
|
||||||
omap (F , A) = Functor.omap F A
|
omap (F , A) = Functor.omap F A
|
||||||
|
|
||||||
-- The exponential object
|
-- The exponential object
|
||||||
object : Categoryℓ
|
object : Categoryℓ
|
||||||
object = Fun
|
object = Fun
|
||||||
|
|
||||||
module _ {dom cod : Functor ℂ 𝔻 × Object ℂ} where
|
module _ {dom cod : Functor ℂ 𝔻 × ℂ.Object} where
|
||||||
|
open Σ dom renaming (proj₁ to F ; proj₂ to A)
|
||||||
|
open Σ cod renaming (proj₁ to G ; proj₂ to B)
|
||||||
private
|
private
|
||||||
F : Functor ℂ 𝔻
|
|
||||||
F = proj₁ dom
|
|
||||||
A : Object ℂ
|
|
||||||
A = proj₂ dom
|
|
||||||
|
|
||||||
G : Functor ℂ 𝔻
|
|
||||||
G = proj₁ cod
|
|
||||||
B : Object ℂ
|
|
||||||
B = proj₂ cod
|
|
||||||
|
|
||||||
module F = Functor F
|
module F = Functor F
|
||||||
module G = Functor G
|
module G = Functor G
|
||||||
|
|
||||||
fmap : (pobj : NaturalTransformation F G × ℂ [ A , B ])
|
fmap : (pobj : NaturalTransformation F G × ℂ [ A , B ])
|
||||||
→ 𝔻 [ F.omap A , G.omap B ]
|
→ 𝔻 [ F.omap A , G.omap B ]
|
||||||
fmap ((θ , θNat) , f) = result
|
fmap ((θ , θNat) , f) = 𝔻 [ θ B ∘ F.fmap f ]
|
||||||
where
|
-- Alternatively:
|
||||||
θA : 𝔻 [ F.omap A , G.omap A ]
|
--
|
||||||
θA = θ A
|
-- fmap ((θ , θNat) , f) = 𝔻 [ G.fmap f ∘ θ A ]
|
||||||
θB : 𝔻 [ F.omap B , G.omap B ]
|
--
|
||||||
θB = θ B
|
-- Since they are equal by naturality of θ.
|
||||||
F→f : 𝔻 [ F.omap A , F.omap B ]
|
|
||||||
F→f = F.fmap f
|
|
||||||
G→f : 𝔻 [ G.omap A , G.omap B ]
|
|
||||||
G→f = G.fmap f
|
|
||||||
l : 𝔻 [ F.omap A , G.omap B ]
|
|
||||||
l = 𝔻 [ θB ∘ F.fmap f ]
|
|
||||||
r : 𝔻 [ F.omap A , G.omap B ]
|
|
||||||
r = 𝔻 [ G.fmap f ∘ θA ]
|
|
||||||
result : 𝔻 [ F.omap A , G.omap B ]
|
|
||||||
result = l
|
|
||||||
|
|
||||||
open CatProduct renaming (object to _⊗_) using ()
|
open CatProduct renaming (object to _⊗_) using ()
|
||||||
|
|
||||||
module _ {c : Functor ℂ 𝔻 × Object ℂ} where
|
module _ {c : Functor ℂ 𝔻 × ℂ.Object} where
|
||||||
private
|
open Σ c renaming (proj₁ to F ; proj₂ to C)
|
||||||
F : Functor ℂ 𝔻
|
|
||||||
F = proj₁ c
|
|
||||||
C : Object ℂ
|
|
||||||
C = proj₂ c
|
|
||||||
|
|
||||||
ident : fmap {c} {c} (NT.identity F , 𝟙 ℂ {A = proj₂ c}) ≡ 𝟙 𝔻
|
ident : fmap {c} {c} (NT.identity F , ℂ.𝟙 {A = snd c}) ≡ 𝔻.𝟙
|
||||||
ident = begin
|
ident = begin
|
||||||
fmap {c} {c} (𝟙 (object ⊗ ℂ) {c}) ≡⟨⟩
|
fmap {c} {c} (Category.𝟙 (object ⊗ ℂ) {c}) ≡⟨⟩
|
||||||
fmap {c} {c} (idN F , 𝟙 ℂ) ≡⟨⟩
|
fmap {c} {c} (idN F , ℂ.𝟙) ≡⟨⟩
|
||||||
𝔻 [ identityTrans F C ∘ F.fmap (𝟙 ℂ)] ≡⟨⟩
|
𝔻 [ identityTrans F C ∘ F.fmap ℂ.𝟙 ] ≡⟨⟩
|
||||||
𝔻 [ 𝟙 𝔻 ∘ F.fmap (𝟙 ℂ)] ≡⟨ proj₂ 𝔻.isIdentity ⟩
|
𝔻 [ 𝔻.𝟙 ∘ F.fmap ℂ.𝟙 ] ≡⟨ 𝔻.leftIdentity ⟩
|
||||||
F.fmap (𝟙 ℂ) ≡⟨ F.isIdentity ⟩
|
F.fmap ℂ.𝟙 ≡⟨ F.isIdentity ⟩
|
||||||
𝟙 𝔻 ∎
|
𝔻.𝟙 ∎
|
||||||
where
|
where
|
||||||
module F = Functor F
|
module F = Functor F
|
||||||
|
|
||||||
module _ {F×A G×B H×C : Functor ℂ 𝔻 × Object ℂ} where
|
module _ {F×A G×B H×C : Functor ℂ 𝔻 × ℂ.Object} where
|
||||||
|
open Σ F×A renaming (proj₁ to F ; proj₂ to A)
|
||||||
|
open Σ G×B renaming (proj₁ to G ; proj₂ to B)
|
||||||
|
open Σ H×C renaming (proj₁ to H ; proj₂ to C)
|
||||||
private
|
private
|
||||||
F = F×A .proj₁
|
|
||||||
A = F×A .proj₂
|
|
||||||
G = G×B .proj₁
|
|
||||||
B = G×B .proj₂
|
|
||||||
H = H×C .proj₁
|
|
||||||
C = H×C .proj₂
|
|
||||||
module F = Functor F
|
module F = Functor F
|
||||||
module G = Functor G
|
module G = Functor G
|
||||||
module H = Functor H
|
module H = Functor H
|
||||||
|
|
||||||
module _
|
module _
|
||||||
-- NaturalTransformation F G × ℂ .Arrow A B
|
|
||||||
{θ×f : NaturalTransformation F G × ℂ [ A , B ]}
|
{θ×f : NaturalTransformation F G × ℂ [ A , B ]}
|
||||||
{η×g : NaturalTransformation G H × ℂ [ B , C ]} where
|
{η×g : NaturalTransformation G H × ℂ [ B , C ]} where
|
||||||
|
open Σ θ×f renaming (proj₁ to θNT ; proj₂ to f)
|
||||||
|
open Σ θNT renaming (proj₁ to θ ; proj₂ to θNat)
|
||||||
|
open Σ η×g renaming (proj₁ to ηNT ; proj₂ to g)
|
||||||
|
open Σ ηNT renaming (proj₁ to η ; proj₂ to ηNat)
|
||||||
private
|
private
|
||||||
θ : Transformation F G
|
|
||||||
θ = proj₁ (proj₁ θ×f)
|
|
||||||
θNat : Natural F G θ
|
|
||||||
θNat = proj₂ (proj₁ θ×f)
|
|
||||||
f : ℂ [ A , B ]
|
|
||||||
f = proj₂ θ×f
|
|
||||||
η : Transformation G H
|
|
||||||
η = proj₁ (proj₁ η×g)
|
|
||||||
ηNat : Natural G H η
|
|
||||||
ηNat = proj₂ (proj₁ η×g)
|
|
||||||
g : ℂ [ B , C ]
|
|
||||||
g = proj₂ η×g
|
|
||||||
|
|
||||||
ηθNT : NaturalTransformation F H
|
ηθNT : NaturalTransformation F H
|
||||||
ηθNT = Category._∘_ Fun {F} {G} {H} (η , ηNat) (θ , θNat)
|
ηθNT = NT[_∘_] {F} {G} {H} ηNT θNT
|
||||||
|
open Σ ηθNT renaming (proj₁ to ηθ ; proj₂ to ηθNat)
|
||||||
ηθ = proj₁ ηθNT
|
|
||||||
ηθNat = proj₂ ηθNT
|
|
||||||
|
|
||||||
isDistributive :
|
isDistributive :
|
||||||
𝔻 [ 𝔻 [ η C ∘ θ C ] ∘ F.fmap ( ℂ [ g ∘ f ] ) ]
|
𝔻 [ 𝔻 [ η C ∘ θ C ] ∘ F.fmap ( ℂ [ g ∘ f ] ) ]
|
||||||
|
|
|
||||||
|
|
@ -1,21 +1,18 @@
|
||||||
{-# OPTIONS --allow-unsolved-metas #-}
|
{-# OPTIONS --allow-unsolved-metas #-}
|
||||||
module Cat.Categories.Cube where
|
module Cat.Categories.Cube where
|
||||||
|
|
||||||
|
open import Cat.Prelude
|
||||||
open import Level
|
open import Level
|
||||||
open import Data.Bool hiding (T)
|
open import Data.Bool hiding (T)
|
||||||
open import Data.Sum hiding ([_,_])
|
open import Data.Sum hiding ([_,_])
|
||||||
open import Data.Unit
|
open import Data.Unit
|
||||||
open import Data.Empty
|
open import Data.Empty
|
||||||
open import Data.Product
|
|
||||||
open import Cubical
|
|
||||||
open import Function
|
open import Function
|
||||||
open import Relation.Nullary
|
open import Relation.Nullary
|
||||||
open import Relation.Nullary.Decidable
|
open import Relation.Nullary.Decidable
|
||||||
|
|
||||||
open import Cat.Category
|
open import Cat.Category
|
||||||
open import Cat.Category.Functor
|
open import Cat.Category.Functor
|
||||||
open import Cat.Equality
|
|
||||||
open Equality.Data.Product
|
|
||||||
|
|
||||||
-- See chapter 1 for a discussion on how presheaf categories are CwF's.
|
-- See chapter 1 for a discussion on how presheaf categories are CwF's.
|
||||||
|
|
||||||
|
|
|
||||||
|
|
@ -1,36 +1,33 @@
|
||||||
module Cat.Categories.CwF where
|
module Cat.Categories.CwF where
|
||||||
|
|
||||||
open import Agda.Primitive
|
open import Cat.Prelude
|
||||||
open import Data.Product
|
|
||||||
|
|
||||||
open import Cat.Category
|
open import Cat.Category
|
||||||
open import Cat.Category.Functor
|
open import Cat.Category.Functor
|
||||||
open import Cat.Categories.Fam
|
open import Cat.Categories.Fam
|
||||||
|
|
||||||
open Category
|
|
||||||
open Functor
|
|
||||||
|
|
||||||
module _ {ℓa ℓb : Level} where
|
module _ {ℓa ℓb : Level} where
|
||||||
record CwF : Set (lsuc (ℓa ⊔ ℓb)) where
|
record CwF : Set (lsuc (ℓa ⊔ ℓb)) where
|
||||||
-- "A category with families consists of"
|
-- "A category with families consists of"
|
||||||
field
|
field
|
||||||
-- "A base category"
|
-- "A base category"
|
||||||
ℂ : Category ℓa ℓb
|
ℂ : Category ℓa ℓb
|
||||||
|
module ℂ = Category ℂ
|
||||||
-- It's objects are called contexts
|
-- It's objects are called contexts
|
||||||
Contexts = Object ℂ
|
Contexts = ℂ.Object
|
||||||
-- It's arrows are called substitutions
|
-- It's arrows are called substitutions
|
||||||
Substitutions = Arrow ℂ
|
Substitutions = ℂ.Arrow
|
||||||
field
|
field
|
||||||
-- A functor T
|
-- A functor T
|
||||||
T : Functor (opposite ℂ) (Fam ℓa ℓb)
|
T : Functor (opposite ℂ) (Fam ℓa ℓb)
|
||||||
-- Empty context
|
-- Empty context
|
||||||
[] : Terminal ℂ
|
[] : ℂ.Terminal
|
||||||
private
|
private
|
||||||
module T = Functor T
|
module T = Functor T
|
||||||
Type : (Γ : Object ℂ) → Set ℓa
|
Type : (Γ : ℂ.Object) → Set ℓa
|
||||||
Type Γ = proj₁ (proj₁ (T.omap Γ))
|
Type Γ = proj₁ (proj₁ (T.omap Γ))
|
||||||
|
|
||||||
module _ {Γ : Object ℂ} {A : Type Γ} where
|
module _ {Γ : ℂ.Object} {A : Type Γ} where
|
||||||
|
|
||||||
-- module _ {A B : Object ℂ} {γ : ℂ [ A , B ]} where
|
-- module _ {A B : Object ℂ} {γ : ℂ [ A , B ]} where
|
||||||
-- k : Σ (proj₁ (omap T B) → proj₁ (omap T A))
|
-- k : Σ (proj₁ (omap T B) → proj₁ (omap T A))
|
||||||
|
|
@ -46,7 +43,7 @@ module _ {ℓa ℓb : Level} where
|
||||||
|
|
||||||
record ContextComprehension : Set (ℓa ⊔ ℓb) where
|
record ContextComprehension : Set (ℓa ⊔ ℓb) where
|
||||||
field
|
field
|
||||||
Γ&A : Object ℂ
|
Γ&A : ℂ.Object
|
||||||
proj1 : ℂ [ Γ&A , Γ ]
|
proj1 : ℂ [ Γ&A , Γ ]
|
||||||
-- proj2 : ????
|
-- proj2 : ????
|
||||||
|
|
||||||
|
|
|
||||||
|
|
@ -1,21 +1,14 @@
|
||||||
{-# OPTIONS --allow-unsolved-metas #-}
|
{-# OPTIONS --allow-unsolved-metas #-}
|
||||||
module Cat.Categories.Fam where
|
module Cat.Categories.Fam where
|
||||||
|
|
||||||
open import Agda.Primitive
|
open import Cat.Prelude
|
||||||
open import Data.Product
|
|
||||||
import Function
|
import Function
|
||||||
|
|
||||||
open import Cubical
|
|
||||||
open import Cubical.Universe
|
|
||||||
|
|
||||||
open import Cat.Category
|
open import Cat.Category
|
||||||
open import Cat.Equality
|
|
||||||
|
|
||||||
open Equality.Data.Product
|
|
||||||
|
|
||||||
module _ (ℓa ℓb : Level) where
|
module _ (ℓa ℓb : Level) where
|
||||||
private
|
private
|
||||||
Object = Σ[ hA ∈ hSet {ℓa} ] (proj₁ hA → hSet {ℓb})
|
Object = Σ[ hA ∈ hSet ℓa ] (proj₁ hA → hSet ℓb)
|
||||||
Arr : Object → Object → Set (ℓa ⊔ ℓb)
|
Arr : Object → Object → Set (ℓa ⊔ ℓb)
|
||||||
Arr ((A , _) , B) ((A' , _) , B') = Σ[ f ∈ (A → A') ] ({x : A} → proj₁ (B x) → proj₁ (B' (f x)))
|
Arr ((A , _) , B) ((A' , _) , B') = Σ[ f ∈ (A → A') ] ({x : A} → proj₁ (B x) → proj₁ (B' (f x)))
|
||||||
𝟙 : {A : Object} → Arr A A
|
𝟙 : {A : Object} → Arr A A
|
||||||
|
|
|
||||||
|
|
@ -1,62 +1,77 @@
|
||||||
{-# OPTIONS --allow-unsolved-metas #-}
|
{-# OPTIONS --allow-unsolved-metas #-}
|
||||||
module Cat.Categories.Free where
|
module Cat.Categories.Free where
|
||||||
|
|
||||||
open import Agda.Primitive
|
open import Cat.Prelude hiding (Path ; empty)
|
||||||
open import Cubical hiding (Path ; isSet ; empty)
|
|
||||||
open import Data.Product
|
open import Relation.Binary
|
||||||
|
|
||||||
open import Cat.Category
|
open import Cat.Category
|
||||||
|
|
||||||
data Path {ℓ ℓ' : Level} {A : Set ℓ} (R : A → A → Set ℓ') : (a b : A) → Set (ℓ ⊔ ℓ') where
|
module _ {ℓ : Level} {A : Set ℓ} {ℓr : Level} where
|
||||||
|
data Path (R : Rel A ℓr) : (a b : A) → Set (ℓ ⊔ ℓr) where
|
||||||
empty : {a : A} → Path R a a
|
empty : {a : A} → Path R a a
|
||||||
cons : {a b c : A} → R b c → Path R a b → Path R a c
|
cons : {a b c : A} → R b c → Path R a b → Path R a c
|
||||||
|
|
||||||
concatenate _++_ : ∀ {ℓ ℓ'} {A : Set ℓ} {a b c : A} {R : A → A → Set ℓ'} → Path R b c → Path R a b → Path R a c
|
module _ {R : Rel A ℓr} where
|
||||||
|
concatenate : {a b c : A} → Path R b c → Path R a b → Path R a c
|
||||||
concatenate empty p = p
|
concatenate empty p = p
|
||||||
concatenate (cons x q) p = cons x (concatenate q p)
|
concatenate (cons x q) p = cons x (concatenate q p)
|
||||||
_++_ = concatenate
|
_++_ : {a b c : A} → Path R b c → Path R a b → Path R a c
|
||||||
|
a ++ b = concatenate a b
|
||||||
|
|
||||||
singleton : ∀ {ℓ} {𝓤 : Set ℓ} {ℓr} {R : 𝓤 → 𝓤 → Set ℓr} {A B : 𝓤} → R A B → Path R A B
|
singleton : {a b : A} → R a b → Path R a b
|
||||||
singleton f = cons f empty
|
singleton f = cons f empty
|
||||||
|
|
||||||
module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
|
module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||||
private
|
private
|
||||||
module ℂ = Category ℂ
|
module ℂ = Category ℂ
|
||||||
open Category ℂ
|
|
||||||
|
|
||||||
p-isAssociative : {A B C D : Object} {r : Path Arrow A B} {q : Path Arrow B C} {p : Path Arrow C D}
|
RawFree : RawCategory ℓa (ℓa ⊔ ℓb)
|
||||||
|
RawCategory.Object RawFree = ℂ.Object
|
||||||
|
RawCategory.Arrow RawFree = Path ℂ.Arrow
|
||||||
|
RawCategory.𝟙 RawFree = empty
|
||||||
|
RawCategory._∘_ RawFree = concatenate
|
||||||
|
|
||||||
|
open RawCategory RawFree
|
||||||
|
|
||||||
|
isAssociative : {A B C D : ℂ.Object} {r : Path ℂ.Arrow A B} {q : Path ℂ.Arrow B C} {p : Path ℂ.Arrow C D}
|
||||||
→ p ++ (q ++ r) ≡ (p ++ q) ++ r
|
→ p ++ (q ++ r) ≡ (p ++ q) ++ r
|
||||||
p-isAssociative {r = r} {q} {empty} = refl
|
isAssociative {r = r} {q} {empty} = refl
|
||||||
p-isAssociative {A} {B} {C} {D} {r = r} {q} {cons x p} = begin
|
isAssociative {A} {B} {C} {D} {r = r} {q} {cons x p} = begin
|
||||||
cons x p ++ (q ++ r) ≡⟨ cong (cons x) lem ⟩
|
cons x p ++ (q ++ r) ≡⟨ cong (cons x) lem ⟩
|
||||||
cons x ((p ++ q) ++ r) ≡⟨⟩
|
cons x ((p ++ q) ++ r) ≡⟨⟩
|
||||||
(cons x p ++ q) ++ r ∎
|
(cons x p ++ q) ++ r ∎
|
||||||
where
|
where
|
||||||
lem : p ++ (q ++ r) ≡ ((p ++ q) ++ r)
|
lem : p ++ (q ++ r) ≡ ((p ++ q) ++ r)
|
||||||
lem = p-isAssociative {r = r} {q} {p}
|
lem = isAssociative {r = r} {q} {p}
|
||||||
|
|
||||||
ident-r : ∀ {A} {B} {p : Path Arrow A B} → concatenate p empty ≡ p
|
ident-r : ∀ {A} {B} {p : Path ℂ.Arrow A B} → concatenate p empty ≡ p
|
||||||
ident-r {p = empty} = refl
|
ident-r {p = empty} = refl
|
||||||
ident-r {p = cons x p} = cong (cons x) ident-r
|
ident-r {p = cons x p} = cong (cons x) ident-r
|
||||||
|
|
||||||
ident-l : ∀ {A} {B} {p : Path Arrow A B} → concatenate empty p ≡ p
|
ident-l : ∀ {A} {B} {p : Path ℂ.Arrow A B} → concatenate empty p ≡ p
|
||||||
ident-l = refl
|
ident-l = refl
|
||||||
|
|
||||||
module _ {A B : Object} where
|
isIdentity : IsIdentity 𝟙
|
||||||
isSet : Cubical.isSet (Path Arrow A B)
|
isIdentity = ident-l , ident-r
|
||||||
isSet a b p q = {!!}
|
|
||||||
|
|
||||||
RawFree : RawCategory ℓ (ℓ ⊔ ℓ')
|
open Univalence isIdentity
|
||||||
RawFree = record
|
|
||||||
{ Object = Object
|
module _ {A B : ℂ.Object} where
|
||||||
; Arrow = Path Arrow
|
arrowsAreSets : isSet (Path ℂ.Arrow A B)
|
||||||
; 𝟙 = empty
|
arrowsAreSets a b p q = {!!}
|
||||||
; _∘_ = concatenate
|
|
||||||
}
|
eqv : isEquiv (A ≡ B) (A ≅ B) (Univalence.id-to-iso isIdentity A B)
|
||||||
RawIsCategoryFree : IsCategory RawFree
|
eqv = {!!}
|
||||||
RawIsCategoryFree = record
|
|
||||||
{ isAssociative = λ { {f = f} {g} {h} → p-isAssociative {r = f} {g} {h}}
|
univalent : Univalent
|
||||||
; isIdentity = ident-r , ident-l
|
univalent = eqv
|
||||||
; arrowsAreSets = {!!}
|
|
||||||
; univalent = {!!}
|
isCategory : IsCategory RawFree
|
||||||
}
|
IsCategory.isAssociative isCategory {f = f} {g} {h} = isAssociative {r = f} {g} {h}
|
||||||
|
IsCategory.isIdentity isCategory = isIdentity
|
||||||
|
IsCategory.arrowsAreSets isCategory = arrowsAreSets
|
||||||
|
IsCategory.univalent isCategory = univalent
|
||||||
|
|
||||||
|
Free : Category _ _
|
||||||
|
Free = record { raw = RawFree ; isCategory = isCategory }
|
||||||
|
|
|
||||||
|
|
@ -1,13 +1,7 @@
|
||||||
{-# OPTIONS --allow-unsolved-metas --cubical #-}
|
{-# OPTIONS --allow-unsolved-metas --cubical #-}
|
||||||
module Cat.Categories.Fun where
|
module Cat.Categories.Fun where
|
||||||
|
|
||||||
open import Agda.Primitive
|
open import Cat.Prelude
|
||||||
open import Data.Product
|
|
||||||
|
|
||||||
|
|
||||||
open import Cubical
|
|
||||||
open import Cubical.GradLemma
|
|
||||||
open import Cubical.NType.Properties
|
|
||||||
|
|
||||||
open import Cat.Category
|
open import Cat.Category
|
||||||
open import Cat.Category.Functor hiding (identity)
|
open import Cat.Category.Functor hiding (identity)
|
||||||
|
|
@ -45,18 +39,18 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
|
||||||
eq-r : ∀ C → (𝔻 [ f' C ∘ identityTrans A C ]) ≡ f' C
|
eq-r : ∀ C → (𝔻 [ f' C ∘ identityTrans A C ]) ≡ f' C
|
||||||
eq-r C = begin
|
eq-r C = begin
|
||||||
𝔻 [ f' C ∘ identityTrans A C ] ≡⟨⟩
|
𝔻 [ f' C ∘ identityTrans A C ] ≡⟨⟩
|
||||||
𝔻 [ f' C ∘ 𝔻.𝟙 ] ≡⟨ proj₁ 𝔻.isIdentity ⟩
|
𝔻 [ f' C ∘ 𝔻.𝟙 ] ≡⟨ 𝔻.rightIdentity ⟩
|
||||||
f' C ∎
|
f' C ∎
|
||||||
eq-l : ∀ C → (𝔻 [ identityTrans B C ∘ f' C ]) ≡ f' C
|
eq-l : ∀ C → (𝔻 [ identityTrans B C ∘ f' C ]) ≡ f' C
|
||||||
eq-l C = proj₂ 𝔻.isIdentity
|
eq-l C = 𝔻.leftIdentity
|
||||||
ident-r : (NT[_∘_] {A} {A} {B} f (NT.identity A)) ≡ f
|
ident-r : (NT[_∘_] {A} {A} {B} f (NT.identity A)) ≡ f
|
||||||
ident-r = lemSig allNatural _ _ (funExt eq-r)
|
ident-r = lemSig allNatural _ _ (funExt eq-r)
|
||||||
ident-l : (NT[_∘_] {A} {B} {B} (NT.identity B) f) ≡ f
|
ident-l : (NT[_∘_] {A} {B} {B} (NT.identity B) f) ≡ f
|
||||||
ident-l = lemSig allNatural _ _ (funExt eq-l)
|
ident-l = lemSig allNatural _ _ (funExt eq-l)
|
||||||
isIdentity
|
isIdentity
|
||||||
: (NT[_∘_] {A} {A} {B} f (NT.identity A)) ≡ f
|
: (NT[_∘_] {A} {B} {B} (NT.identity B) f) ≡ f
|
||||||
× (NT[_∘_] {A} {B} {B} (NT.identity B) f) ≡ f
|
× (NT[_∘_] {A} {A} {B} f (NT.identity A)) ≡ f
|
||||||
isIdentity = ident-r , ident-l
|
isIdentity = ident-l , ident-r
|
||||||
-- Functor categories. Objects are functors, arrows are natural transformations.
