Include appendices
This commit is contained in:
parent
4e7506f06a
commit
e5d55c7b2b
|
|
@ -10,6 +10,9 @@ Cubical Agda extends the underlying type system (\TODO{Cite someone smarter than
|
|||
me with a good resource on this}) but introduces a data-type within the
|
||||
languages.
|
||||
|
||||
Exceprts of the source code relevant to this section can be found in appendix
|
||||
\ref{sec:app-cubical}.
|
||||
|
||||
\subsection{The equality type}
|
||||
The usual notion of judgmental equality says that given a type $A \tp \MCU$ and
|
||||
two points of $A$; $a_0, a_1 \tp A$ we can form the type:
|
||||
|
|
|
|||
|
|
@ -50,6 +50,11 @@ where one can reason about two categories by simply focusing on the data. This
|
|||
is achieved by creating a function embodying the ``equality principle'' for a
|
||||
given type.
|
||||
|
||||
For the rest of this chapter I will present some of these results. For didactic
|
||||
reasons no source-code has been included in this chapter. To see the formal
|
||||
definitions excerpts of the implementation have been included in appendix
|
||||
\ref{ch:app-sources}.
|
||||
|
||||
\section{Categories}
|
||||
The data for a category consist of a type for the sort of objects; a type for
|
||||
the sort of arrows; an identity arrow and a composition operation for arrows.
|
||||
|
|
|
|||
|
|
@ -195,7 +195,7 @@ Name & Agda & Notation \\
|
|||
\nomen{Type} & \texttt{Set} & $\Type$ \\
|
||||
\nomen{Set} & \texttt{Σ Set IsSet} & $\Set$ \\
|
||||
Function, morphism, map & \texttt{A → B} & $A → B$ \\
|
||||
Dependent- ditto & \texttt{(a \tp A) → B} & $∏_{a \tp A} B$ \\
|
||||
Dependent- ditto & \texttt{(a : A) → B} & $∏_{a \tp A} B$ \\
|
||||
\nomen{Arrow} & \texttt{Arrow A B} & $\Arrow\ A\ B$ \\
|
||||
\nomen{Object} & \texttt{C.Object} & $̱ℂ.Object$ \\
|
||||
Definition & \texttt{=} & $̱\defeq$ \\
|
||||
|
|
|
|||
|
|
@ -69,6 +69,7 @@
|
|||
\begin{appendices}
|
||||
\setcounter{page}{1}
|
||||
\pagenumbering{roman}
|
||||
\input{sources.tex}
|
||||
%% \input{planning.tex}
|
||||
%% \input{halftime.tex}
|
||||
\end{appendices}
|
||||
|
|
|
|||
|
|
@ -15,7 +15,7 @@
|
|||
\usepackage{amssymb,amsmath,amsthm,stmaryrd,mathrsfs,wasysym}
|
||||
\usepackage[toc,page]{appendix}
|
||||
\usepackage{xspace}
|
||||
\usepackage{geometry}
|
||||
\usepackage[a4paper]{geometry}
|
||||
|
||||
% \setlength{\parskip}{10pt}
|
||||
|
||||
|
|
@ -37,6 +37,7 @@
|
|||
\usepackage{lmodern}
|
||||
|
||||
\usepackage{enumerate}
|
||||
\usepackage{verbatim}
|
||||
|
||||
\usepackage{fontspec}
|
||||
\usepackage[light]{sourcecodepro}
|
||||
|
|
@ -56,3 +57,31 @@
|
|||
%% \RequirePackage{kvoptions}
|
||||
|
||||
\usepackage{pgffor}
|
||||
\lstset
|
||||
{basicstyle=\ttfamily
|
||||
,columns=fullflexible
|
||||
,breaklines=true
|
||||
,inputencoding=utf8
|
||||
,extendedchars=true
|
||||
%% ,literate={á}{{\'a}}1 {ã}{{\~a}}1 {é}{{\'e}}1
|
||||
}
|
||||
|
||||
\usepackage{newunicodechar}
|
||||
|
||||
%% \setmainfont{PT Serif}
|
||||
\newfontfamily{\fallbackfont}{FreeMono.otf}[Scale=MatchLowercase]
|
||||
%% \setmonofont[Mapping=tex-text]{FreeMono.otf}
|
||||
\DeclareTextFontCommand{\textfallback}{\fallbackfont}
|
||||
\newunicodechar{∨}{\textfallback{∨}}
|
||||
\newunicodechar{∧}{\textfallback{∧}}
|
||||
\newunicodechar{⊔}{\textfallback{⊔}}
|
||||
\newunicodechar{≊}{\textfallback{≊}}
|
||||
\newunicodechar{∈}{\textfallback{∈}}
|
||||
\newunicodechar{ℂ}{\textfallback{ℂ}}
|
||||
\newunicodechar{∘}{\textfallback{∘}}
|
||||
\newunicodechar{⟨}{\textfallback{⟨}}
|
||||
\newunicodechar{⟩}{\textfallback{⟩}}
|
||||
\newunicodechar{∎}{\textfallback{∎}}
|
||||
\newunicodechar{𝒜}{\textfallback{?}}
|
||||
\newunicodechar{ℬ}{\textfallback{?}}
|
||||
%% \newunicodechar{≊}{\textfallback{≊}}
|
||||
|
|
|
|||
428
doc/sources.tex
Normal file
428
doc/sources.tex
Normal file
|
|
@ -0,0 +1,428 @@
|
|||
\chapter{Source code}
|
||||
\label{ch:app-sources}
|
||||
In the following a few of the definitions mentioned in the report are included.