|
-- Functor categories. Objects are functors, arrows are natural transformations.
|
||||||
RawFun : RawCategory (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') (ℓc ⊔ ℓc' ⊔ ℓd')
|
RawFun : RawCategory (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') (ℓc ⊔ ℓc' ⊔ ℓd')
|
||||||
RawFun = record
|
RawFun = record
|
||||||
|
|
@ -67,7 +61,9 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
|
||||||
}
|
}
|
||||||
|
|
||||||
open RawCategory RawFun
|
open RawCategory RawFun
|
||||||
open Univalence RawFun
|
open Univalence (λ {A} {B} {f} → isIdentity {A} {B} {f})
|
||||||
|
|
||||||
|
private
|
||||||
module _ {A B : Functor ℂ 𝔻} where
|
module _ {A B : Functor ℂ 𝔻} where
|
||||||
module A = Functor A
|
module A = Functor A
|
||||||
module B = Functor B
|
module B = Functor B
|
||||||
|
|
@ -145,10 +141,10 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
|
||||||
re-ve : (x : A ≡ B) → reverse (obverse x) ≡ x
|
re-ve : (x : A ≡ B) → reverse (obverse x) ≡ x
|
||||||
re-ve = {!!}
|
re-ve = {!!}
|
||||||
|
|
||||||
done : isEquiv (A ≡ B) (A ≅ B) (id-to-iso (λ { {A} {B} → isIdentity {A} {B}}) A B)
|
done : isEquiv (A ≡ B) (A ≅ B) (Univalence.id-to-iso (λ { {A} {B} → isIdentity {A} {B}}) A B)
|
||||||
done = {!gradLemma obverse reverse ve-re re-ve!}
|
done = {!gradLemma obverse reverse ve-re re-ve!}
|
||||||
|
|
||||||
univalent : Univalent (λ{ {A} {B} → isIdentity {A} {B}})
|
univalent : Univalent
|
||||||
univalent = done
|
univalent = done
|
||||||
|
|
||||||
instance
|
instance
|
||||||
|
|
|
||||||
|
|
@ -1,12 +1,8 @@
|
||||||
{-# OPTIONS --cubical --allow-unsolved-metas #-}
|
{-# OPTIONS --cubical --allow-unsolved-metas #-}
|
||||||
module Cat.Categories.Rel where
|
module Cat.Categories.Rel where
|
||||||
|
|
||||||
open import Cubical
|
open import Cat.Prelude renaming (proj₁ to fst ; proj₂ to snd)
|
||||||
open import Cubical.GradLemma
|
|
||||||
open import Agda.Primitive
|
|
||||||
open import Data.Product renaming (proj₁ to fst ; proj₂ to snd)
|
|
||||||
open import Function
|
open import Function
|
||||||
import Cubical.FromStdLib
|
|
||||||
|
|
||||||
open import Cat.Category
|
open import Cat.Category
|
||||||
|
|
||||||
|
|
@ -74,11 +70,11 @@ module _ {A B : Set} {S : Subset (A × B)} (ab : A × B) where
|
||||||
|
|
||||||
equi : (Σ[ a' ∈ A ] (a , a') ∈ Diag A × (a' , b) ∈ S)
|
equi : (Σ[ a' ∈ A ] (a , a') ∈ Diag A × (a' , b) ∈ S)
|
||||||
≃ (a , b) ∈ S
|
≃ (a , b) ∈ S
|
||||||
equi = backwards Cubical.FromStdLib., isequiv
|
equi = backwards , isequiv
|
||||||
|
|
||||||
ident-l : (Σ[ a' ∈ A ] (a , a') ∈ Diag A × (a' , b) ∈ S)
|
ident-r : (Σ[ a' ∈ A ] (a , a') ∈ Diag A × (a' , b) ∈ S)
|
||||||
≡ (a , b) ∈ S
|
≡ (a , b) ∈ S
|
||||||
ident-l = equivToPath equi
|
ident-r = equivToPath equi
|
||||||
|
|
||||||
module _ where
|
module _ where
|
||||||
private
|
private
|
||||||
|
|
@ -108,11 +104,11 @@ module _ {A B : Set} {S : Subset (A × B)} (ab : A × B) where
|
||||||
|
|
||||||
equi : (Σ[ b' ∈ B ] (a , b') ∈ S × (b' , b) ∈ Diag B)
|
equi : (Σ[ b' ∈ B ] (a , b') ∈ S × (b' , b) ∈ Diag B)
|
||||||
≃ ab ∈ S
|
≃ ab ∈ S
|
||||||
equi = backwards Cubical.FromStdLib., isequiv
|
equi = backwards , isequiv
|
||||||
|
|
||||||
ident-r : (Σ[ b' ∈ B ] (a , b') ∈ S × (b' , b) ∈ Diag B)
|
ident-l : (Σ[ b' ∈ B ] (a , b') ∈ S × (b' , b) ∈ Diag B)
|
||||||
≡ ab ∈ S
|
≡ ab ∈ S
|
||||||
ident-r = equivToPath equi
|
ident-l = equivToPath equi
|
||||||
|
|
||||||
module _ {A B C D : Set} {S : Subset (A × B)} {R : Subset (B × C)} {Q : Subset (C × D)} (ad : A × D) where
|
module _ {A B C D : Set} {S : Subset (A × B)} {R : Subset (B × C)} {Q : Subset (C × D)} (ad : A × D) where
|
||||||
private
|
private
|
||||||
|
|
@ -146,7 +142,7 @@ module _ {A B C D : Set} {S : Subset (A × B)} {R : Subset (B × C)} {Q : Subset
|
||||||
|
|
||||||
equi : (Σ[ c ∈ C ] (Σ[ b ∈ B ] (a , b) ∈ S × (b , c) ∈ R) × (c , d) ∈ Q)
|
equi : (Σ[ c ∈ C ] (Σ[ b ∈ B ] (a , b) ∈ S × (b , c) ∈ R) × (c , d) ∈ Q)
|
||||||
≃ (Σ[ b ∈ B ] (a , b) ∈ S × (Σ[ c ∈ C ] (b , c) ∈ R × (c , d) ∈ Q))
|
≃ (Σ[ b ∈ B ] (a , b) ∈ S × (Σ[ c ∈ C ] (b , c) ∈ R × (c , d) ∈ Q))
|
||||||
equi = fwd Cubical.FromStdLib., isequiv
|
equi = fwd , isequiv
|
||||||
|
|
||||||
-- isAssociativec : Q + (R + S) ≡ (Q + R) + S
|
-- isAssociativec : Q + (R + S) ≡ (Q + R) + S
|
||||||
is-isAssociative : (Σ[ c ∈ C ] (Σ[ b ∈ B ] (a , b) ∈ S × (b , c) ∈ R) × (c , d) ∈ Q)
|
is-isAssociative : (Σ[ c ∈ C ] (Σ[ b ∈ B ] (a , b) ∈ S × (b , c) ∈ R) × (c , d) ∈ Q)
|
||||||
|
|
|
||||||
|
|
@ -1,166 +1,289 @@
|
||||||
-- | The category of homotopy sets
|
-- | The category of homotopy sets
|
||||||
{-# OPTIONS --allow-unsolved-metas --cubical #-}
|
{-# OPTIONS --allow-unsolved-metas --cubical --caching #-}
|
||||||
module Cat.Categories.Sets where
|
module Cat.Categories.Sets where
|
||||||
|
|
||||||
open import Agda.Primitive
|
open import Cat.Prelude hiding (_≃_)
|
||||||
open import Data.Product
|
import Data.Product
|
||||||
open import Function using (_∘_)
|
|
||||||
|
|
||||||
open import Cubical hiding (_≃_ ; inverse)
|
open import Function using (_∘_ ; _∘′_)
|
||||||
open import Cubical.Equivalence
|
|
||||||
renaming
|
open import Cubical.Univalence using (univalence ; con ; _≃_ ; idtoeqv ; ua)
|
||||||
( _≅_ to _A≅_ )
|
|
||||||
using
|
|
||||||
(_≃_ ; con ; AreInverses)
|
|
||||||
open import Cubical.Univalence
|
|
||||||
open import Cubical.GradLemma
|
|
||||||
|
|
||||||
open import Cat.Category
|
open import Cat.Category
|
||||||
open import Cat.Category.Functor
|
open import Cat.Category.Functor
|
||||||
open import Cat.Category.Product
|
open import Cat.Category.Product
|
||||||
open import Cat.Wishlist
|
open import Cat.Wishlist
|
||||||
|
open import Cat.Equivalence as Eqv using (AreInverses ; module Equiv≃ ; module NoEta)
|
||||||
|
|
||||||
|
open NoEta
|
||||||
|
|
||||||
|
module Equivalence = Equivalence′
|
||||||
|
|
||||||
|
_⊙_ : {ℓa ℓb ℓc : Level} {A : Set ℓa} {B : Set ℓb} {C : Set ℓc} → (A ≃ B) → (B ≃ C) → A ≃ C
|
||||||
|
eqA ⊙ eqB = Equivalence.compose eqA eqB
|
||||||
|
|
||||||
|
sym≃ : ∀ {ℓa ℓb} {A : Set ℓa} {B : Set ℓb} → A ≃ B → B ≃ A
|
||||||
|
sym≃ = Equivalence.symmetry
|
||||||
|
|
||||||
|
infixl 10 _⊙_
|
||||||
|
|
||||||
|
module _ {ℓ : Level} {A : Set ℓ} {a : A} where
|
||||||
|
id-coe : coe refl a ≡ a
|
||||||
|
id-coe = begin
|
||||||
|
coe refl a ≡⟨⟩
|
||||||
|
pathJ (λ y x → A) _ A refl ≡⟨ pathJprop {x = a} (λ y x → A) _ ⟩
|
||||||
|
_ ≡⟨ pathJprop {x = a} (λ y x → A) _ ⟩
|
||||||
|
a ∎
|
||||||
|
|
||||||
|
module _ {ℓ : Level} {A B : Set ℓ} {a : A} where
|
||||||
|
inv-coe : (p : A ≡ B) → coe (sym p) (coe p a) ≡ a
|
||||||
|
inv-coe p =
|
||||||
|
let
|
||||||
|
D : (y : Set ℓ) → _ ≡ y → Set _
|
||||||
|
D _ q = coe (sym q) (coe q a) ≡ a
|
||||||
|
d : D A refl
|
||||||
|
d = begin
|
||||||
|
coe (sym refl) (coe refl a) ≡⟨⟩
|
||||||
|
coe refl (coe refl a) ≡⟨ id-coe ⟩
|
||||||
|
coe refl a ≡⟨ id-coe ⟩
|
||||||
|
a ∎
|
||||||
|
in pathJ D d B p
|
||||||
|
inv-coe' : (p : B ≡ A) → coe p (coe (sym p) a) ≡ a
|
||||||
|
inv-coe' p =
|
||||||
|
let
|
||||||
|
D : (y : Set ℓ) → _ ≡ y → Set _
|
||||||
|
D _ q = coe (sym q) (coe q a) ≡ a
|
||||||
|
k : coe p (coe (sym p) a) ≡ a
|
||||||
|
k = pathJ D (trans id-coe id-coe) B (sym p)
|
||||||
|
in k
|
||||||
|
|
||||||
module _ (ℓ : Level) where
|
module _ (ℓ : Level) where
|
||||||
private
|
private
|
||||||
open import Cubical.Univalence
|
|
||||||
open import Cubical.NType.Properties
|
|
||||||
open import Cubical.Universe
|
|
||||||
|
|
||||||
SetsRaw : RawCategory (lsuc ℓ) ℓ
|
SetsRaw : RawCategory (lsuc ℓ) ℓ
|
||||||
RawCategory.Object SetsRaw = hSet
|
RawCategory.Object SetsRaw = hSet ℓ
|
||||||
RawCategory.Arrow SetsRaw (T , _) (U , _) = T → U
|
RawCategory.Arrow SetsRaw (T , _) (U , _) = T → U
|
||||||
RawCategory.𝟙 SetsRaw = Function.id
|
RawCategory.𝟙 SetsRaw = Function.id
|
||||||
RawCategory._∘_ SetsRaw = Function._∘′_
|
RawCategory._∘_ SetsRaw = Function._∘′_
|
||||||
|
|
||||||
open RawCategory SetsRaw hiding (_∘_)
|
open RawCategory SetsRaw hiding (_∘_)
|
||||||
open Univalence SetsRaw
|
|
||||||
|
|
||||||
isIdentity : IsIdentity Function.id
|
isIdentity : IsIdentity Function.id
|
||||||
proj₁ isIdentity = funExt λ _ → refl
|
proj₁ isIdentity = funExt λ _ → refl
|
||||||
proj₂ isIdentity = funExt λ _ → refl
|
proj₂ isIdentity = funExt λ _ → refl
|
||||||
|
|
||||||
|
open Univalence (λ {A} {B} {f} → isIdentity {A} {B} {f})
|
||||||
|
|
||||||
arrowsAreSets : ArrowsAreSets
|
arrowsAreSets : ArrowsAreSets
|
||||||
arrowsAreSets {B = (_ , s)} = setPi λ _ → s
|
arrowsAreSets {B = (_ , s)} = setPi λ _ → s
|
||||||
|
|
||||||
|
isIso = Eqv.Isomorphism
|
||||||
|
module _ {hA hB : hSet ℓ} where
|
||||||
|
open Σ hA renaming (proj₁ to A ; proj₂ to sA)
|
||||||
|
open Σ hB renaming (proj₁ to B ; proj₂ to sB)
|
||||||
|
lem1 : (f : A → B) → isSet A → isSet B → isProp (isIso f)
|
||||||
|
lem1 f sA sB = res
|
||||||
|
where
|
||||||
|
module _ (x y : isIso f) where
|
||||||
|
module x = Σ x renaming (proj₁ to inverse ; proj₂ to areInverses)
|
||||||
|
module y = Σ y renaming (proj₁ to inverse ; proj₂ to areInverses)
|
||||||
|
module xA = AreInverses x.areInverses
|
||||||
|
module yA = AreInverses y.areInverses
|
||||||
|
-- I had a lot of difficulty using the corresponding proof where
|
||||||
|
-- AreInverses is defined. This is sadly a bit anti-modular. The
|
||||||
|
-- reason for my troubles is probably related to the type of objects
|
||||||
|
-- being hSet's rather than sets.
|
||||||
|
p : ∀ {f} g → isProp (AreInverses {A = A} {B} f g)
|
||||||
|
p {f} g xx yy i = record
|
||||||
|
{ verso-recto = ve-re
|
||||||
|
; recto-verso = re-ve
|
||||||
|
}
|
||||||
|
where
|
||||||
|
module xxA = AreInverses xx
|
||||||
|
module yyA = AreInverses yy
|
||||||
|
ve-re : g ∘ f ≡ Function.id
|
||||||
|
ve-re = arrowsAreSets {A = hA} {B = hA} _ _ xxA.verso-recto yyA.verso-recto i
|
||||||
|
re-ve : f ∘ g ≡ Function.id
|
||||||
|
re-ve = arrowsAreSets {A = hB} {B = hB} _ _ xxA.recto-verso yyA.recto-verso i
|
||||||
|
1eq : x.inverse ≡ y.inverse
|
||||||
|
1eq = begin
|
||||||
|
x.inverse ≡⟨⟩
|
||||||
|
x.inverse ∘ Function.id ≡⟨ cong (λ φ → x.inverse ∘ φ) (sym yA.recto-verso) ⟩
|
||||||
|
x.inverse ∘ (f ∘ y.inverse) ≡⟨⟩
|
||||||
|
(x.inverse ∘ f) ∘ y.inverse ≡⟨ cong (λ φ → φ ∘ y.inverse) xA.verso-recto ⟩
|
||||||
|
Function.id ∘ y.inverse ≡⟨⟩
|
||||||
|
y.inverse ∎
|
||||||
|
2eq : (λ i → AreInverses f (1eq i)) [ x.areInverses ≡ y.areInverses ]
|
||||||
|
2eq = lemPropF p 1eq
|
||||||
|
res : x ≡ y
|
||||||
|
res i = 1eq i , 2eq i
|
||||||
|
module _ {ℓa ℓb : Level} {A : Set ℓa} {P : A → Set ℓb} where
|
||||||
|
lem2 : ((x : A) → isProp (P x)) → (p q : Σ A P)
|
||||||
|
→ (p ≡ q) ≃ (proj₁ p ≡ proj₁ q)
|
||||||
|
lem2 pA p q = fromIsomorphism iso
|
||||||
|
where
|
||||||
|
f : ∀ {p q} → p ≡ q → proj₁ p ≡ proj₁ q
|
||||||
|
f e i = proj₁ (e i)
|
||||||
|
g : ∀ {p q} → proj₁ p ≡ proj₁ q → p ≡ q
|
||||||
|
g {p} {q} = lemSig pA p q
|
||||||
|
ve-re : (e : p ≡ q) → (g ∘ f) e ≡ e
|
||||||
|
ve-re = pathJ (\ q (e : p ≡ q) → (g ∘ f) e ≡ e)
|
||||||
|
(\ i j → p .proj₁ , propSet (pA (p .proj₁)) (p .proj₂) (p .proj₂) (λ i → (g {p} {p} ∘ f) (λ i₁ → p) i .proj₂) (λ i → p .proj₂) i j ) q
|
||||||
|
re-ve : (e : proj₁ p ≡ proj₁ q) → (f {p} {q} ∘ g {p} {q}) e ≡ e
|
||||||
|
re-ve e = refl
|
||||||
|
inv : AreInverses (f {p} {q}) (g {p} {q})
|
||||||
|
inv = record
|
||||||
|
{ verso-recto = funExt ve-re
|
||||||
|
; recto-verso = funExt re-ve
|
||||||
|
}
|
||||||
|
iso : (p ≡ q) Eqv.≅ (proj₁ p ≡ proj₁ q)
|
||||||
|
iso = f , g , inv
|
||||||
|
|
||||||
|
lem3 : ∀ {ℓc} {Q : A → Set (ℓc ⊔ ℓb)}
|
||||||
|
→ ((a : A) → P a ≃ Q a) → Σ A P ≃ Σ A Q
|
||||||
|
lem3 {Q = Q} eA = res
|
||||||
|
where
|
||||||
|
f : Σ A P → Σ A Q
|
||||||
|
f (a , pA) = a , _≃_.eqv (eA a) pA
|
||||||
|
g : Σ A Q → Σ A P
|
||||||
|
g (a , qA) = a , g' qA
|
||||||
|
where
|
||||||
|
k : Eqv.Isomorphism _
|
||||||
|
k = Equiv≃.toIso _ _ (_≃_.isEqv (eA a))
|
||||||
|
open Σ k renaming (proj₁ to g')
|
||||||
|
ve-re : (x : Σ A P) → (g ∘ f) x ≡ x
|
||||||
|
ve-re x i = proj₁ x , eq i
|
||||||
|
where
|
||||||
|
eq : proj₂ ((g ∘ f) x) ≡ proj₂ x
|
||||||
|
eq = begin
|
||||||
|
proj₂ ((g ∘ f) x) ≡⟨⟩
|
||||||
|
proj₂ (g (f (a , pA))) ≡⟨⟩
|
||||||
|
g' (_≃_.eqv (eA a) pA) ≡⟨ lem ⟩
|
||||||
|
pA ∎
|
||||||
|
where
|
||||||
|
open Σ x renaming (proj₁ to a ; proj₂ to pA)
|
||||||
|
k : Eqv.Isomorphism _
|
||||||
|
k = Equiv≃.toIso _ _ (_≃_.isEqv (eA a))
|
||||||
|
open Σ k renaming (proj₁ to g' ; proj₂ to inv)
|
||||||
|
module A = AreInverses inv
|
||||||
|
-- anti-funExt
|
||||||
|
lem : (g' ∘ (_≃_.eqv (eA a))) pA ≡ pA
|
||||||
|
lem i = A.verso-recto i pA
|
||||||
|
re-ve : (x : Σ A Q) → (f ∘ g) x ≡ x
|
||||||
|
re-ve x i = proj₁ x , eq i
|
||||||
|
where
|
||||||
|
open Σ x renaming (proj₁ to a ; proj₂ to qA)
|
||||||
|
eq = begin
|
||||||
|
proj₂ ((f ∘ g) x) ≡⟨⟩
|
||||||
|
_≃_.eqv (eA a) (g' qA) ≡⟨ (λ i → A.recto-verso i qA) ⟩
|
||||||
|
qA ∎
|
||||||
|
where
|
||||||
|
k : Eqv.Isomorphism _
|
||||||
|
k = Equiv≃.toIso _ _ (_≃_.isEqv (eA a))
|
||||||
|
open Σ k renaming (proj₁ to g' ; proj₂ to inv)
|
||||||
|
module A = AreInverses inv
|
||||||
|
inv : AreInverses f g
|
||||||
|
inv = record
|
||||||
|
{ verso-recto = funExt ve-re
|
||||||
|
; recto-verso = funExt re-ve
|
||||||
|
}
|
||||||
|
iso : Σ A P Eqv.≅ Σ A Q
|
||||||
|
iso = f , g , inv
|
||||||
|
res : Σ A P ≃ Σ A Q
|
||||||
|
res = fromIsomorphism iso
|
||||||
|
|
||||||
|
module _ {ℓa ℓb : Level} {A : Set ℓa} {B : Set ℓb} where
|
||||||
|
lem4 : isSet A → isSet B → (f : A → B)
|
||||||
|
→ isEquiv A B f ≃ isIso f
|
||||||
|
lem4 sA sB f =
|
||||||
|
let
|
||||||
|
obv : isEquiv A B f → isIso f
|
||||||
|
obv = Equiv≃.toIso A B
|
||||||
|
inv : isIso f → isEquiv A B f
|
||||||
|
inv = Equiv≃.fromIso A B
|
||||||
|
re-ve : (x : isEquiv A B f) → (inv ∘ obv) x ≡ x
|
||||||
|
re-ve = Equiv≃.inverse-from-to-iso A B
|
||||||
|
ve-re : (x : isIso f) → (obv ∘ inv) x ≡ x
|
||||||
|
ve-re = Equiv≃.inverse-to-from-iso A B sA sB
|
||||||
|
iso : isEquiv A B f Eqv.≅ isIso f
|
||||||
|
iso = obv , inv ,
|
||||||
|
record
|
||||||
|
{ verso-recto = funExt re-ve
|
||||||
|
; recto-verso = funExt ve-re
|
||||||
|
}
|
||||||
|
in fromIsomorphism iso
|
||||||
|
|
||||||
module _ {hA hB : Object} where
|
module _ {hA hB : Object} where
|
||||||
private
|
open Σ hA renaming (proj₁ to A ; proj₂ to sA)
|
||||||
A = proj₁ hA
|
open Σ hB renaming (proj₁ to B ; proj₂ to sB)
|
||||||
isSetA : isSet A
|
|
||||||
isSetA = proj₂ hA
|
|
||||||
B = proj₁ hB
|
|
||||||
isSetB : isSet B
|
|
||||||
isSetB = proj₂ hB
|
|
||||||
|
|
||||||
toIsomorphism : A ≃ B → hA ≅ hB
|
-- lem3 and the equivalence from lem4
|
||||||
toIsomorphism e = obverse , inverse , verso-recto , recto-verso
|
step0 : Σ (A → B) isIso ≃ Σ (A → B) (isEquiv A B)
|
||||||
|
step0 = lem3 {ℓc = lzero} (λ f → sym≃ (lem4 sA sB f))
|
||||||
|
|
||||||
|
-- univalence
|
||||||
|
step1 : Σ (A → B) (isEquiv A B) ≃ (A ≡ B)
|
||||||
|
step1 = hh ⊙ h
|
||||||
where
|
where
|
||||||
open _≃_ e
|
h : (A ≃ B) ≃ (A ≡ B)
|
||||||
|
h = sym≃ (univalence {A = A} {B})
|
||||||
|
obv : Σ (A → B) (isEquiv A B) → A ≃ B
|
||||||
|
obv = Eqv.deEta
|
||||||
|
inv : A ≃ B → Σ (A → B) (isEquiv A B)
|
||||||
|
inv = Eqv.doEta
|
||||||
|
re-ve : (x : _) → (inv ∘ obv) x ≡ x
|
||||||
|
re-ve x = refl
|
||||||
|
-- Because _≃_ does not have eta equality!
|
||||||
|
ve-re : (x : _) → (obv ∘ inv) x ≡ x
|
||||||
|
ve-re (con eqv isEqv) i = con eqv isEqv
|
||||||
|
areInv : AreInverses obv inv
|
||||||
|
areInv = record { verso-recto = funExt re-ve ; recto-verso = funExt ve-re }
|
||||||
|
eqv : Σ (A → B) (isEquiv A B) Eqv.≅ (A ≃ B)
|
||||||
|
eqv = obv , inv , areInv
|
||||||
|
hh : Σ (A → B) (isEquiv A B) ≃ (A ≃ B)
|
||||||
|
hh = fromIsomorphism eqv
|
||||||
|
|
||||||
fromIsomorphism : hA ≅ hB → A ≃ B
|
-- lem2 with propIsSet
|
||||||
fromIsomorphism iso = con obverse (gradLemma obverse inverse recto-verso verso-recto)
|
step2 : (A ≡ B) ≃ (hA ≡ hB)
|
||||||
|
step2 = sym≃ (lem2 (λ A → isSetIsProp) hA hB)
|
||||||
|
|
||||||
|
-- Go from an isomorphism on sets to an isomorphism on homotopic sets
|
||||||
|
trivial? : (hA ≅ hB) ≃ (A Eqv.≅ B)
|
||||||
|
trivial? = sym≃ (fromIsomorphism res)
|
||||||
where
|
where
|
||||||
obverse : A → B
|
fwd : Σ (A → B) isIso → hA ≅ hB
|
||||||
obverse = proj₁ iso
|
fwd (f , g , inv) = f , g , inv.toPair
|
||||||
inverse : B → A
|
|
||||||
inverse = proj₁ (proj₂ iso)
|
|
||||||
-- FIXME IsInverseOf should change name to AreInverses and the
|
|
||||||
-- ordering should be swapped.
|
|
||||||
areInverses : IsInverseOf {A = hA} {hB} obverse inverse
|
|
||||||
areInverses = proj₂ (proj₂ iso)
|
|
||||||
verso-recto : ∀ a → (inverse ∘ obverse) a ≡ a
|
|
||||||
verso-recto a i = proj₁ areInverses i a
|
|
||||||
recto-verso : ∀ b → (obverse Function.∘ inverse) b ≡ b
|
|
||||||
recto-verso b i = proj₂ areInverses i b
|
|
||||||
|
|
||||||
private
|
|
||||||
univIso : (A ≡ B) A≅ (A ≃ B)
|
|
||||||
univIso = _≃_.toIsomorphism univalence
|
|
||||||
obverse' : A ≡ B → A ≃ B
|
|
||||||
obverse' = proj₁ univIso
|
|
||||||
inverse' : A ≃ B → A ≡ B
|
|
||||||
inverse' = proj₁ (proj₂ univIso)
|
|
||||||
-- Drop proof of being a set from both sides of an equality.
|
|
||||||
dropP : hA ≡ hB → A ≡ B
|
|
||||||
dropP eq i = proj₁ (eq i)
|
|
||||||
-- Add proof of being a set to both sides of a set-theoretic equivalence
|
|
||||||
-- returning a category-theoretic equivalence.
|
|
||||||
addE : A A≅ B → hA ≅ hB
|
|
||||||
addE eqv = proj₁ eqv , (proj₁ (proj₂ eqv)) , asPair
|
|
||||||
where
|
where
|
||||||
areeqv = proj₂ (proj₂ eqv)
|
module inv = AreInverses inv
|
||||||
asPair =
|
bwd : hA ≅ hB → Σ (A → B) isIso
|
||||||
let module Inv = AreInverses areeqv
|
bwd (f , g , x , y) = f , g , record { verso-recto = x ; recto-verso = y }
|
||||||
in Inv.verso-recto , Inv.recto-verso
|
res : Σ (A → B) isIso Eqv.≅ (hA ≅ hB)
|
||||||
|
res = fwd , bwd , record { verso-recto = refl ; recto-verso = refl }
|
||||||
|
|
||||||
obverse : hA ≡ hB → hA ≅ hB
|
conclusion : (hA ≅ hB) ≃ (hA ≡ hB)
|
||||||
obverse = addE ∘ _≃_.toIsomorphism ∘ obverse' ∘ dropP
|
conclusion = trivial? ⊙ step0 ⊙ step1 ⊙ step2
|
||||||
|
|
||||||
-- Drop proof of being a set form both sides of a category-theoretic
|
univ≃ : (hA ≅ hB) ≃ (hA ≡ hB)
|
||||||
-- equivalence returning a set-theoretic equivalence.
|
univ≃ = trivial? ⊙ step0 ⊙ step1 ⊙ step2
|
||||||
dropE : hA ≅ hB → A A≅ B
|
|
||||||
dropE eqv = obv , inv , asAreInverses
|
module _ (hA : Object) where
|
||||||
|
open Σ hA renaming (proj₁ to A)
|
||||||
|
|
||||||
|
eq1 : (Σ[ hB ∈ Object ] hA ≅ hB) ≡ (Σ[ hB ∈ Object ] hA ≡ hB)
|
||||||
|
eq1 = ua (lem3 (\ hB → univ≃))
|
||||||
|
|
||||||
|
univalent[Contr] : isContr (Σ[ hB ∈ Object ] hA ≅ hB)
|
||||||
|
univalent[Contr] = subst {P = isContr} (sym eq1) tres
|
||||||
where
|
where
|
||||||
obv = proj₁ eqv
|
module _ (y : Σ[ hB ∈ Object ] hA ≡ hB) where
|
||||||
inv = proj₁ (proj₂ eqv)
|
open Σ y renaming (proj₁ to hB ; proj₂ to hA≡hB)
|
||||||
areEq = proj₂ (proj₂ eqv)
|
qres : (hA , refl) ≡ (hB , hA≡hB)
|
||||||
asAreInverses : AreInverses A B obv inv
|
qres = contrSingl hA≡hB
|
||||||
asAreInverses = record { verso-recto = proj₁ areEq ; recto-verso = proj₂ areEq }
|
|
||||||
|
|
||||||
-- Dunno if this is a thing.
|
tres : isContr (Σ[ hB ∈ Object ] hA ≡ hB)
|
||||||
isoToEquiv : A A≅ B → A ≃ B
|
tres = (hA , refl) , qres
|
||||||
isoToEquiv = {!!}
|
|
||||||
-- Add proof of being a set to both sides of an equality.
|
|
||||||
addP : A ≡ B → hA ≡ hB
|
|
||||||
addP p = lemSig (λ X → propPi λ x → propPi (λ y → propIsProp)) hA hB p
|
|
||||||
inverse : hA ≅ hB → hA ≡ hB
|
|
||||||
inverse = addP ∘ inverse' ∘ isoToEquiv ∘ dropE
|
|
||||||
|
|
||||||
-- open AreInverses (proj₂ (proj₂ univIso)) renaming
|
univalent : Univalent
|
||||||
-- ( verso-recto to verso-recto'
|
univalent = from[Contr] univalent[Contr]
|
||||||
-- ; recto-verso to recto-verso'
|
|
||||||
-- )
|
|
||||||
-- I can just open them but I wanna be able to see the type annotations.
|
|
||||||
verso-recto' : inverse' ∘ obverse' ≡ Function.id
|
|
||||||
verso-recto' = AreInverses.verso-recto (proj₂ (proj₂ univIso))
|
|
||||||
recto-verso' : obverse' ∘ inverse' ≡ Function.id
|
|
||||||
recto-verso' = AreInverses.recto-verso (proj₂ (proj₂ univIso))
|
|
||||||
verso-recto : (iso : hA ≅ hB) → obverse (inverse iso) ≡ iso
|
|
||||||
verso-recto iso = begin
|
|
||||||
obverse (inverse iso) ≡⟨⟩
|
|
||||||
( addE ∘ _≃_.toIsomorphism
|
|
||||||
∘ obverse' ∘ dropP ∘ addP
|
|
||||||
∘ inverse' ∘ isoToEquiv
|
|
||||||
∘ dropE) iso
|
|
||||||
≡⟨⟩
|
|
||||||
( addE ∘ _≃_.toIsomorphism
|
|
||||||
∘ obverse'
|
|
||||||
∘ inverse' ∘ isoToEquiv
|
|
||||||
∘ dropE) iso
|
|
||||||
≡⟨ {!!} ⟩ -- obverse' inverse' are inverses
|
|
||||||
( addE ∘ _≃_.toIsomorphism ∘ isoToEquiv ∘ dropE) iso
|
|
||||||
≡⟨ {!!} ⟩ -- should be easy to prove
|
|
||||||
-- _≃_.toIsomorphism ∘ isoToEquiv ≡ id
|
|
||||||
(addE ∘ dropE) iso
|
|
||||||
≡⟨⟩
|
|
||||||
iso ∎
|
|
||||||
|
|
||||||
-- Similar to above.
|
|
||||||
recto-verso : (eq : hA ≡ hB) → inverse (obverse eq) ≡ eq
|
|
||||||
recto-verso eq = begin
|
|
||||||
inverse (obverse eq) ≡⟨ {!!} ⟩
|
|
||||||
eq ∎
|
|
||||||
|
|
||||||
-- Use the fact that being an h-level is a mere proposition.
|
|
||||||
-- This is almost provable using `Wishlist.isSetIsProp` - although
|
|
||||||
-- this creates homogenous paths.