|
||||
The following is \emph{not} a verbatim copy of individual modules from the
|
||||
implementation. Rather, this is a redacted and pre-formatted designed for
|
||||
presentation purposes. It's provided here as a convenience. The actual sources
|
||||
are the only authoritative source. Is something is not clear, please refer to
|
||||
those.
|
||||
\clearpage
|
||||
\section{Cubical}
|
||||
\label{sec:app-cubical}
|
||||
\begin{figure}[h]
|
||||
\begin{Verbatim}
|
||||
postulate
|
||||
PathP : ∀ {ℓ} (A : I → Set ℓ) → A i0 → A i1 → Set ℓ
|
||||
|
||||
module _ {ℓ} {A : Set ℓ} where
|
||||
_≡_ : A → A → Set ℓ
|
||||
_≡_ {A = A} = PathP (λ _ → A)
|
||||
|
||||
refl : {x : A} → x ≡ x
|
||||
refl {x = x} = λ _ → x
|
||||
\end{Verbatim}
|
||||
\caption{Excerpt from the cubical library. Definition of the path-type.}
|
||||
\end{figure}
|
||||
%
|
||||
\begin{figure}[h]
|
||||
\begin{Verbatim}
|
||||
module _ {ℓ : Level} (A : Set ℓ) where
|
||||
isContr : Set ℓ
|
||||
isContr = Σ[ x ∈ A ] (∀ y → x ≡ y)
|
||||
|
||||
isProp : Set ℓ
|
||||
isProp = (x y : A) → x ≡ y
|
||||
|
||||
isSet : Set ℓ
|
||||
isSet = (x y : A) → (p q : x ≡ y) → p ≡ q
|
||||
|
||||
isGrpd : Set ℓ
|
||||
isGrpd = (x y : A) → (p q : x ≡ y) → (a b : p ≡ q) → a ≡ b
|
||||
\end{Verbatim}
|
||||
\caption{Excerpt from the cubical library. Definition of the first few
|
||||
homotopy-levels.}
|
||||
\end{figure}
|
||||
%
|
||||
\begin{figure}[h]
|
||||
\begin{Verbatim}
|
||||
module _ {ℓ ℓ'} {A : Set ℓ} {x : A}
|
||||
(P : ∀ y → x ≡ y → Set ℓ') (d : P x ((λ i → x))) where
|
||||
pathJ : (y : A) → (p : x ≡ y) → P y p
|
||||
pathJ _ p = transp (λ i → uncurry P (contrSingl p i)) d
|
||||
\end{Verbatim}
|
||||
\clearpage
|
||||
\caption{Excerpt from the cubical library. Definition of based path-induction.}
|
||||
\end{figure}
|
||||
%
|
||||
\begin{figure}[h]
|
||||
\begin{Verbatim}
|
||||
module _ {ℓ ℓ'} {A : Set ℓ} {B : A → Set ℓ'} where
|
||||
propPi : (h : (x : A) → isProp (B x)) → isProp ((x : A) → B x)
|
||||
propPi h f0 f1 = λ i → λ x → (h x (f0 x) (f1 x)) i
|
||||
|
||||
lemPropF : (P : (x : A) → isProp (B x)) {a0 a1 : A}
|
||||
(p : a0 ≡ a1) {b0 : B a0} {b1 : B a1} → PathP (λ i → B (p i)) b0 b1
|
||||
lemPropF P p {b0} {b1} = λ i → P (p i)
|
||||
(primComp (λ j → B (p (i ∧ j)) ) (~ i) (λ _ → λ { (i = i0) → b0 }) b0)
|
||||
(primComp (λ j → B (p (i ∨ ~ j)) ) (i) (λ _ → λ{ (i = i1) → b1 }) b1) i
|
||||
|
||||
lemSig : (pB : (x : A) → isProp (B x))
|
||||
(u v : Σ A B) (p : fst u ≡ fst v) → u ≡ v
|
||||
lemSig pB u v p = λ i → (p i) , ((lemPropF pB p) {snd u} {snd v} i)
|
||||
|
||||