|
|
||||||
isSetEq : (p : A ≡ B) → (λ i → isSet (p i)) [ isSetA ≡ isSetB ]
|
|
||||||
isSetEq = {!!}
|
|
||||||
|
|
||||||
res : hA ≡ hB
|
|
||||||
proj₁ (res i) = {!!}
|
|
||||||
proj₂ (res i) = isSetEq {!!} i
|
|
||||||
univalent : isEquiv (hA ≡ hB) (hA ≅ hB) (id-to-iso (λ {A} {B} → isIdentity {A} {B}) hA hB)
|
|
||||||
univalent = {!gradLemma obverse inverse verso-recto recto-verso!}
|
|
||||||
|
|
||||||
SetsIsCategory : IsCategory SetsRaw
|
SetsIsCategory : IsCategory SetsRaw
|
||||||
IsCategory.isAssociative SetsIsCategory = refl
|
IsCategory.isAssociative SetsIsCategory = refl
|
||||||
|
|
@ -179,39 +302,35 @@ module _ {ℓ : Level} where
|
||||||
open Category 𝓢
|
open Category 𝓢
|
||||||
open import Cubical.Sigma
|
open import Cubical.Sigma
|
||||||
|
|
||||||
module _ (0A 0B : Object) where
|
module _ (hA hB : Object) where
|
||||||
|
open Σ hA renaming (proj₁ to A ; proj₂ to sA)
|
||||||
|
open Σ hB renaming (proj₁ to B ; proj₂ to sB)
|
||||||
|
|
||||||
private
|
private
|
||||||
A : Set ℓ
|
productObject : Object
|
||||||
A = proj₁ 0A
|
productObject = (A × B) , sigPresSet sA λ _ → sB
|
||||||
sA : isSet A
|
|
||||||
sA = proj₂ 0A
|
|
||||||
B : Set ℓ
|
|
||||||
B = proj₁ 0B
|
|
||||||
sB : isSet B
|
|
||||||
sB = proj₂ 0B
|
|
||||||
0A×0B : Object
|
|
||||||
0A×0B = (A × B) , sigPresSet sA λ _ → sB
|
|
||||||
|
|
||||||
module _ {X A B : Set ℓ} (f : X → A) (g : X → B) where
|
module _ {X A B : Set ℓ} (f : X → A) (g : X → B) where
|
||||||
_&&&_ : (X → A × B)
|
_&&&_ : (X → A × B)
|
||||||
_&&&_ x = f x , g x
|
_&&&_ x = f x , g x
|
||||||
module _ {0X : Object} where
|
|
||||||
X = proj₁ 0X
|
|
||||||
module _ (f : X → A ) (g : X → B) where
|
|
||||||
lem : proj₁ Function.∘′ (f &&& g) ≡ f × proj₂ Function.∘′ (f &&& g) ≡ g
|
|
||||||
proj₁ lem = refl
|
|
||||||
proj₂ lem = refl
|
|
||||||
|
|
||||||
rawProduct : RawProduct 𝓢 0A 0B
|
module _ (hX : Object) where
|
||||||
RawProduct.object rawProduct = 0A×0B
|
open Σ hX renaming (proj₁ to X)
|
||||||
|
module _ (f : X → A ) (g : X → B) where
|
||||||
|
ump : proj₁ Function.∘′ (f &&& g) ≡ f × proj₂ Function.∘′ (f &&& g) ≡ g
|
||||||
|
proj₁ ump = refl
|
||||||
|
proj₂ ump = refl
|
||||||
|
|
||||||
|
rawProduct : RawProduct 𝓢 hA hB
|
||||||
|
RawProduct.object rawProduct = productObject
|
||||||
RawProduct.proj₁ rawProduct = Data.Product.proj₁
|
RawProduct.proj₁ rawProduct = Data.Product.proj₁
|
||||||
RawProduct.proj₂ rawProduct = Data.Product.proj₂
|
RawProduct.proj₂ rawProduct = Data.Product.proj₂
|
||||||
|
|
||||||
isProduct : IsProduct 𝓢 _ _ rawProduct
|
isProduct : IsProduct 𝓢 _ _ rawProduct
|
||||||
IsProduct.isProduct isProduct {X = X} f g
|
IsProduct.ump isProduct {X = hX} f g
|
||||||
= (f &&& g) , lem {0X = X} f g
|
= (f &&& g) , ump hX f g
|
||||||
|
|
||||||
product : Product 𝓢 0A 0B
|
product : Product 𝓢 hA hB
|
||||||
Product.raw product = rawProduct
|
Product.raw product = rawProduct
|
||||||
Product.isProduct product = isProduct
|
Product.isProduct product = isProduct
|
||||||
|
|
||||||
|
|
@ -220,6 +339,8 @@ module _ {ℓ : Level} where
|
||||||
SetsHasProducts = record { product = product }
|
SetsHasProducts = record { product = product }
|
||||||
|
|
||||||
module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||||
|
open Category ℂ
|
||||||
|
|
||||||
-- Covariant Presheaf
|
-- Covariant Presheaf
|
||||||
Representable : Set (ℓa ⊔ lsuc ℓb)
|
Representable : Set (ℓa ⊔ lsuc ℓb)
|
||||||
Representable = Functor ℂ (𝓢𝓮𝓽 ℓb)
|
Representable = Functor ℂ (𝓢𝓮𝓽 ℓb)
|
||||||
|
|
@ -228,8 +349,6 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||||
Presheaf : Set (ℓa ⊔ lsuc ℓb)
|
Presheaf : Set (ℓa ⊔ lsuc ℓb)
|
||||||
Presheaf = Functor (opposite ℂ) (𝓢𝓮𝓽 ℓb)
|
Presheaf = Functor (opposite ℂ) (𝓢𝓮𝓽 ℓb)
|
||||||
|
|
||||||
open Category ℂ
|
|
||||||
|
|
||||||
-- The "co-yoneda" embedding.
|
-- The "co-yoneda" embedding.
|
||||||
representable : Category.Object ℂ → Representable
|
representable : Category.Object ℂ → Representable
|
||||||
representable A = record
|
representable A = record
|
||||||
|
|
@ -238,7 +357,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||||
; fmap = ℂ [_∘_]
|
; fmap = ℂ [_∘_]
|
||||||
}
|
}
|
||||||
; isFunctor = record
|
; isFunctor = record
|
||||||
{ isIdentity = funExt λ _ → proj₂ isIdentity
|
{ isIdentity = funExt λ _ → leftIdentity
|
||||||
; isDistributive = funExt λ x → sym isAssociative
|
; isDistributive = funExt λ x → sym isAssociative
|
||||||
}
|
}
|
||||||
}
|
}
|
||||||
|
|
@ -251,7 +370,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||||
; fmap = λ f g → ℂ [ g ∘ f ]
|
; fmap = λ f g → ℂ [ g ∘ f ]
|
||||||
}
|
}
|
||||||
; isFunctor = record
|
; isFunctor = record
|
||||||
{ isIdentity = funExt λ x → proj₁ isIdentity
|
{ isIdentity = funExt λ x → rightIdentity
|
||||||
; isDistributive = funExt λ x → isAssociative
|
; isDistributive = funExt λ x → isAssociative
|
||||||
}
|
}
|
||||||
}
|
}
|
||||||
|
|
|
||||||
|
|
@ -28,37 +28,17 @@
|
||||||
|
|
||||||
module Cat.Category where
|
module Cat.Category where
|
||||||
|
|
||||||
open import Agda.Primitive
|
open import Cat.Prelude
|
||||||
open import Data.Unit.Base
|
renaming
|
||||||
open import Data.Product renaming
|
|
||||||
( proj₁ to fst
|
( proj₁ to fst
|
||||||
; proj₂ to snd
|
; proj₂ to snd
|
||||||
; ∃! to ∃!≈
|
|
||||||
)
|
)
|
||||||
open import Data.Empty
|
|
||||||
import Function
|
import Function
|
||||||
open import Cubical
|
|
||||||
open import Cubical.NType.Properties using ( propIsEquiv ; lemPropF )
|
|
||||||
|
|
||||||
open import Cat.Wishlist
|
------------------
|
||||||
|
|
||||||
-----------------
|
|
||||||
-- * Utilities --
|
|
||||||
-----------------
|
|
||||||
|
|
||||||
-- | Unique existensials.
|
|
||||||
∃! : ∀ {a b} {A : Set a}
|
|
||||||
→ (A → Set b) → Set (a ⊔ b)
|
|
||||||
∃! = ∃!≈ _≡_
|
|
||||||
|
|
||||||
∃!-syntax : ∀ {a b} {A : Set a} → (A → Set b) → Set (a ⊔ b)
|
|
||||||
∃!-syntax = ∃
|
|
||||||
|
|
||||||
syntax ∃!-syntax (λ x → B) = ∃![ x ] B
|
|
||||||
|
|
||||||
-----------------
|
|
||||||
-- * Categories --
|
-- * Categories --
|
||||||
-----------------
|
------------------
|
||||||
|
|
||||||
-- | Raw categories
|
-- | Raw categories
|
||||||
--
|
--
|
||||||
|
|
@ -96,7 +76,7 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
|
||||||
|
|
||||||
IsIdentity : ({A : Object} → Arrow A A) → Set (ℓa ⊔ ℓb)
|
IsIdentity : ({A : Object} → Arrow A A) → Set (ℓa ⊔ ℓb)
|
||||||
IsIdentity id = {A B : Object} {f : Arrow A B}
|
IsIdentity id = {A B : Object} {f : Arrow A B}
|
||||||
→ f ∘ id ≡ f × id ∘ f ≡ f
|
→ id ∘ f ≡ f × f ∘ id ≡ f
|
||||||
|
|
||||||
ArrowsAreSets : Set (ℓa ⊔ ℓb)
|
ArrowsAreSets : Set (ℓa ⊔ ℓb)
|
||||||
ArrowsAreSets = ∀ {A B : Object} → isSet (Arrow A B)
|
ArrowsAreSets = ∀ {A B : Object} → isSet (Arrow A B)
|
||||||
|
|
@ -130,21 +110,33 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
|
||||||
Terminal = Σ Object IsTerminal
|
Terminal = Σ Object IsTerminal
|
||||||
|
|
||||||
-- | Univalence is indexed by a raw category as well as an identity proof.
|
-- | Univalence is indexed by a raw category as well as an identity proof.
|
||||||
--
|
module Univalence (isIdentity : IsIdentity 𝟙) where
|
||||||
-- FIXME Put this in `RawCategory` and index it on the witness to `isIdentity`.
|
-- | The identity isomorphism
|
||||||
module Univalence {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
|
|
||||||
open RawCategory ℂ
|
|
||||||
module _ (isIdentity : IsIdentity 𝟙) where
|
|
||||||
idIso : (A : Object) → A ≅ A
|
idIso : (A : Object) → A ≅ A
|
||||||
idIso A = 𝟙 , (𝟙 , isIdentity)
|
idIso A = 𝟙 , (𝟙 , isIdentity)
|
||||||
|
|
||||||
-- Lemma 9.1.4 in [HoTT]
|
-- | Extract an isomorphism from an equality
|
||||||
|
--
|
||||||
|
-- [HoTT §9.1.4]
|
||||||
id-to-iso : (A B : Object) → A ≡ B → A ≅ B
|
id-to-iso : (A B : Object) → A ≡ B → A ≅ B
|
||||||
id-to-iso A B eq = transp (\ i → A ≅ eq i) (idIso A)
|
id-to-iso A B eq = transp (\ i → A ≅ eq i) (idIso A)
|
||||||
|
|
||||||
Univalent : Set (ℓa ⊔ ℓb)
|
Univalent : Set (ℓa ⊔ ℓb)
|
||||||
Univalent = {A B : Object} → isEquiv (A ≡ B) (A ≅ B) (id-to-iso A B)
|
Univalent = {A B : Object} → isEquiv (A ≡ B) (A ≅ B) (id-to-iso A B)
|
||||||
|
|
||||||
|
-- A perhaps more readable version of univalence:
|
||||||
|
Univalent≃ = {A B : Object} → (A ≡ B) ≃ (A ≅ B)
|
||||||
|
|
||||||
|
-- | Equivalent formulation of univalence.
|
||||||
|
Univalent[Contr] : Set _
|
||||||
|
Univalent[Contr] = ∀ A → isContr (Σ[ X ∈ Object ] A ≅ X)
|
||||||
|
|
||||||
|
-- From: Thierry Coquand <Thierry.Coquand@cse.gu.se>
|
||||||
|
-- Date: Wed, Mar 21, 2018 at 3:12 PM
|
||||||
|
--
|
||||||
|
-- This is not so straight-forward so you can assume it
|
||||||
|
postulate from[Contr] : Univalent[Contr] → Univalent
|
||||||
|
|
||||||
-- | The mere proposition of being a category.
|
-- | The mere proposition of being a category.
|
||||||
--
|
--
|
||||||
-- Also defines a few lemmas:
|
-- Also defines a few lemmas:
|
||||||
|
|
@ -156,52 +148,59 @@ module Univalence {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
|
||||||
-- [HoTT].
|
-- [HoTT].
|
||||||
record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc (ℓa ⊔ ℓb)) where
|
record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc (ℓa ⊔ ℓb)) where
|
||||||
open RawCategory ℂ public
|
open RawCategory ℂ public
|
||||||
open Univalence ℂ public
|
|
||||||
field
|
field
|
||||||
isAssociative : IsAssociative
|
isAssociative : IsAssociative
|
||||||
isIdentity : IsIdentity 𝟙
|
isIdentity : IsIdentity 𝟙
|
||||||
arrowsAreSets : ArrowsAreSets
|
arrowsAreSets : ArrowsAreSets
|
||||||
univalent : Univalent isIdentity
|
open Univalence isIdentity public
|
||||||
|
field
|
||||||
|
univalent : Univalent
|
||||||
|
|
||||||
-- Some common lemmas about categories.
|
leftIdentity : {A B : Object} {f : Arrow A B} → 𝟙 ∘ f ≡ f
|
||||||
|
leftIdentity {A} {B} {f} = fst (isIdentity {A = A} {B} {f})
|
||||||
|
|
||||||
|
rightIdentity : {A B : Object} {f : Arrow A B} → f ∘ 𝟙 ≡ f
|
||||||
|
rightIdentity {A} {B} {f} = snd (isIdentity {A = A} {B} {f})
|
||||||
|
|
||||||
|
------------
|
||||||
|
-- Lemmas --
|
||||||
|
------------
|
||||||
|
|
||||||
|
-- | Relation between iso- epi- and mono- morphisms.
|
||||||
module _ {A B : Object} {X : Object} (f : Arrow A B) where
|
module _ {A B : Object} {X : Object} (f : Arrow A B) where
|
||||||
iso-is-epi : Isomorphism f → Epimorphism {X = X} f
|
iso→epi : Isomorphism f → Epimorphism {X = X} f
|
||||||
iso-is-epi (f- , left-inv , right-inv) g₀ g₁ eq = begin
|
iso→epi (f- , left-inv , right-inv) g₀ g₁ eq = begin
|
||||||
g₀ ≡⟨ sym (fst isIdentity) ⟩
|
g₀ ≡⟨ sym rightIdentity ⟩
|
||||||
g₀ ∘ 𝟙 ≡⟨ cong (_∘_ g₀) (sym right-inv) ⟩
|
g₀ ∘ 𝟙 ≡⟨ cong (_∘_ g₀) (sym right-inv) ⟩
|
||||||
g₀ ∘ (f ∘ f-) ≡⟨ isAssociative ⟩
|
g₀ ∘ (f ∘ f-) ≡⟨ isAssociative ⟩
|
||||||
(g₀ ∘ f) ∘ f- ≡⟨ cong (λ φ → φ ∘ f-) eq ⟩
|
(g₀ ∘ f) ∘ f- ≡⟨ cong (λ φ → φ ∘ f-) eq ⟩
|
||||||
(g₁ ∘ f) ∘ f- ≡⟨ sym isAssociative ⟩
|
(g₁ ∘ f) ∘ f- ≡⟨ sym isAssociative ⟩
|
||||||
g₁ ∘ (f ∘ f-) ≡⟨ cong (_∘_ g₁) right-inv ⟩
|
g₁ ∘ (f ∘ f-) ≡⟨ cong (_∘_ g₁) right-inv ⟩
|
||||||
g₁ ∘ 𝟙 ≡⟨ fst isIdentity ⟩
|
g₁ ∘ 𝟙 ≡⟨ rightIdentity ⟩
|
||||||
g₁ ∎
|
g₁ ∎
|
||||||
|
|
||||||
iso-is-mono : Isomorphism f → Monomorphism {X = X} f
|
iso→mono : Isomorphism f → Monomorphism {X = X} f
|
||||||
iso-is-mono (f- , (left-inv , right-inv)) g₀ g₁ eq =
|
iso→mono (f- , left-inv , right-inv) g₀ g₁ eq =
|
||||||
begin
|
begin
|
||||||
g₀ ≡⟨ sym (snd isIdentity) ⟩
|
g₀ ≡⟨ sym leftIdentity ⟩
|
||||||
𝟙 ∘ g₀ ≡⟨ cong (λ φ → φ ∘ g₀) (sym left-inv) ⟩
|
𝟙 ∘ g₀ ≡⟨ cong (λ φ → φ ∘ g₀) (sym left-inv) ⟩
|
||||||
(f- ∘ f) ∘ g₀ ≡⟨ sym isAssociative ⟩
|
(f- ∘ f) ∘ g₀ ≡⟨ sym isAssociative ⟩
|
||||||
f- ∘ (f ∘ g₀) ≡⟨ cong (_∘_ f-) eq ⟩
|
f- ∘ (f ∘ g₀) ≡⟨ cong (_∘_ f-) eq ⟩
|
||||||
f- ∘ (f ∘ g₁) ≡⟨ isAssociative ⟩
|
f- ∘ (f ∘ g₁) ≡⟨ isAssociative ⟩
|
||||||
(f- ∘ f) ∘ g₁ ≡⟨ cong (λ φ → φ ∘ g₁) left-inv ⟩
|
(f- ∘ f) ∘ g₁ ≡⟨ cong (λ φ → φ ∘ g₁) left-inv ⟩
|
||||||
𝟙 ∘ g₁ ≡⟨ snd isIdentity ⟩
|
𝟙 ∘ g₁ ≡⟨ leftIdentity ⟩
|
||||||
g₁ ∎
|
g₁ ∎
|
||||||
|
|
||||||
iso-is-epi-mono : Isomorphism f → Epimorphism {X = X} f × Monomorphism {X = X} f
|
iso→epi×mono : Isomorphism f → Epimorphism {X = X} f × Monomorphism {X = X} f
|
||||||
iso-is-epi-mono iso = iso-is-epi iso , iso-is-mono iso
|
iso→epi×mono iso = iso→epi iso , iso→mono iso
|
||||||
|
|
||||||
-- | Propositionality of being a category
|
-- | The formulation of univalence expressed with _≃_ is trivially admissable -
|
||||||
--
|
-- just "forget" the equivalence.
|
||||||
-- Proves that all projections of `IsCategory` are mere propositions as well as
|
univalent≃ : Univalent≃
|
||||||
-- `IsCategory` itself being a mere proposition.
|
univalent≃ = _ , univalent
|
||||||
module Propositionality {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
|
|
||||||
open RawCategory ℂ
|
|
||||||
module _ (ℂ : IsCategory ℂ) where
|
|
||||||
open IsCategory ℂ using (isAssociative ; arrowsAreSets ; isIdentity ; Univalent)
|
|
||||||
open import Cubical.NType
|
|
||||||
open import Cubical.NType.Properties
|
|
||||||
|
|
||||||
|
-- | All projections are propositions.
|
||||||
|
module Propositionality where
|
||||||
propIsAssociative : isProp IsAssociative
|
propIsAssociative : isProp IsAssociative
|
||||||
propIsAssociative x y i = arrowsAreSets _ _ x y i
|
propIsAssociative x y i = arrowsAreSets _ _ x y i
|
||||||
|
|
||||||
|
|
@ -227,31 +226,34 @@ module Propositionality {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
|
||||||
isoIsProp a@(g , η , ε) a'@(g' , η' , ε') =
|
isoIsProp a@(g , η , ε) a'@(g' , η' , ε') =
|
||||||
lemSig (λ g → propIsInverseOf) a a' geq
|
lemSig (λ g → propIsInverseOf) a a' geq
|
||||||
where
|
where
|
||||||
open Cubical.NType.Properties
|
|
||||||
geq : g ≡ g'
|
geq : g ≡ g'
|
||||||
geq = begin
|
geq = begin
|
||||||
g ≡⟨ sym (fst isIdentity) ⟩
|
g ≡⟨ sym rightIdentity ⟩
|
||||||
g ∘ 𝟙 ≡⟨ cong (λ φ → g ∘ φ) (sym ε') ⟩
|
g ∘ 𝟙 ≡⟨ cong (λ φ → g ∘ φ) (sym ε') ⟩
|
||||||
g ∘ (f ∘ g') ≡⟨ isAssociative ⟩
|
g ∘ (f ∘ g') ≡⟨ isAssociative ⟩
|
||||||
(g ∘ f) ∘ g' ≡⟨ cong (λ φ → φ ∘ g') η ⟩
|
(g ∘ f) ∘ g' ≡⟨ cong (λ φ → φ ∘ g') η ⟩
|
||||||
𝟙 ∘ g' ≡⟨ snd isIdentity ⟩
|
𝟙 ∘ g' ≡⟨ leftIdentity ⟩
|
||||||
g' ∎
|
g' ∎
|
||||||
|
|
||||||
propUnivalent : isProp (Univalent isIdentity)
|
propUnivalent : isProp Univalent
|
||||||
propUnivalent a b i = propPi (λ iso → propHasLevel ⟨-2⟩) a b i
|
propUnivalent a b i = propPi (λ iso → propIsContr) a b i
|
||||||
|
|
||||||
|
-- | Propositionality of being a category
|
||||||
|
module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
|
||||||
|
open RawCategory ℂ
|
||||||
|
open Univalence
|
||||||
private
|
private
|
||||||
module _ (x y : IsCategory ℂ) where
|
module _ (x y : IsCategory ℂ) where
|
||||||
module IC = IsCategory
|
|
||||||
module X = IsCategory x
|
module X = IsCategory x
|
||||||
module Y = IsCategory y
|
module Y = IsCategory y
|
||||||
open Univalence ℂ
|
|
||||||
-- In a few places I use the result of propositionality of the various
|
-- In a few places I use the result of propositionality of the various
|
||||||
-- projections of `IsCategory` - I've arbitrarily chosed to use this
|
-- projections of `IsCategory` - Here I arbitrarily chose to use this
|
||||||
-- result from `x : IsCategory C`. I don't know which (if any) possibly
|
-- result from `x : IsCategory C`. I don't know which (if any) possibly
|
||||||
-- adverse effects this may have.
|
-- adverse effects this may have.
|
||||||
|
module Prop = X.Propositionality
|
||||||
|
|
||||||
isIdentity : (λ _ → IsIdentity 𝟙) [ X.isIdentity ≡ Y.isIdentity ]
|
isIdentity : (λ _ → IsIdentity 𝟙) [ X.isIdentity ≡ Y.isIdentity ]
|
||||||
isIdentity = propIsIdentity x X.isIdentity Y.isIdentity
|
isIdentity = Prop.propIsIdentity X.isIdentity Y.isIdentity
|
||||||
U : ∀ {a : IsIdentity 𝟙}
|
U : ∀ {a : IsIdentity 𝟙}
|
||||||
→ (λ _ → IsIdentity 𝟙) [ X.isIdentity ≡ a ]
|
→ (λ _ → IsIdentity 𝟙) [ X.isIdentity ≡ a ]
|
||||||
→ (b : Univalent a)
|
→ (b : Univalent a)
|
||||||
|
|
@ -264,16 +266,16 @@ module Propositionality {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
|
||||||
P y eq = ∀ (univ : Univalent y) → U eq univ
|
P y eq = ∀ (univ : Univalent y) → U eq univ
|
||||||
p : ∀ (b' : Univalent X.isIdentity)
|
p : ∀ (b' : Univalent X.isIdentity)
|
||||||
→ (λ _ → Univalent X.isIdentity) [ X.univalent ≡ b' ]
|
→ (λ _ → Univalent X.isIdentity) [ X.univalent ≡ b' ]
|
||||||
p univ = propUnivalent x X.univalent univ
|
p univ = Prop.propUnivalent X.univalent univ
|
||||||
helper : P Y.isIdentity isIdentity
|
helper : P Y.isIdentity isIdentity
|
||||||
helper = pathJ P p Y.isIdentity isIdentity
|
helper = pathJ P p Y.isIdentity isIdentity
|
||||||
eqUni : U isIdentity Y.univalent
|
eqUni : U isIdentity Y.univalent
|
||||||
eqUni = helper Y.univalent
|
eqUni = helper Y.univalent
|
||||||
done : x ≡ y
|
done : x ≡ y
|
||||||
IC.isAssociative (done i) = propIsAssociative x X.isAssociative Y.isAssociative i
|
IsCategory.isAssociative (done i) = Prop.propIsAssociative X.isAssociative Y.isAssociative i
|
||||||
IC.isIdentity (done i) = isIdentity i
|
IsCategory.isIdentity (done i) = isIdentity i
|
||||||
IC.arrowsAreSets (done i) = propArrowIsSet x X.arrowsAreSets Y.arrowsAreSets i
|
IsCategory.arrowsAreSets (done i) = Prop.propArrowIsSet X.arrowsAreSets Y.arrowsAreSets i
|
||||||
IC.univalent (done i) = eqUni i
|
IsCategory.univalent (done i) = eqUni i
|
||||||
|
|
||||||
propIsCategory : isProp (IsCategory ℂ)
|
propIsCategory : isProp (IsCategory ℂ)
|
||||||
propIsCategory = done
|
propIsCategory = done
|
||||||
|
|
@ -298,7 +300,7 @@ module _ {ℓa ℓb : Level} {ℂ 𝔻 : Category ℓa ℓb} where
|
||||||
module _ (rawEq : ℂ.raw ≡ 𝔻.raw) where
|
module _ (rawEq : ℂ.raw ≡ 𝔻.raw) where
|
||||||
private
|
private
|
||||||
isCategoryEq : (λ i → IsCategory (rawEq i)) [ ℂ.isCategory ≡ 𝔻.isCategory ]
|
isCategoryEq : (λ i → IsCategory (rawEq i)) [ ℂ.isCategory ≡ 𝔻.isCategory ]
|
||||||
isCategoryEq = lemPropF Propositionality.propIsCategory rawEq
|
isCategoryEq = lemPropF propIsCategory rawEq
|
||||||
|
|
||||||
Category≡ : ℂ ≡ 𝔻
|
Category≡ : ℂ ≡ 𝔻
|
||||||
Category≡ i = record
|
Category≡ i = record
|
||||||
|
|
@ -330,21 +332,22 @@ module Opposite {ℓa ℓb : Level} where
|
||||||
RawCategory._∘_ opRaw = Function.flip ℂ._∘_
|
RawCategory._∘_ opRaw = Function.flip ℂ._∘_
|
||||||
|
|
||||||
open RawCategory opRaw
|
open RawCategory opRaw
|
||||||
open Univalence opRaw
|
|
||||||
|
|
||||||
isIdentity : IsIdentity 𝟙
|
isIdentity : IsIdentity 𝟙
|
||||||
isIdentity = swap ℂ.isIdentity
|
isIdentity = swap ℂ.isIdentity
|
||||||
|
|
||||||
|
open Univalence isIdentity
|
||||||
|
|
||||||
module _ {A B : ℂ.Object} where
|
module _ {A B : ℂ.Object} where
|
||||||
univalent : isEquiv (A ≡ B) (A ≅ B)
|
univalent : isEquiv (A ≡ B) (A ≅ B)
|
||||||
(id-to-iso (swap ℂ.isIdentity) A B)
|
(Univalence.id-to-iso (swap ℂ.isIdentity) A B)
|
||||||
fst (univalent iso) = flipFiber (fst (ℂ.univalent (flipIso iso)))
|
fst (univalent iso) = flipFiber (fst (ℂ.univalent (flipIso iso)))
|
||||||
where
|
where
|
||||||
flipIso : A ≅ B → B ℂ.≅ A
|
flipIso : A ≅ B → B ℂ.≅ A
|
||||||
flipIso (f , f~ , iso) = f , f~ , swap iso
|
flipIso (f , f~ , iso) = f , f~ , swap iso
|
||||||
flipFiber
|
flipFiber
|
||||||
: fiber (ℂ.id-to-iso ℂ.isIdentity B A) (flipIso iso)
|
: fiber (ℂ.id-to-iso B A) (flipIso iso)
|
||||||
→ fiber ( id-to-iso isIdentity A B) iso
|
→ fiber ( id-to-iso A B) iso
|
||||||
flipFiber (eq , eqIso) = sym eq , {!!}
|
flipFiber (eq , eqIso) = sym eq , {!!}
|
||||||
snd (univalent iso) = {!!}
|
snd (univalent iso) = {!!}
|
||||||
|
|
||||||
|
|
|
||||||
|
|
@ -1,8 +1,6 @@
|
||||||
module Cat.Category.Exponential where
|
module Cat.Category.Exponential where
|
||||||
|
|
||||||
open import Agda.Primitive
|
open import Cat.Prelude hiding (_×_)
|
||||||
open import Data.Product hiding (_×_)
|
|
||||||
open import Cubical
|
|
||||||
|
|
||||||
open import Cat.Category
|
open import Cat.Category
|
||||||
open import Cat.Category.Product
|
open import Cat.Category.Product
|
||||||
|
|
|
||||||
|
|
@ -2,9 +2,11 @@
|
||||||
module Cat.Category.Functor where
|
module Cat.Category.Functor where
|
||||||
|
|
||||||
open import Agda.Primitive
|
open import Agda.Primitive
|
||||||
open import Cubical
|
|
||||||
open import Function
|
open import Function
|
||||||
|
|
||||||
|
open import Cubical
|
||||||
|
open import Cubical.NType.Properties using (lemPropF)
|
||||||
|
|
||||||
open import Cat.Category
|
open import Cat.Category
|
||||||
|
|
||||||
open Category hiding (_∘_ ; raw ; IsIdentity)
|
open Category hiding (_∘_ ; raw ; IsIdentity)
|
||||||
|
|
@ -72,14 +74,12 @@ module _ {ℓc ℓc' ℓd ℓd'}
|
||||||
|
|
||||||
open IsFunctor isFunctor public
|
open IsFunctor isFunctor public
|
||||||
|
|
||||||
open Functor
|
|
||||||
|
|
||||||
EndoFunctor : ∀ {ℓa ℓb} (ℂ : Category ℓa ℓb) → Set _
|
EndoFunctor : ∀ {ℓa ℓb} (ℂ : Category ℓa ℓb) → Set _
|
||||||
EndoFunctor ℂ = Functor ℂ ℂ
|
EndoFunctor ℂ = Functor ℂ ℂ
|
||||||
|
|
||||||
module _
|
module _
|
||||||
{ℓa ℓb : Level}
|
{ℓc ℓc' ℓd ℓd' : Level}
|
||||||
{ℂ 𝔻 : Category ℓa ℓb}
|
{ℂ : Category ℓc ℓc'} {𝔻 : Category ℓd ℓd'}
|
||||||
(F : RawFunctor ℂ 𝔻)
|
(F : RawFunctor ℂ 𝔻)
|
||||||
where
|
where
|
||||||
private
|
private
|
||||||
|
|
@ -96,8 +96,7 @@ module _
|
||||||
|
|
||||||
-- Alternate version of above where `F` is indexed by an interval
|
-- Alternate version of above where `F` is indexed by an interval
|
||||||
module _
|
module _
|
||||||
{ℓa ℓb : Level}
|
{ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Category ℓd ℓd'}
|
||||||
{ℂ 𝔻 : Category ℓa ℓb}
|
|
||||||
{F : I → RawFunctor ℂ 𝔻}
|
{F : I → RawFunctor ℂ 𝔻}
|
||||||
where
|
where
|
||||||
private
|
private
|
||||||
|
|
@ -108,15 +107,14 @@ module _
|
||||||
IsFunctorIsProp' : IsProp' λ i → IsFunctor _ _ (F i)
|
IsFunctorIsProp' : IsProp' λ i → IsFunctor _ _ (F i)
|
||||||
IsFunctorIsProp' isF0 isF1 = lemPropF {B = IsFunctor ℂ 𝔻}
|
IsFunctorIsProp' isF0 isF1 = lemPropF {B = IsFunctor ℂ 𝔻}
|
||||||
(\ F → propIsFunctor F) (\ i → F i)
|
(\ F → propIsFunctor F) (\ i → F i)
|
||||||
where
|
|
||||||
open import Cubical.NType.Properties using (lemPropF)
|
|
||||||
|
|
||||||
module _ {ℓ ℓ' : Level} {ℂ 𝔻 : Category ℓ ℓ'} where
|
module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Category ℓd ℓd'} where
|
||||||
|
open Functor
|
||||||
Functor≡ : {F G : Functor ℂ 𝔻}
|
Functor≡ : {F G : Functor ℂ 𝔻}
|
||||||
→ raw F ≡ raw G
|
→ Functor.raw F ≡ Functor.raw G
|
||||||
→ F ≡ G
|
→ F ≡ G
|
||||||
raw (Functor≡ eq i) = eq i
|
Functor.raw (Functor≡ eq i) = eq i
|
||||||
isFunctor (Functor≡ {F} {G} eq i)
|
Functor.isFunctor (Functor≡ {F} {G} eq i)
|
||||||
= res i
|
= res i
|
||||||
where
|
where
|
||||||
res : (λ i → IsFunctor ℂ 𝔻 (eq i)) [ isFunctor F ≡ isFunctor G ]
|
res : (λ i → IsFunctor ℂ 𝔻 (eq i)) [ isFunctor F ≡ isFunctor G ]
|
||||||
|
|
@ -124,35 +122,37 @@ module _ {ℓ ℓ' : Level} {ℂ 𝔻 : Category ℓ ℓ'} where
|
||||||
|
|
||||||
module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : Functor A B) where
|
module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : Functor A B) where
|
||||||
private
|
private
|
||||||
F* = omap F
|
module F = Functor F
|
||||||
F→ = fmap F
|
module G = Functor G
|
||||||
G* = omap G
|
|
||||||
G→ = fmap G
|
|
||||||
module _ {a0 a1 a2 : Object A} {α0 : A [ a0 , a1 ]} {α1 : A [ a1 , a2 ]} where
|
module _ {a0 a1 a2 : Object A} {α0 : A [ a0 , a1 ]} {α1 : A [ a1 , a2 ]} where
|
||||||
|
dist : (F.fmap ∘ G.fmap) (A [ α1 ∘ α0 ]) ≡ C [ (F.fmap ∘ G.fmap) α1 ∘ (F.fmap ∘ G.fmap) α0 ]
|
||||||
dist : (F→ ∘ G→) (A [ α1 ∘ α0 ]) ≡ C [ (F→ ∘ G→) α1 ∘ (F→ ∘ G→) α0 ]
|
|
||||||
dist = begin
|
dist = begin
|
||||||
(F→ ∘ G→) (A [ α1 ∘ α0 ]) ≡⟨ refl ⟩
|
(F.fmap ∘ G.fmap) (A [ α1 ∘ α0 ])
|
||||||
F→ (G→ (A [ α1 ∘ α0 ])) ≡⟨ cong F→ (isDistributive G) ⟩
|
≡⟨ refl ⟩
|
||||||
F→ (B [ G→ α1 ∘ G→ α0 ]) ≡⟨ isDistributive F ⟩
|
F.fmap (G.fmap (A [ α1 ∘ α0 ]))
|
||||||
C [ (F→ ∘ G→) α1 ∘ (F→ ∘ G→) α0 ] ∎
|
≡⟨ cong F.fmap G.isDistributive ⟩
|
||||||
|
F.fmap (B [ G.fmap α1 ∘ G.fmap α0 ])
|
||||||
|
≡⟨ F.isDistributive ⟩
|
||||||
|
C [ (F.fmap ∘ G.fmap) α1 ∘ (F.fmap ∘ G.fmap) α0 ]
|
||||||
|
∎
|
||||||
|
|
||||||
_∘fr_ : RawFunctor A C
|
raw : RawFunctor A C
|
||||||
RawFunctor.omap _∘fr_ = F* ∘ G*
|
RawFunctor.omap raw = F.omap ∘ G.omap
|
||||||
RawFunctor.fmap _∘fr_ = F→ ∘ G→
|
RawFunctor.fmap raw = F.fmap ∘ G.fmap
|
||||||
instance
|
|
||||||
isFunctor' : IsFunctor A C _∘fr_
|
isFunctor : IsFunctor A C raw
|
||||||
isFunctor' = record
|
isFunctor = record
|
||||||
{ isIdentity = begin
|
{ isIdentity = begin
|
||||||
(F→ ∘ G→) (𝟙 A) ≡⟨ refl ⟩
|
(F.fmap ∘ G.fmap) (𝟙 A) ≡⟨ refl ⟩
|
||||||
F→ (G→ (𝟙 A)) ≡⟨ cong F→ (isIdentity G)⟩
|
F.fmap (G.fmap (𝟙 A)) ≡⟨ cong F.fmap (G.isIdentity)⟩
|
||||||
F→ (𝟙 B) ≡⟨ isIdentity F ⟩
|
F.fmap (𝟙 B) ≡⟨ F.isIdentity ⟩
|
||||||
𝟙 C ∎
|
𝟙 C ∎
|
||||||
; isDistributive = dist
|
; isDistributive = dist
|
||||||
}
|
}
|
||||||
|
|
||||||
F[_∘_] : Functor A C
|
F[_∘_] : Functor A C
|
||||||
raw F[_∘_] = _∘fr_
|
Functor.raw F[_∘_] = raw
|
||||||
|
Functor.isFunctor F[_∘_] = isFunctor
|
||||||
|
|
||||||
-- The identity functor
|
-- The identity functor
|
||||||
identity : ∀ {ℓ ℓ'} → {C : Category ℓ ℓ'} → Functor C C
|
identity : ∀ {ℓ ℓ'} → {C : Category ℓ ℓ'} → Functor C C
|
||||||
|
|
|
||||||
|
|
@ -20,23 +20,19 @@ The monoidal representation is exposed by default from this module.
|
||||||
{-# OPTIONS --cubical --allow-unsolved-metas #-}
|
{-# OPTIONS --cubical --allow-unsolved-metas #-}
|
||||||
module Cat.Category.Monad where
|
module Cat.Category.Monad where
|
||||||
|
|
||||||
open import Agda.Primitive
|
open import Cat.Prelude
|
||||||
|
|
||||||
open import Data.Product
|
|
||||||
|
|
||||||
open import Cubical
|
|
||||||
open import Cubical.NType.Properties using (lemPropF ; lemSig ; lemSigP)
|
|
||||||
open import Cubical.GradLemma using (gradLemma)
|
|
||||||
|
|
||||||
open import Cat.Category
|
open import Cat.Category
|
||||||
open import Cat.Category.Functor as F
|
open import Cat.Category.Functor as F
|
||||||
open import Cat.Category.NaturalTransformation
|
open import Cat.Category.NaturalTransformation
|
||||||
open import Cat.Category.Monad.Monoidal as Monoidal public
|
import Cat.Category.Monad.Monoidal
|
||||||
open import Cat.Category.Monad.Kleisli as Kleisli
|
import Cat.Category.Monad.Kleisli
|
||||||
open import Cat.Categories.Fun
|
open import Cat.Categories.Fun
|
||||||
|
|
||||||
|
module Monoidal = Cat.Category.Monad.Monoidal
|
||||||
|
module Kleisli = Cat.Category.Monad.Kleisli
|
||||||
|
|
||||||
-- | The monoidal- and kleisli presentation of monads are equivalent.
|
-- | The monoidal- and kleisli presentation of monads are equivalent.
|
||||||
module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
|
module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||||
private
|
private
|
||||||
module ℂ = Category ℂ
|
module ℂ = Category ℂ
|
||||||
open ℂ using (Object ; Arrow ; 𝟙 ; _∘_ ; _>>>_)
|
open ℂ using (Object ; Arrow ; 𝟙 ; _∘_ ; _>>>_)
|
||||||
|
|
@ -121,7 +117,7 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
|
||||||
bind (f >>> (pure >>> bind 𝟙))
|
bind (f >>> (pure >>> bind 𝟙))
|
||||||
≡⟨ cong (λ φ → bind (f >>> φ)) (isNatural _) ⟩
|
≡⟨ cong (λ φ → bind (f >>> φ)) (isNatural _) ⟩
|
||||||
bind (f >>> 𝟙)
|
bind (f >>> 𝟙)
|
||||||
≡⟨ cong bind (proj₂ ℂ.isIdentity) ⟩
|
≡⟨ cong bind ℂ.leftIdentity ⟩
|
||||||
bind f ∎
|
bind f ∎
|
||||||
|
|
||||||
forthRawEq : forthRaw (backRaw m) ≡ K.Monad.raw m
|
forthRawEq : forthRaw (backRaw m) ≡ K.Monad.raw m
|
||||||
|
|
@ -152,7 +148,7 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
|
||||||
KM.bind 𝟙 ≡⟨⟩
|
KM.bind 𝟙 ≡⟨⟩
|
||||||
bind 𝟙 ≡⟨⟩
|
bind 𝟙 ≡⟨⟩
|
||||||
joinT X ∘ Rfmap 𝟙 ≡⟨ cong (λ φ → _ ∘ φ) R.isIdentity ⟩
|
joinT X ∘ Rfmap 𝟙 ≡⟨ cong (λ φ → _ ∘ φ) R.isIdentity ⟩
|
||||||
joinT X ∘ 𝟙 ≡⟨ proj₁ ℂ.isIdentity ⟩
|
joinT X ∘ 𝟙 ≡⟨ ℂ.rightIdentity ⟩
|
||||||
joinT X ∎
|
joinT X ∎
|
||||||
|
|
||||||
fmapEq : ∀ {A B} → KM.fmap {A} {B} ≡ Rfmap
|
fmapEq : ∀ {A B} → KM.fmap {A} {B} ≡ Rfmap
|
||||||
|
|
@ -164,7 +160,7 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
|
||||||
Rfmap (f >>> pureT B) >>> joinT B ≡⟨ cong (λ φ → φ >>> joinT B) R.isDistributive ⟩
|
Rfmap (f >>> pureT B) >>> joinT B ≡⟨ cong (λ φ → φ >>> joinT B) R.isDistributive ⟩
|
||||||
Rfmap f >>> Rfmap (pureT B) >>> joinT B ≡⟨ ℂ.isAssociative ⟩
|
Rfmap f >>> Rfmap (pureT B) >>> joinT B ≡⟨ ℂ.isAssociative ⟩
|
||||||
joinT B ∘ Rfmap (pureT B) ∘ Rfmap f ≡⟨ cong (λ φ → φ ∘ Rfmap f) (proj₂ isInverse) ⟩
|
joinT B ∘ Rfmap (pureT B) ∘ Rfmap f ≡⟨ cong (λ φ → φ ∘ Rfmap f) (proj₂ isInverse) ⟩
|
||||||
𝟙 ∘ Rfmap f ≡⟨ proj₂ ℂ.isIdentity ⟩
|
𝟙 ∘ Rfmap f ≡⟨ ℂ.leftIdentity ⟩
|
||||||
Rfmap f ∎
|
Rfmap f ∎
|
||||||
)
|
)
|
||||||
|
|
||||||
|
|
@ -189,7 +185,7 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
|
||||||
M.RawMonad.joinT (backRaw (forth m)) X ≡⟨⟩
|
M.RawMonad.joinT (backRaw (forth m)) X ≡⟨⟩
|
||||||
KM.join ≡⟨⟩
|
KM.join ≡⟨⟩
|
||||||
joinT X ∘ Rfmap 𝟙 ≡⟨ cong (λ φ → joinT X ∘ φ) R.isIdentity ⟩
|
joinT X ∘ Rfmap 𝟙 ≡⟨ cong (λ φ → joinT X ∘ φ) R.isIdentity ⟩
|
||||||
joinT X ∘ 𝟙 ≡⟨ proj₁ ℂ.isIdentity ⟩
|
joinT X ∘ 𝟙 ≡⟨ ℂ.rightIdentity ⟩
|
||||||
joinT X ∎)
|
joinT X ∎)
|
||||||
|
|
||||||
joinNTEq : (λ i → NaturalTransformation F[ Req i ∘ Req i ] (Req i))
|
joinNTEq : (λ i → NaturalTransformation F[ Req i ∘ Req i ] (Req i))
|
||||||
|
|
@ -207,5 +203,10 @@ module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} where
|
||||||
eqv : isEquiv M.Monad K.Monad forth
|
eqv : isEquiv M.Monad K.Monad forth
|
||||||
eqv = gradLemma forth back fortheq backeq
|
eqv = gradLemma forth back fortheq backeq
|
||||||
|
|
||||||
|
open import Cat.Equivalence
|
||||||
|
|
||||||
|
Monoidal≅Kleisli : M.Monad ≅ K.Monad
|
||||||
|
Monoidal≅Kleisli = forth , (back , (record { verso-recto = funExt backeq ; recto-verso = funExt fortheq }))
|
||||||
|
|
||||||
Monoidal≃Kleisli : M.Monad ≃ K.Monad
|
Monoidal≃Kleisli : M.Monad ≃ K.Monad
|
||||||
Monoidal≃Kleisli = forth , eqv
|
Monoidal≃Kleisli = forth , eqv
|
||||||
|
|
|
||||||
|
|
@ -4,11 +4,7 @@ The Kleisli formulation of monads
|
||||||
{-# OPTIONS --cubical --allow-unsolved-metas #-}
|
{-# OPTIONS --cubical --allow-unsolved-metas #-}
|
||||||
open import Agda.Primitive
|
open import Agda.Primitive
|
||||||
|
|
||||||
open import Data.Product
|
open import Cat.Prelude
|
||||||
|
|
||||||
open import Cubical
|
|
||||||
open import Cubical.NType.Properties using (lemPropF ; lemSig ; lemSigP)
|
|
||||||
open import Cubical.GradLemma using (gradLemma)
|
|
||||||
|
|
||||||
open import Cat.Category
|
open import Cat.Category
|
||||||
open import Cat.Category.Functor as F
|
open import Cat.Category.Functor as F
|
||||||
|
|
@ -104,7 +100,7 @@ record IsMonad (raw : RawMonad) : Set ℓ where
|
||||||
|
|
||||||
isFunctorR : IsFunctor ℂ ℂ rawR
|
isFunctorR : IsFunctor ℂ ℂ rawR
|
||||||
IsFunctor.isIdentity isFunctorR = begin
|
IsFunctor.isIdentity isFunctorR = begin
|
||||||
bind (pure ∘ 𝟙) ≡⟨ cong bind (proj₁ ℂ.isIdentity) ⟩
|
bind (pure ∘ 𝟙) ≡⟨ cong bind (ℂ.rightIdentity) ⟩
|
||||||
bind pure ≡⟨ isIdentity ⟩
|
bind pure ≡⟨ isIdentity ⟩
|
||||||
𝟙 ∎
|
𝟙 ∎
|
||||||
|
|
||||||
|
|
@ -156,9 +152,9 @@ record IsMonad (raw : RawMonad) : Set ℓ where
|
||||||
bind (bind (f >>> pure) >>> (pure >>> bind 𝟙))
|
bind (bind (f >>> pure) >>> (pure >>> bind 𝟙))
|
||||||
≡⟨ cong (λ φ → bind (bind (f >>> pure) >>> φ)) (isNatural _) ⟩
|
≡⟨ cong (λ φ → bind (bind (f >>> pure) >>> φ)) (isNatural _) ⟩
|
||||||
bind (bind (f >>> pure) >>> 𝟙)
|
bind (bind (f >>> pure) >>> 𝟙)
|
||||||
≡⟨ cong bind (proj₂ ℂ.isIdentity) ⟩
|
≡⟨ cong bind ℂ.leftIdentity ⟩
|
||||||
bind (bind (f >>> pure))
|
bind (bind (f >>> pure))
|
||||||
≡⟨ cong bind (sym (proj₁ ℂ.isIdentity)) ⟩
|
≡⟨ cong bind (sym ℂ.rightIdentity) ⟩
|
||||||
bind (𝟙 >>> bind (f >>> pure)) ≡⟨⟩
|
bind (𝟙 >>> bind (f >>> pure)) ≡⟨⟩
|
||||||
bind (𝟙 >=> (f >>> pure))
|
bind (𝟙 >=> (f >>> pure))
|
||||||
≡⟨ sym (isDistributive _ _) ⟩
|
≡⟨ sym (isDistributive _ _) ⟩
|
||||||
|
|
@ -186,10 +182,10 @@ record IsMonad (raw : RawMonad) : Set ℓ where
|
||||||
bind (join >>> (pure >>> bind 𝟙))
|
bind (join >>> (pure >>> bind 𝟙))
|
||||||
≡⟨ cong (λ φ → bind (join >>> φ)) (isNatural _) ⟩
|
≡⟨ cong (λ φ → bind (join >>> φ)) (isNatural _) ⟩
|
||||||
bind (join >>> 𝟙)
|
bind (join >>> 𝟙)
|
||||||
≡⟨ cong bind (proj₂ ℂ.isIdentity) ⟩
|
≡⟨ cong bind ℂ.leftIdentity ⟩
|
||||||
bind join ≡⟨⟩
|
bind join ≡⟨⟩
|
||||||
bind (bind 𝟙)
|
bind (bind 𝟙)
|
||||||
≡⟨ cong bind (sym (proj₁ ℂ.isIdentity)) ⟩
|
≡⟨ cong bind (sym ℂ.rightIdentity) ⟩
|
||||||
bind (𝟙 >>> bind 𝟙) ≡⟨⟩
|
bind (𝟙 >>> bind 𝟙) ≡⟨⟩
|
||||||
bind (𝟙 >=> 𝟙) ≡⟨ sym (isDistributive _ _) ⟩
|
bind (𝟙 >=> 𝟙) ≡⟨ sym (isDistributive _ _) ⟩
|
||||||
bind 𝟙 >>> bind 𝟙 ≡⟨⟩
|
bind 𝟙 >>> bind 𝟙 ≡⟨⟩
|
||||||
|
|
@ -212,7 +208,7 @@ record IsMonad (raw : RawMonad) : Set ℓ where
|
||||||
bind (pure >>> (pure >>> bind 𝟙))
|
bind (pure >>> (pure >>> bind 𝟙))
|
||||||
≡⟨ cong (λ φ → bind (pure >>> φ)) (isNatural _) ⟩
|
≡⟨ cong (λ φ → bind (pure >>> φ)) (isNatural _) ⟩
|
||||||
bind (pure >>> 𝟙)
|
bind (pure >>> 𝟙)
|
||||||
≡⟨ cong bind (proj₂ ℂ.isIdentity) ⟩
|
≡⟨ cong bind ℂ.leftIdentity ⟩
|
||||||
bind pure ≡⟨ isIdentity ⟩
|
bind pure ≡⟨ isIdentity ⟩
|
||||||
𝟙 ∎
|
𝟙 ∎
|
||||||
|
|
||||||
|
|
|
||||||
|
|
@ -4,11 +4,7 @@ Monoidal formulation of monads
|
||||||
{-# OPTIONS --cubical --allow-unsolved-metas #-}
|
{-# OPTIONS --cubical --allow-unsolved-metas #-}
|
||||||
open import Agda.Primitive
|
open import Agda.Primitive
|
||||||
|
|
||||||
open import Data.Product
|
open import Cat.Prelude
|
||||||
|
|
||||||
open import Cubical
|
|
||||||
open import Cubical.NType.Properties using (lemPropF ; lemSig ; lemSigP)
|
|
||||||
open import Cubical.GradLemma using (gradLemma)
|
|
||||||
|
|
||||||
open import Cat.Category
|
open import Cat.Category
|
||||||
open import Cat.Category.Functor as F
|
open import Cat.Category.Functor as F
|
||||||
|
|
@ -75,7 +71,7 @@ record IsMonad (raw : RawMonad) : Set ℓ where
|
||||||
joinT Y ∘ (R.fmap f ∘ pureT X) ≡⟨ cong (λ φ → joinT Y ∘ φ) (sym (pureN f)) ⟩
|
joinT Y ∘ (R.fmap f ∘ pureT X) ≡⟨ cong (λ φ → joinT Y ∘ φ) (sym (pureN f)) ⟩
|
||||||
joinT Y ∘ (pureT (R.omap Y) ∘ f) ≡⟨ ℂ.isAssociative ⟩
|
joinT Y ∘ (pureT (R.omap Y) ∘ f) ≡⟨ ℂ.isAssociative ⟩
|
||||||
joinT Y ∘ pureT (R.omap Y) ∘ f ≡⟨ cong (λ φ → φ ∘ f) (proj₁ isInverse) ⟩
|
joinT Y ∘ pureT (R.omap Y) ∘ f ≡⟨ cong (λ φ → φ ∘ f) (proj₁ isInverse) ⟩
|
||||||
𝟙 ∘ f ≡⟨ proj₂ ℂ.isIdentity ⟩
|
𝟙 ∘ f ≡⟨ ℂ.leftIdentity ⟩
|
||||||
f ∎
|
f ∎
|
||||||
|
|
||||||
isDistributive : IsDistributive
|
isDistributive : IsDistributive
|
||||||
|
|
|
||||||
|
|
@ -4,22 +4,15 @@ This module provides construction 2.3 in [voe]
|
||||||
{-# OPTIONS --cubical --allow-unsolved-metas --caching #-}
|
{-# OPTIONS --cubical --allow-unsolved-metas --caching #-}
|
||||||
module Cat.Category.Monad.Voevodsky where
|
module Cat.Category.Monad.Voevodsky where
|
||||||
|
|
||||||
open import Agda.Primitive
|
open import Cat.Prelude
|
||||||
|
open import Function
|
||||||
open import Data.Product
|
|
||||||
|
|
||||||
open import Cubical
|
|
||||||
open import Cubical.NType.Properties using (lemPropF ; lemSig ; lemSigP)
|
|
||||||
open import Cubical.GradLemma using (gradLemma)
|
|
||||||
|
|
||||||
open import Cat.Category
|
open import Cat.Category
|
||||||
open import Cat.Category.Functor as F
|
open import Cat.Category.Functor as F
|
||||||
open import Cat.Category.NaturalTransformation
|
open import Cat.Category.NaturalTransformation
|
||||||
open import Cat.Category.Monad using (Monoidal≃Kleisli)
|
open import Cat.Category.Monad
|
||||||
import Cat.Category.Monad.Monoidal as Monoidal
|
|
||||||
import Cat.Category.Monad.Kleisli as Kleisli
|
|
||||||
open import Cat.Categories.Fun
|
open import Cat.Categories.Fun
|
||||||
open import Function using (_∘′_)
|
open import Cat.Equivalence
|
||||||
|
|
||||||
module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||||
private
|
private
|
||||||
|
|
@ -27,11 +20,10 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||||
module ℂ = Category ℂ
|
module ℂ = Category ℂ
|
||||||
open ℂ using (Object ; Arrow)
|
open ℂ using (Object ; Arrow)
|
||||||
open NaturalTransformation ℂ ℂ
|
open NaturalTransformation ℂ ℂ
|
||||||
open import Function using (_∘_ ; _$_)
|
|
||||||
module M = Monoidal ℂ
|
module M = Monoidal ℂ
|
||||||
module K = Kleisli ℂ
|
module K = Kleisli ℂ
|
||||||
|
|
||||||
module §2-3 (omap : Omap ℂ ℂ) (pure : {X : Object} → Arrow X (omap X)) where
|
module §2-3 (omap : Object → Object) (pure : {X : Object} → Arrow X (omap X)) where
|
||||||
record §1 : Set ℓ where
|
record §1 : Set ℓ where
|
||||||
open M
|
open M
|
||||||
|
|
||||||
|
|
@ -134,13 +126,23 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||||
; isMnd = K.Monad.isMonad m
|
; isMnd = K.Monad.isMonad m
|
||||||
}
|
}
|
||||||
|
|
||||||
|
-- | In the following we seek to transform the equivalence `Monoidal≃Kleisli`
|
||||||
|
-- | to talk about voevodsky's construction.
|
||||||
module _ (omap : Omap ℂ ℂ) (pure : {X : Object} → Arrow X (omap X)) where
|
module _ (omap : Omap ℂ ℂ) (pure : {X : Object} → Arrow X (omap X)) where
|
||||||
private
|
private
|
||||||
|
module E = AreInverses (Monoidal≅Kleisli ℂ .proj₂ .proj₂)
|
||||||
|
|
||||||
Monoidal→Kleisli : M.Monad → K.Monad
|
Monoidal→Kleisli : M.Monad → K.Monad
|
||||||
Monoidal→Kleisli = proj₁ Monoidal≃Kleisli
|
Monoidal→Kleisli = E.obverse
|
||||||
|
|
||||||
Kleisli→Monoidal : K.Monad → M.Monad
|
Kleisli→Monoidal : K.Monad → M.Monad
|
||||||
Kleisli→Monoidal = inverse Monoidal≃Kleisli
|
Kleisli→Monoidal = E.reverse
|
||||||
|
|
||||||
|
ve-re : Kleisli→Monoidal ∘ Monoidal→Kleisli ≡ Function.id
|
||||||
|
ve-re = E.verso-recto
|
||||||
|
|
||||||
|
re-ve : Monoidal→Kleisli ∘ Kleisli→Monoidal ≡ Function.id
|
||||||
|
re-ve = E.recto-verso
|
||||||
|
|
||||||
forth : §2-3.§1 omap pure → §2-3.§2 omap pure
|
forth : §2-3.§1 omap pure → §2-3.§2 omap pure
|
||||||
forth = §2-fromMonad ∘ Monoidal→Kleisli ∘ §2-3.§1.toMonad
|
forth = §2-fromMonad ∘ Monoidal→Kleisli ∘ §2-3.§1.toMonad
|
||||||
|
|
@ -163,7 +165,7 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||||
∘ Monoidal→Kleisli
|
∘ Monoidal→Kleisli
|
||||||
∘ Kleisli→Monoidal
|
∘ Kleisli→Monoidal
|
||||||
∘ §2-3.§2.toMonad
|
∘ §2-3.§2.toMonad
|
||||||
) m ≡⟨ u ⟩
|
) m ≡⟨ cong (λ φ → φ m) t ⟩
|
||||||
-- Monoidal→Kleisli and Kleisli→Monoidal are inverses
|
-- Monoidal→Kleisli and Kleisli→Monoidal are inverses
|
||||||
-- I should be able to prove this using congruence and `lem` below.
|
-- I should be able to prove this using congruence and `lem` below.
|
||||||
( §2-fromMonad
|
( §2-fromMonad
|
||||||
|
|
@ -174,23 +176,18 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||||
) m ≡⟨⟩ -- fromMonad and toMonad are inverses
|
) m ≡⟨⟩ -- fromMonad and toMonad are inverses
|
||||||
m ∎
|
m ∎
|
||||||
where
|
where
|
||||||
lem : Monoidal→Kleisli ∘ Kleisli→Monoidal ≡ Function.id
|
|
||||||
lem = {!!} -- verso-recto Monoidal≃Kleisli
|
|
||||||
t' : ((Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad {omap} {pure})
|
t' : ((Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad {omap} {pure})
|
||||||
≡ §2-3.§2.toMonad
|
≡ §2-3.§2.toMonad
|
||||||
t' = cong (\ φ → φ ∘ §2-3.§2.toMonad) lem
|
cong-d : ∀ {ℓ} {A : Set ℓ} {ℓ'} {B : A → Set ℓ'} {x y : A}
|
||||||
cong-d : ∀ {ℓ} {A : Set ℓ} {ℓ'} {B : A → Set ℓ'} {x y : A} → (f : (x : A) → B x) → (eq : x ≡ y) → PathP (\ i → B (eq i)) (f x) (f y)
|
→ (f : (x : A) → B x) → (eq : x ≡ y) → PathP (\ i → B (eq i)) (f x) (f y)
|
||||||
cong-d f p = λ i → f (p i)
|
cong-d f p = λ i → f (p i)
|
||||||
|
t' = cong (\ φ → φ ∘ §2-3.§2.toMonad) re-ve
|
||||||
t : (§2-fromMonad ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad {omap} {pure})
|
t : (§2-fromMonad ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad {omap} {pure})
|
||||||
≡ (§2-fromMonad ∘ §2-3.§2.toMonad)
|
≡ (§2-fromMonad ∘ §2-3.§2.toMonad)
|
||||||
t = cong-d (\ f → §2-fromMonad ∘ f) t'
|
t = cong-d (\ f → §2-fromMonad ∘ f) t'
|
||||||
u : (§2-fromMonad ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad) m
|
u : (§2-fromMonad ∘ (Monoidal→Kleisli ∘ Kleisli→Monoidal) ∘ §2-3.§2.toMonad) m
|
||||||
≡ (§2-fromMonad ∘ §2-3.§2.toMonad) m
|
≡ (§2-fromMonad ∘ §2-3.§2.toMonad) m
|
||||||
u = cong (\ f → f m) t
|
u = cong (\ φ → φ m) t
|
||||||
{-
|
|
||||||
(K.RawMonad.omap (K.Monad.raw (?0 ℂ omap pure m i (§2-3.§2.toMonad m))) x) = (omap x) : Object
|
|
||||||
(K.RawMonad.pure (K.Monad.raw (?0 ℂ omap pure m x (§2-3.§2.toMonad x)))) = pure : Arrow X (_350 ℂ omap pure m x x X)
|
|
||||||
-}
|
|
||||||
|
|
||||||
backEq : ∀ m → (back ∘ forth) m ≡ m
|
backEq : ∀ m → (back ∘ forth) m ≡ m
|
||||||
backEq m = begin
|
backEq m = begin
|
||||||
|
|
@ -212,7 +209,10 @@ module voe {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||||
) m ≡⟨⟩ -- fromMonad and toMonad are inverses
|
) m ≡⟨⟩ -- fromMonad and toMonad are inverses
|
||||||
m ∎
|
m ∎
|
||||||
where
|
where
|
||||||
t = {!!} -- cong (λ φ → voe-2-3-1-fromMonad ∘ φ ∘ voe-2-3.voe-2-3-1.toMonad) (recto-verso Monoidal≃Kleisli)
|
t : §1-fromMonad ∘ Kleisli→Monoidal ∘ Monoidal→Kleisli ∘ §2-3.§1.toMonad
|
||||||
|
≡ §1-fromMonad ∘ §2-3.§1.toMonad
|
||||||
|
-- Why does `re-ve` not satisfy this goal?
|
||||||
|
t i m = §1-fromMonad (ve-re i (§2-3.§1.toMonad m))
|
||||||
|
|
||||||
voe-isEquiv : isEquiv (§2-3.§1 omap pure) (§2-3.§2 omap pure) forth
|
voe-isEquiv : isEquiv (§2-3.§1 omap pure) (§2-3.§2 omap pure) forth
|
||||||
voe-isEquiv = gradLemma forth back forthEq backEq
|
voe-isEquiv = gradLemma forth back forthEq backEq
|
||||||
|
|
|
||||||
|
|
@ -19,7 +19,7 @@ module _ (ℓa ℓb : Level) where
|
||||||
--
|
--
|
||||||
-- Since it doesn't we'll make the following (definitionally equivalent) ad-hoc definition.
|
-- Since it doesn't we'll make the following (definitionally equivalent) ad-hoc definition.
|
||||||
_×_ : ∀ {ℓa ℓb} → Category ℓa ℓb → Category ℓa ℓb → Category ℓa ℓb
|
_×_ : ∀ {ℓa ℓb} → Category ℓa ℓb → Category ℓa ℓb → Category ℓa ℓb
|
||||||
ℂ × 𝔻 = Cat.CatProduct.obj ℂ 𝔻
|
ℂ × 𝔻 = Cat.CatProduct.object ℂ 𝔻
|
||||||
|
|
||||||
record RawMonoidalCategory : Set ℓ where
|
record RawMonoidalCategory : Set ℓ where
|
||||||
field
|
field
|
||||||
|
|
|
||||||
|
|
@ -19,15 +19,12 @@
|
||||||
-- * A composition operator.
|
-- * A composition operator.
|
||||||
{-# OPTIONS --allow-unsolved-metas --cubical #-}
|
{-# OPTIONS --allow-unsolved-metas --cubical #-}
|
||||||
module Cat.Category.NaturalTransformation where
|
module Cat.Category.NaturalTransformation where
|
||||||
open import Agda.Primitive
|
|
||||||
open import Data.Product
|
open import Cat.Prelude
|
||||||
|
|
||||||
open import Data.Nat using (_≤_ ; z≤n ; s≤s)
|
open import Data.Nat using (_≤_ ; z≤n ; s≤s)
|
||||||
module Nat = Data.Nat
|
module Nat = Data.Nat
|
||||||
|
|
||||||
open import Cubical
|
|
||||||
open import Cubical.Sigma
|
|
||||||
open import Cubical.NType.Properties
|
|
||||||
|
|
||||||
open import Cat.Category
|
open import Cat.Category
|
||||||
open import Cat.Category.Functor hiding (identity)
|
open import Cat.Category.Functor hiding (identity)
|
||||||
open import Cat.Wishlist
|
open import Cat.Wishlist
|
||||||
|
|
@ -72,8 +69,8 @@ module NaturalTransformation {ℓc ℓc' ℓd ℓd' : Level}
|
||||||
identityNatural : (F : Functor ℂ 𝔻) → Natural F F (identityTrans F)
|
identityNatural : (F : Functor ℂ 𝔻) → Natural F F (identityTrans F)
|
||||||
identityNatural F {A = A} {B = B} f = begin
|
identityNatural F {A = A} {B = B} f = begin
|
||||||
𝔻 [ identityTrans F B ∘ F→ f ] ≡⟨⟩
|
𝔻 [ identityTrans F B ∘ F→ f ] ≡⟨⟩
|
||||||
𝔻 [ 𝟙 𝔻 ∘ F→ f ] ≡⟨ proj₂ 𝔻.isIdentity ⟩
|
𝔻 [ 𝟙 𝔻 ∘ F→ f ] ≡⟨ 𝔻.leftIdentity ⟩
|
||||||
F→ f ≡⟨ sym (proj₁ 𝔻.isIdentity) ⟩
|
F→ f ≡⟨ sym 𝔻.rightIdentity ⟩
|
||||||
𝔻 [ F→ f ∘ 𝟙 𝔻 ] ≡⟨⟩
|
𝔻 [ F→ f ∘ 𝟙 𝔻 ] ≡⟨⟩
|
||||||
𝔻 [ F→ f ∘ identityTrans F A ] ∎
|
𝔻 [ F→ f ∘ identityTrans F A ] ∎
|
||||||
where
|
where
|
||||||
|
|
@ -113,8 +110,11 @@ module NaturalTransformation {ℓc ℓc' ℓd ℓd' : Level}
|
||||||
lem : (λ _ → Natural F G θ) [ (λ f → θNat f) ≡ (λ f → θNat' f) ]
|
lem : (λ _ → Natural F G θ) [ (λ f → θNat f) ≡ (λ f → θNat' f) ]
|
||||||
lem = λ i f → 𝔻.arrowsAreSets _ _ (θNat f) (θNat' f) i
|
lem = λ i f → 𝔻.arrowsAreSets _ _ (θNat f) (θNat' f) i
|
||||||
|
|
||||||
naturalTransformationIsSet : isSet (NaturalTransformation F G)
|
naturalIsSet : (θ : Transformation F G) → isSet (Natural F G θ)
|
||||||
naturalTransformationIsSet = sigPresSet transformationIsSet
|
naturalIsSet θ =
|
||||||
λ θ → ntypeCommulative
|
ntypeCommulative
|
||||||
(s≤s {n = Nat.suc Nat.zero} z≤n)
|
(s≤s {n = Nat.suc Nat.zero} z≤n)
|
||||||
(naturalIsProp θ)
|
(naturalIsProp θ)
|
||||||
|
|
||||||
|
naturalTransformationIsSet : isSet (NaturalTransformation F G)
|
||||||
|
naturalTransformationIsSet = sigPresSet transformationIsSet naturalIsSet
|
||||||
|
|
|
||||||
|
|
@ -1,11 +1,10 @@
|
||||||
{-# OPTIONS --allow-unsolved-metas #-}
|
{-# OPTIONS --allow-unsolved-metas #-}
|
||||||
module Cat.Category.Product where
|
module Cat.Category.Product where
|
||||||
|
|
||||||
open import Agda.Primitive
|
open import Cat.Prelude hiding (_×_ ; proj₁ ; proj₂)
|
||||||
open import Cubical
|
import Data.Product as P
|
||||||
open import Data.Product as P hiding (_×_ ; proj₁ ; proj₂)
|
|
||||||
|
|
||||||
open import Cat.Category hiding (module Propositionality)
|
open import Cat.Category
|
||||||
|
|
||||||
module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||||
|
|
||||||
|
|
@ -24,13 +23,13 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||||
record IsProduct (raw : RawProduct) : Set (ℓa ⊔ ℓb) where
|
record IsProduct (raw : RawProduct) : Set (ℓa ⊔ ℓb) where
|
||||||
open RawProduct raw public
|
open RawProduct raw public
|
||||||
field
|
field
|
||||||
isProduct : ∀ {X : Object} (f : ℂ [ X , A ]) (g : ℂ [ X , B ])
|
ump : ∀ {X : Object} (f : ℂ [ X , A ]) (g : ℂ [ X , B ])
|
||||||
→ ∃![ f×g ] (ℂ [ proj₁ ∘ f×g ] ≡ f P.× ℂ [ proj₂ ∘ f×g ] ≡ g)
|
→ ∃![ f×g ] (ℂ [ proj₁ ∘ f×g ] ≡ f P.× ℂ [ proj₂ ∘ f×g ] ≡ g)
|
||||||
|
|
||||||
-- | Arrow product
|
-- | Arrow product
|
||||||
_P[_×_] : ∀ {X} → (π₁ : ℂ [ X , A ]) (π₂ : ℂ [ X , B ])
|
_P[_×_] : ∀ {X} → (π₁ : ℂ [ X , A ]) (π₂ : ℂ [ X , B ])
|
||||||
→ ℂ [ X , object ]
|
→ ℂ [ X , object ]
|
||||||
_P[_×_] π₁ π₂ = P.proj₁ (isProduct π₁ π₂)
|
_P[_×_] π₁ π₂ = P.proj₁ (ump π₁ π₂)
|
||||||
|
|
||||||
record Product : Set (ℓa ⊔ ℓb) where
|
record Product : Set (ℓa ⊔ ℓb) where
|
||||||
field
|
field
|
||||||
|
|
@ -59,10 +58,63 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||||
× ℂ [ g ∘ snd ]
|
× ℂ [ g ∘ snd ]
|
||||||
]
|
]
|
||||||
|
|
||||||
module Propositionality {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} {A B : Category.Object ℂ} where
|
module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} {A B : Category.Object ℂ} where
|
||||||
-- TODO I'm not sure this is actually provable. Check with Thierry.
|
private
|
||||||
propProduct : isProp (Product ℂ A B)
|
open Category ℂ
|
||||||
propProduct = {!!}
|
module _ (raw : RawProduct ℂ A B) where
|
||||||
|
module _ (x y : IsProduct ℂ A B raw) where
|
||||||
|
private
|
||||||
|
module x = IsProduct x
|
||||||
|
module y = IsProduct y
|
||||||
|
|
||||||
|
module _ {X : Object} (f : ℂ [ X , A ]) (g : ℂ [ X , B ]) where
|
||||||
|
prodAux : x.ump f g ≡ y.ump f g
|
||||||
|
prodAux = {!!}
|
||||||
|
|
||||||
|
propIsProduct' : x ≡ y
|
||||||
|
propIsProduct' i = record { ump = λ f g → prodAux f g i }
|
||||||
|
|
||||||
|
propIsProduct : isProp (IsProduct ℂ A B raw)
|
||||||
|
propIsProduct = propIsProduct'
|
||||||
|
|
||||||
|
Product≡ : {x y : Product ℂ A B} → (Product.raw x ≡ Product.raw y) → x ≡ y
|
||||||
|
Product≡ {x} {y} p i = record { raw = p i ; isProduct = q i }
|
||||||
|
where
|
||||||
|
q : (λ i → IsProduct ℂ A B (p i)) [ Product.isProduct x ≡ Product.isProduct y ]
|
||||||
|
q = lemPropF propIsProduct p
|
||||||
|
|
||||||
|
module _ {ℓa ℓb : Level} {ℂ : Category ℓa ℓb} {A B : Category.Object ℂ} where
|
||||||
|
open Category ℂ
|
||||||
|
private
|
||||||
|
module _ (x y : HasProducts ℂ) where
|
||||||
|
private
|
||||||
|
module x = HasProducts x
|
||||||
|
module y = HasProducts y
|
||||||
|
module _ (A B : Object) where
|
||||||
|
module pX = Product (x.product A B)
|
||||||
|
module pY = Product (y.product A B)
|
||||||
|
objEq : pX.object ≡ pY.object
|
||||||
|
objEq = {!!}
|
||||||
|
proj₁Eq : (λ i → ℂ [ objEq i , A ]) [ pX.proj₁ ≡ pY.proj₁ ]
|
||||||
|
proj₁Eq = {!!}
|
||||||
|
proj₂Eq : (λ i → ℂ [ objEq i , B ]) [ pX.proj₂ ≡ pY.proj₂ ]
|
||||||
|
proj₂Eq = {!!}
|
||||||
|
rawEq : pX.raw ≡ pY.raw
|
||||||
|
RawProduct.object (rawEq i) = objEq i
|
||||||
|
RawProduct.proj₁ (rawEq i) = {!!}
|
||||||
|
RawProduct.proj₂ (rawEq i) = {!!}
|
||||||
|
|
||||||
|
isEq : (λ i → IsProduct ℂ A B (rawEq i)) [ pX.isProduct ≡ pY.isProduct ]
|
||||||
|
isEq = {!!}
|
||||||
|
|
||||||
|
appEq : x.product A B ≡ y.product A B
|
||||||
|
appEq = Product≡ rawEq
|
||||||
|
|
||||||
|
productEq : x.product ≡ y.product
|
||||||
|
productEq i = λ A B → appEq A B i
|
||||||
|
|
||||||
|
propHasProducts' : x ≡ y
|
||||||
|
propHasProducts' i = record { product = productEq i }
|
||||||
|
|
||||||
propHasProducts : isProp (HasProducts ℂ)
|
propHasProducts : isProp (HasProducts ℂ)
|
||||||
propHasProducts = {!!}
|
propHasProducts = propHasProducts'
|
||||||
|
|
|
||||||
|
|
@ -2,61 +2,61 @@
|
||||||
|
|
||||||
module Cat.Category.Yoneda where
|
module Cat.Category.Yoneda where
|
||||||
|
|
||||||
open import Agda.Primitive
|
open import Cat.Prelude
|
||||||
open import Data.Product
|
|
||||||
open import Cubical
|
|
||||||
open import Cubical.NType.Properties
|
|
||||||
|
|
||||||
open import Cat.Category
|
open import Cat.Category
|
||||||
open import Cat.Category.Functor
|
open import Cat.Category.Functor
|
||||||
open import Cat.Equality
|
|
||||||
|
|
||||||
open import Cat.Categories.Fun
|
open import Cat.Categories.Fun
|
||||||
open import Cat.Categories.Sets
|
open import Cat.Categories.Sets hiding (presheaf)
|
||||||
open import Cat.Categories.Cat
|
|
||||||
|
|
||||||
module _ {ℓ : Level} {ℂ : Category ℓ ℓ} where
|
|
||||||
private
|
|
||||||
𝓢 = Sets ℓ
|
|
||||||
open Fun (opposite ℂ) 𝓢
|
|
||||||
prshf = presheaf ℂ
|
|
||||||
module ℂ = Category ℂ
|
|
||||||
|
|
||||||
-- There is no (small) category of categories. So we won't use _⇑_ from
|
-- There is no (small) category of categories. So we won't use _⇑_ from
|
||||||
-- `HasExponential`
|
-- `HasExponential`
|
||||||
--
|
--
|
||||||
-- open HasExponentials (Cat.hasExponentials ℓ unprovable) using (_⇑_)
|
-- open HasExponentials (Cat.hasExponentials ℓ unprovable) using (_⇑_)
|
||||||
--
|
--
|
||||||
-- In stead we'll use an ad-hoc definition -- which is definitionally
|
-- In stead we'll use an ad-hoc definition -- which is definitionally equivalent
|
||||||
-- equivalent to that other one.
|
-- to that other one - even without mentioning the category of categories.
|
||||||
_⇑_ = CatExponential.object
|
_⇑_ : {ℓ : Level} → Category ℓ ℓ → Category ℓ ℓ → Category ℓ ℓ
|
||||||
|
_⇑_ = Fun.Fun
|
||||||
|
|
||||||
|
module _ {ℓ : Level} {ℂ : Category ℓ ℓ} where
|
||||||
|
private
|
||||||
|
𝓢 = Sets ℓ
|
||||||
|
open Fun (opposite ℂ) 𝓢
|
||||||
|
|
||||||
|
module ℂ = Category ℂ
|
||||||
|
|
||||||
|
presheaf : ℂ.Object → Presheaf ℂ
|
||||||
|
presheaf = Cat.Categories.Sets.presheaf ℂ
|
||||||
|
|
||||||
module _ {A B : ℂ.Object} (f : ℂ [ A , B ]) where
|
module _ {A B : ℂ.Object} (f : ℂ [ A , B ]) where
|
||||||
fmap : Transformation (prshf A) (prshf B)
|
fmap : Transformation (presheaf A) (presheaf B)
|
||||||
fmap C x = ℂ [ f ∘ x ]
|
fmap C x = ℂ [ f ∘ x ]
|
||||||
|
|
||||||
fmapNatural : Natural (prshf A) (prshf B) fmap
|
fmapNatural : Natural (presheaf A) (presheaf B) fmap
|
||||||
fmapNatural g = funExt λ _ → ℂ.isAssociative
|
fmapNatural g = funExt λ _ → ℂ.isAssociative
|
||||||
|
|
||||||
fmapNT : NaturalTransformation (prshf A) (prshf B)
|
fmapNT : NaturalTransformation (presheaf A) (presheaf B)
|
||||||
fmapNT = fmap , fmapNatural
|
fmapNT = fmap , fmapNatural
|
||||||
|
|
||||||
rawYoneda : RawFunctor ℂ Fun
|
rawYoneda : RawFunctor ℂ Fun
|
||||||
RawFunctor.omap rawYoneda = prshf
|
RawFunctor.omap rawYoneda = presheaf
|
||||||
RawFunctor.fmap rawYoneda = fmapNT
|
RawFunctor.fmap rawYoneda = fmapNT
|
||||||
|
|
||||||
open RawFunctor rawYoneda hiding (fmap)
|
open RawFunctor rawYoneda hiding (fmap)
|
||||||
|
|
||||||
isIdentity : IsIdentity
|
isIdentity : IsIdentity
|
||||||
isIdentity {c} = lemSig (naturalIsProp {F = prshf c} {prshf c}) _ _ eq
|
isIdentity {c} = lemSig (naturalIsProp {F = presheaf c} {presheaf c}) _ _ eq
|
||||||
where
|
where
|
||||||
eq : (λ C x → ℂ [ ℂ.𝟙 ∘ x ]) ≡ identityTrans (prshf c)
|
eq : (λ C x → ℂ [ ℂ.𝟙 ∘ x ]) ≡ identityTrans (presheaf c)
|
||||||
eq = funExt λ A → funExt λ B → proj₂ ℂ.isIdentity
|
eq = funExt λ A → funExt λ B → ℂ.leftIdentity
|
||||||
|
|
||||||
isDistributive : IsDistributive
|
isDistributive : IsDistributive
|
||||||
isDistributive {A} {B} {C} {f = f} {g}
|
isDistributive {A} {B} {C} {f = f} {g}
|
||||||
= lemSig (propIsNatural (prshf A) (prshf C)) _ _ eq
|
= lemSig (propIsNatural (presheaf A) (presheaf C)) _ _ eq
|
||||||
where
|
where
|
||||||
T[_∘_]' = T[_∘_] {F = prshf A} {prshf B} {prshf C}
|
T[_∘_]' = T[_∘_] {F = presheaf A} {presheaf B} {presheaf C}
|
||||||
eqq : (X : ℂ.Object) → (x : ℂ [ X , A ])
|
eqq : (X : ℂ.Object) → (x : ℂ [ X , A ])
|
||||||
→ fmap (ℂ [ g ∘ f ]) X x ≡ T[ fmap g ∘ fmap f ]' X x
|
→ fmap (ℂ [ g ∘ f ]) X x ≡ T[ fmap g ∘ fmap f ]' X x
|
||||||
eqq X x = begin
|
eqq X x = begin
|
||||||
|
|
|
||||||
|
|
@ -1,22 +0,0 @@
|
||||||
{-# OPTIONS --cubical #-}
|
|
||||||
-- Defines equality-principles for data-types from the standard library.
|
|
||||||
|
|
||||||
module Cat.Equality where
|
|
||||||
|
|
||||||
open import Level
|
|
||||||
open import Cubical
|
|
||||||
|
|
||||||
-- _[_≡_] = PathP
|
|
||||||
|
|
||||||
module Equality where
|
|
||||||
module Data where
|
|
||||||
module Product where
|
|
||||||
open import Data.Product
|
|
||||||
|
|
||||||
module _ {ℓa ℓb : Level} {A : Set ℓa} {B : A → Set ℓb} {a b : Σ A B}
|
|
||||||
(proj₁≡ : (λ _ → A) [ proj₁ a ≡ proj₁ b ])
|
|
||||||
(proj₂≡ : (λ i → B (proj₁≡ i)) [ proj₂ a ≡ proj₂ b ]) where
|
|
||||||
|
|
||||||
Σ≡ : a ≡ b
|
|
||||||
proj₁ (Σ≡ i) = proj₁≡ i
|
|
||||||
proj₂ (Σ≡ i) = proj₂≡ i
|
|
||||||
290
src/Cat/Equivalence.agda
Normal file
290
src/Cat/Equivalence.agda
Normal file
|
|
@ -0,0 +1,290 @@
|
||||||
|
{-# OPTIONS --allow-unsolved-metas --cubical #-}
|
||||||
|
module Cat.Equivalence where
|
||||||
|
|
||||||
|
open import Cubical.Primitives
|
||||||
|
open import Cubical.FromStdLib renaming (ℓ-max to _⊔_)
|
||||||
|
open import Cubical.PathPrelude hiding (inverse ; _≃_)
|
||||||
|
open import Cubical.PathPrelude using (isEquiv ; isContr ; fiber) public
|
||||||
|
open import Cubical.GradLemma
|
||||||
|
|
||||||
|
module _ {ℓa ℓb : Level} where
|
||||||
|
private
|
||||||
|
ℓ = ℓa ⊔ ℓb
|
||||||
|
|
||||||
|
module _ {A : Set ℓa} {B : Set ℓb} where
|
||||||
|
-- Quasi-inverse in [HoTT] §2.4.6
|
||||||
|
-- FIXME Maybe rename?
|
||||||
|
record AreInverses (f : A → B) (g : B → A) : Set ℓ where
|
||||||
|
field
|
||||||
|
verso-recto : g ∘ f ≡ idFun A
|
||||||
|
recto-verso : f ∘ g ≡ idFun B
|
||||||
|
obverse = f
|
||||||
|
reverse = g
|
||||||
|
inverse = reverse
|
||||||
|
toPair : Σ _ _
|
||||||
|
toPair = verso-recto , recto-verso
|
||||||
|
|
||||||
|
Isomorphism : (f : A → B) → Set _
|
||||||
|
Isomorphism f = Σ (B → A) λ g → AreInverses f g
|
||||||
|
|
||||||
|
module _ {f : A → B} {g : B → A}
|
||||||
|
(inv : (g ∘ f) ≡ idFun A
|
||||||
|
× (f ∘ g) ≡ idFun B) where
|
||||||
|
open Σ inv renaming (fst to ve-re ; snd to re-ve)
|
||||||
|
toAreInverses : AreInverses f g
|
||||||
|
toAreInverses = record
|
||||||
|
{ verso-recto = ve-re
|
||||||
|
; recto-verso = re-ve
|
||||||
|
}
|
||||||
|
|
||||||
|
_≅_ : Set ℓa → Set ℓb → Set _
|
||||||
|
A ≅ B = Σ (A → B) Isomorphism
|
||||||
|
|
||||||
|
module _ {ℓ : Level} {A B : Set ℓ} {f : A → B}
|
||||||
|
(g : B → A) (s : {A B : Set ℓ} → isSet (A → B)) where
|
||||||
|
|
||||||
|
propAreInverses : isProp (AreInverses {A = A} {B} f g)
|
||||||
|
propAreInverses x y i = record
|
||||||
|
{ verso-recto = ve-re
|
||||||
|
; recto-verso = re-ve
|
||||||
|
}
|
||||||
|
where
|
||||||
|
open AreInverses
|
||||||
|
ve-re : g ∘ f ≡ idFun A
|
||||||
|
ve-re = s (g ∘ f) (idFun A) (verso-recto x) (verso-recto y) i
|
||||||
|
re-ve : f ∘ g ≡ idFun B
|
||||||
|
re-ve = s (f ∘ g) (idFun B) (recto-verso x) (recto-verso y) i
|
||||||
|
|
||||||
|
-- In HoTT they generalize an equivalence to have the following 3 properties:
|
||||||
|
module _ {ℓa ℓb ℓ : Level} (A : Set ℓa) (B : Set ℓb) where
|
||||||
|
record Equiv (iseqv : (A → B) → Set ℓ) : Set (ℓa ⊔ ℓb ⊔ ℓ) where
|
||||||
|
field
|
||||||
|
fromIso : {f : A → B} → Isomorphism f → iseqv f
|
||||||
|
toIso : {f : A → B} → iseqv f → Isomorphism f
|
||||||
|
propIsEquiv : (f : A → B) → isProp (iseqv f)
|
||||||
|
|
||||||
|
-- You're alerady assuming here that we don't need eta-equality on the
|
||||||
|
-- equivalence!
|
||||||
|
_~_ : Set ℓa → Set ℓb → Set _
|
||||||
|
A ~ B = Σ _ iseqv
|
||||||
|
|
||||||
|
inverse-from-to-iso : ∀ {f} (x : _) → (fromIso {f} ∘ toIso {f}) x ≡ x
|
||||||
|
inverse-from-to-iso x = begin
|
||||||
|
(fromIso ∘ toIso) x ≡⟨⟩
|
||||||
|
fromIso (toIso x) ≡⟨ propIsEquiv _ (fromIso (toIso x)) x ⟩
|
||||||
|
x ∎
|
||||||
|
|
||||||
|
-- `toIso` is abstract - so I probably can't close this proof.
|
||||||
|
module _ (sA : isSet A) (sB : isSet B) where
|
||||||
|
module _ {f : A → B} (iso : Isomorphism f) where
|
||||||
|
module _ (iso-x iso-y : Isomorphism f) where
|
||||||
|
open Σ iso-x renaming (fst to x ; snd to inv-x)
|
||||||
|
open Σ iso-y renaming (fst to y ; snd to inv-y)
|
||||||
|
module inv-x = AreInverses inv-x
|
||||||
|
module inv-y = AreInverses inv-y
|
||||||
|
|
||||||
|
fx≡fy : x ≡ y
|
||||||
|
fx≡fy = begin
|
||||||
|
x ≡⟨ cong (λ φ → x ∘ φ) (sym inv-y.recto-verso) ⟩
|
||||||
|
x ∘ (f ∘ y) ≡⟨⟩
|
||||||
|
(x ∘ f) ∘ y ≡⟨ cong (λ φ → φ ∘ y) inv-x.verso-recto ⟩
|
||||||
|
y ∎
|
||||||
|
|
||||||
|
open import Cat.Prelude
|
||||||
|
|
||||||
|
propInv : ∀ g → isProp (AreInverses f g)
|
||||||
|
propInv g t u i = record { verso-recto = a i ; recto-verso = b i }
|
||||||
|
where
|
||||||
|
module t = AreInverses t
|
||||||
|
module u = AreInverses u
|
||||||
|
a : t.verso-recto ≡ u.verso-recto
|
||||||
|
a i = h
|
||||||
|
where
|
||||||
|
hh : ∀ a → (g ∘ f) a ≡ a
|
||||||
|
hh a = sA ((g ∘ f) a) a (λ i → t.verso-recto i a) (λ i → u.verso-recto i a) i
|
||||||
|
h : g ∘ f ≡ idFun A
|
||||||
|
h i a = hh a i
|
||||||
|
b : t.recto-verso ≡ u.recto-verso
|
||||||
|
b i = h
|
||||||
|
where
|
||||||
|
hh : ∀ b → (f ∘ g) b ≡ b
|
||||||
|
hh b = sB _ _ (λ i → t.recto-verso i b) (λ i → u.recto-verso i b) i
|
||||||
|
h : f ∘ g ≡ idFun B
|
||||||
|
h i b = hh b i
|
||||||
|
|
||||||
|
inx≡iny : (λ i → AreInverses f (fx≡fy i)) [ inv-x ≡ inv-y ]
|
||||||
|
inx≡iny = lemPropF propInv fx≡fy
|
||||||
|
|
||||||
|
propIso : iso-x ≡ iso-y
|
||||||
|
propIso i = fx≡fy i , inx≡iny i
|
||||||
|
|
||||||
|
inverse-to-from-iso : (toIso {f} ∘ fromIso {f}) iso ≡ iso
|
||||||
|
inverse-to-from-iso = begin
|
||||||
|
(toIso ∘ fromIso) iso ≡⟨⟩
|
||||||
|
toIso (fromIso iso) ≡⟨ propIso _ _ ⟩
|
||||||
|
iso ∎
|
||||||
|
|
||||||
|
fromIsomorphism : A ≅ B → A ~ B
|
||||||
|
fromIsomorphism (f , iso) = f , fromIso iso
|
||||||
|
|
||||||
|
toIsomorphism : A ~ B → A ≅ B
|
||||||
|
toIsomorphism (f , eqv) = f , toIso eqv
|
||||||
|
|
||||||
|
module _ {ℓa ℓb : Level} (A : Set ℓa) (B : Set ℓb) where
|
||||||
|
-- A wrapper around PathPrelude.≃
|
||||||
|
open Cubical.PathPrelude using (_≃_ ; isEquiv)
|
||||||
|
private
|
||||||
|
module _ {obverse : A → B} (e : isEquiv A B obverse) where
|
||||||
|
inverse : B → A
|
||||||
|
inverse b = fst (fst (e b))
|
||||||
|
|
||||||
|
reverse : B → A
|
||||||
|
reverse = inverse
|
||||||
|
|
||||||
|
areInverses : AreInverses obverse inverse
|
||||||
|
areInverses = record
|
||||||
|
{ verso-recto = funExt verso-recto
|
||||||
|
; recto-verso = funExt recto-verso
|
||||||
|
}
|
||||||
|
where
|
||||||
|
recto-verso : ∀ b → (obverse ∘ inverse) b ≡ b
|
||||||
|
recto-verso b = begin
|
||||||
|
(obverse ∘ inverse) b ≡⟨ sym (μ b) ⟩
|
||||||
|
b ∎
|
||||||
|
where
|
||||||
|
μ : (b : B) → b ≡ obverse (inverse b)
|
||||||
|
μ b = snd (fst (e b))
|
||||||
|
verso-recto : ∀ a → (inverse ∘ obverse) a ≡ a
|
||||||
|
verso-recto a = begin
|
||||||
|
(inverse ∘ obverse) a ≡⟨ sym h ⟩
|
||||||
|
a' ≡⟨ u' ⟩
|
||||||
|
a ∎
|
||||||
|
where
|
||||||
|
c : isContr (fiber obverse (obverse a))
|
||||||
|
c = e (obverse a)
|
||||||
|
fbr : fiber obverse (obverse a)
|
||||||
|
fbr = fst c
|
||||||
|
a' : A
|
||||||
|
a' = fst fbr
|
||||||
|
allC : (y : fiber obverse (obverse a)) → fbr ≡ y
|
||||||
|
allC = snd c
|
||||||
|
k : fbr ≡ (inverse (obverse a), _)
|
||||||
|
k = allC (inverse (obverse a) , sym (recto-verso (obverse a)))
|
||||||
|
h : a' ≡ inverse (obverse a)
|
||||||
|
h i = fst (k i)
|
||||||
|
u : fbr ≡ (a , refl)
|
||||||
|
u = allC (a , refl)
|
||||||
|
u' : a' ≡ a
|
||||||
|
u' i = fst (u i)
|
||||||
|
|
||||||
|
iso : Isomorphism obverse
|
||||||
|
iso = reverse , areInverses
|
||||||
|
|
||||||
|
toIsomorphism : {f : A → B} → isEquiv A B f → Isomorphism f
|
||||||
|
toIsomorphism = iso
|
||||||
|
|
||||||
|
≃isEquiv : Equiv A B (isEquiv A B)
|
||||||
|
Equiv.fromIso ≃isEquiv {f} (f~ , iso) = gradLemma f f~ rv vr
|
||||||
|
where
|
||||||
|
open AreInverses iso
|
||||||
|
rv : (b : B) → _ ≡ b
|
||||||
|
rv b i = recto-verso i b
|
||||||
|
vr : (a : A) → _ ≡ a
|
||||||
|
vr a i = verso-recto i a
|
||||||
|
Equiv.toIso ≃isEquiv = toIsomorphism
|
||||||
|
Equiv.propIsEquiv ≃isEquiv = P.propIsEquiv
|
||||||
|
where
|
||||||
|
import Cubical.NType.Properties as P
|
||||||
|
|
||||||
|
module Equiv≃ where
|
||||||
|
open Equiv ≃isEquiv public
|
||||||
|
|
||||||
|
module _ {ℓa ℓb : Level} {A : Set ℓa} {B : Set ℓb} where
|
||||||
|
open Cubical.PathPrelude using (_≃_)
|
||||||
|
|
||||||
|
-- Gives the quasi inverse from an equivalence.
|
||||||
|
module Equivalence (e : A ≃ B) where
|
||||||
|
open Equiv≃ A B public
|
||||||
|
private
|
||||||
|
iso : Isomorphism (fst e)
|
||||||
|
iso = snd (toIsomorphism e)
|
||||||
|
|
||||||
|
open AreInverses (snd iso) public
|
||||||
|
|
||||||
|
composeIso : {ℓc : Level} {C : Set ℓc} → (B ≅ C) → A ≅ C
|
||||||
|
composeIso {C = C} (g , g' , iso-g) = g ∘ obverse , inverse ∘ g' , inv
|
||||||
|
where
|
||||||
|
module iso-g = AreInverses iso-g
|
||||||
|
inv : AreInverses (g ∘ obverse) (inverse ∘ g')
|
||||||
|
AreInverses.verso-recto inv = begin
|
||||||
|
(inverse ∘ g') ∘ (g ∘ obverse) ≡⟨⟩
|
||||||
|
(inverse ∘ (g' ∘ g) ∘ obverse)
|
||||||
|
≡⟨ cong (λ φ → φ ∘ obverse) (cong (λ φ → inverse ∘ φ) iso-g.verso-recto) ⟩
|
||||||
|
(inverse ∘ idFun B ∘ obverse) ≡⟨⟩
|
||||||
|
(inverse ∘ obverse) ≡⟨ verso-recto ⟩
|
||||||
|
idFun A ∎
|
||||||
|
AreInverses.recto-verso inv = begin
|
||||||
|
g ∘ obverse ∘ inverse ∘ g'
|
||||||
|
≡⟨ cong (λ φ → φ ∘ g') (cong (λ φ → g ∘ φ) recto-verso) ⟩
|
||||||
|
g ∘ idFun B ∘ g' ≡⟨⟩
|
||||||
|
g ∘ g' ≡⟨ iso-g.recto-verso ⟩
|
||||||
|
idFun C ∎
|
||||||
|
|
||||||
|
compose : {ℓc : Level} {C : Set ℓc} → (B ≃ C) → A ≃ C
|
||||||
|
compose {C = C} e = A≃C.fromIsomorphism is
|
||||||
|
where
|
||||||
|
module B≃C = Equiv≃ B C
|
||||||
|
module A≃C = Equiv≃ A C
|
||||||
|
is : A ≅ C
|
||||||
|
is = composeIso (B≃C.toIsomorphism e)
|
||||||
|
|
||||||
|
symmetryIso : B ≅ A
|
||||||
|
symmetryIso
|
||||||
|
= inverse
|
||||||
|
, obverse
|
||||||
|
, record
|
||||||
|
{ verso-recto = recto-verso
|
||||||
|
; recto-verso = verso-recto
|
||||||
|
}
|
||||||
|
|
||||||
|
symmetry : B ≃ A
|
||||||
|
symmetry = B≃A.fromIsomorphism symmetryIso
|
||||||
|
where
|
||||||
|
module B≃A = Equiv≃ B A
|
||||||
|
|
||||||
|
module _ {ℓa ℓb : Level} {A : Set ℓa} {B : Set ℓb} where
|
||||||
|
open import Cubical.PathPrelude renaming (_≃_ to _≃η_)
|
||||||
|
open import Cubical.Univalence using (_≃_)
|
||||||
|
|
||||||
|
doEta : A ≃ B → A ≃η B
|
||||||
|
doEta (_≃_.con eqv isEqv) = eqv , isEqv
|
||||||
|
|
||||||
|
deEta : A ≃η B → A ≃ B
|
||||||
|
deEta (eqv , isEqv) = _≃_.con eqv isEqv
|
||||||
|
|
||||||
|
module NoEta {ℓa ℓb : Level} {A : Set ℓa} {B : Set ℓb} where
|
||||||
|
open import Cubical.PathPrelude renaming (_≃_ to _≃η_)
|
||||||
|
open import Cubical.Univalence using (_≃_)
|
||||||
|
|
||||||
|
module Equivalence′ (e : A ≃ B) where
|
||||||
|
open Equivalence (doEta e) hiding
|
||||||
|
( toIsomorphism ; fromIsomorphism ; _~_
|
||||||
|
; compose ; symmetryIso ; symmetry ) public
|
||||||
|
|
||||||
|
compose : {ℓc : Level} {C : Set ℓc} → (B ≃ C) → A ≃ C
|
||||||
|
compose ee = deEta (Equivalence.compose (doEta e) (doEta ee))
|
||||||
|
|
||||||
|
symmetry : B ≃ A
|
||||||
|
symmetry = deEta (Equivalence.symmetry (doEta e))
|
||||||
|
|
||||||
|
-- fromIso : {f : A → B} → Isomorphism f → isEquiv f
|
||||||
|
-- fromIso = ?
|
||||||
|
|
||||||
|
-- toIso : {f : A → B} → isEquiv f → Isomorphism f
|
||||||
|
-- toIso = ?
|
||||||
|
|
||||||
|
fromIsomorphism : A ≅ B → A ≃ B
|
||||||
|
fromIsomorphism (f , iso) = _≃_.con f (Equiv≃.fromIso _ _ iso)
|
||||||
|
|
||||||
|
toIsomorphism : A ≃ B → A ≅ B
|
||||||
|
toIsomorphism (_≃_.con f eqv) = f , Equiv≃.toIso _ _ eqv
|
||||||
65
src/Cat/Prelude.agda
Normal file
65
src/Cat/Prelude.agda
Normal file
|
|
@ -0,0 +1,65 @@
|
||||||
|
-- | Custom prelude for this module
|
||||||
|
module Cat.Prelude where
|
||||||
|
|
||||||
|
open import Agda.Primitive public
|
||||||
|
-- FIXME Use:
|
||||||
|
-- open import Agda.Builtin.Sigma public
|
||||||
|
-- Rather than
|
||||||
|
open import Data.Product public
|
||||||
|
renaming (∃! to ∃!≈)
|
||||||
|
|
||||||
|
-- TODO Import Data.Function under appropriate names.
|
||||||
|
|
||||||
|
open import Cubical public
|
||||||
|
-- FIXME rename `gradLemma` to `fromIsomorphism` - perhaps just use wrapper
|
||||||
|
-- module.
|
||||||
|
open import Cubical.GradLemma
|
||||||
|
using (gradLemma)
|
||||||
|
public
|
||||||
|
open import Cubical.NType
|
||||||
|
using (⟨-2⟩ ; ⟨-1⟩ ; ⟨0⟩ ; TLevel ; HasLevel)
|
||||||
|
public
|
||||||
|
open import Cubical.NType.Properties
|
||||||
|
using
|
||||||
|
( lemPropF ; lemSig ; lemSigP ; isSetIsProp
|
||||||
|
; propPi ; propHasLevel ; setPi ; propSet)
|
||||||
|
public
|
||||||
|
|
||||||
|
propIsContr : {ℓ : Level} → {A : Set ℓ} → isProp (isContr A)
|
||||||
|
propIsContr = propHasLevel ⟨-2⟩
|
||||||
|
|
||||||
|
open import Cubical.Sigma using (setSig ; sigPresSet) public
|
||||||
|
|
||||||
|
module _ (ℓ : Level) where
|
||||||
|
-- FIXME Ask if we can push upstream.
|
||||||
|
-- A redefinition of `Cubical.Universe` with an explicit parameter
|
||||||
|
_-type : TLevel → Set (lsuc ℓ)
|
||||||
|
n -type = Σ (Set ℓ) (HasLevel n)
|
||||||
|
|
||||||
|
hSet : Set (lsuc ℓ)
|
||||||
|
hSet = ⟨0⟩ -type
|
||||||
|
|
||||||
|
Prop : Set (lsuc ℓ)
|
||||||
|
Prop = ⟨-1⟩ -type
|
||||||
|
|
||||||
|
-----------------
|
||||||
|
-- * Utilities --
|
||||||
|
-----------------
|
||||||
|
|
||||||
|
-- | Unique existensials.
|
||||||
|
∃! : ∀ {a b} {A : Set a}
|
||||||
|
→ (A → Set b) → Set (a ⊔ b)
|
||||||
|
∃! = ∃!≈ _≡_
|
||||||
|
|
||||||
|
∃!-syntax : ∀ {a b} {A : Set a} → (A → Set b) → Set (a ⊔ b)
|
||||||
|
∃!-syntax = ∃
|
||||||
|
|
||||||
|
syntax ∃!-syntax (λ x → B) = ∃![ x ] B
|
||||||
|
|
||||||
|
module _ {ℓa ℓb : Level} {A : Set ℓa} {B : A → Set ℓb} {a b : Σ A B}
|
||||||
|
(proj₁≡ : (λ _ → A) [ proj₁ a ≡ proj₁ b ])
|
||||||
|
(proj₂≡ : (λ i → B (proj₁≡ i)) [ proj₂ a ≡ proj₂ b ]) where
|
||||||
|
|
||||||
|
Σ≡ : a ≡ b
|
||||||
|
proj₁ (Σ≡ i) = proj₁≡ i
|
||||||
|
proj₂ (Σ≡ i) = proj₂≡ i
|
||||||
Loading…
Reference in a new issue