propSig : (pA : isProp A) (pB : (x : A) → isProp (B x)) → isProp (Σ A B)
|
||||
propSig pA pB t u = lemSig pB t u (pA (fst t) (fst u))
|
||||
\end{Verbatim}
|
||||
\caption{Excerpt from the cubical library. A few combinators.}
|
||||
\end{figure}
|
||||
\clearpage
|
||||
\section{Categories}
|
||||
\label{sec:app-categories}
|
||||
\begin{figure}[h]
|
||||
\begin{Verbatim}
|
||||
record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
|
||||
field
|
||||
Object : Set ℓa
|
||||
Arrow : Object → Object → Set ℓb
|
||||
identity : {A : Object} → Arrow A A
|
||||
_<<<_ : {A B C : Object} → Arrow B C → Arrow A B → Arrow A C
|
||||
|
||||
_>>>_ : {A B C : Object} → (Arrow A B) → (Arrow B C) → Arrow A C
|
||||
f >>> g = g <<< f
|
||||
|
||||
IsAssociative : Set (ℓa ⊔ ℓb)
|
||||
IsAssociative = ∀ {A B C D} {f : Arrow A B} {g : Arrow B C} {h : Arrow C D}
|
||||
→ h <<< (g <<< f) ≡ (h <<< g) <<< f
|
||||
|
||||
IsIdentity : ({A : Object} → Arrow A A) → Set (ℓa ⊔ ℓb)
|
||||
IsIdentity id = {A B : Object} {f : Arrow A B}
|
||||
→ id <<< f ≡ f × f <<< id ≡ f
|
||||
|
||||
ArrowsAreSets : Set (ℓa ⊔ ℓb)
|
||||
ArrowsAreSets = ∀ {A B : Object} → isSet (Arrow A B)
|
||||
|
||||
IsInverseOf : ∀ {A B} → (Arrow A B) → (Arrow B A) → Set ℓb
|
||||
IsInverseOf = λ f g → g <<< f ≡ identity × f <<< g ≡ identity
|
||||
|
||||
Isomorphism : ∀ {A B} → (f : Arrow A B) → Set ℓb
|
||||
Isomorphism {A} {B} f = Σ[ g ∈ Arrow B A ] IsInverseOf f g
|
||||
|
||||
_≊_ : (A B : Object) → Set ℓb
|
||||
_≊_ A B = Σ[ f ∈ Arrow A B ] (Isomorphism f)
|
||||
|
||||
IsTerminal : Object → Set (ℓa ⊔ ℓb)
|
||||
IsTerminal T = {X : Object} → isContr (Arrow X T)
|
||||
|
||||
Terminal : Set (ℓa ⊔ ℓb)
|
||||
Terminal = Σ Object IsTerminal
|
||||
\end{Verbatim}
|
||||
\caption{Excerpt from module \texttt{Cat.Category}. The definition of a category.}
|
||||
\end{figure}
|
||||
\clearpage
|
||||
\begin{figure}[h]
|
||||
\begin{Verbatim}
|
||||
module Univalence (isIdentity : IsIdentity identity) where
|
||||
idIso : (A : Object) → A ≊ A
|
||||
idIso A = identity , identity , isIdentity
|
||||
|
||||
idToIso : (A B : Object) → A ≡ B → A ≊ B
|
||||
idToIso A B eq = subst eq (idIso A)
|
||||
|
||||
Univalent : Set (ℓa ⊔ ℓb)
|
||||
Univalent = {A B : Object} → isEquiv (A ≡ B) (A ≊ B) (idToIso A B)
|
||||
\end{Verbatim}
|
||||
\caption{Excerpt from module \texttt{Cat.Category}. Continued from previous. Definition of univalence.}
|
||||
\end{figure}
|
||||
\begin{figure}[h]
|
||||
\begin{Verbatim}
|
||||
module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
|
||||
record IsPreCategory : Set (lsuc (ℓa ⊔ ℓb)) where
|
||||
open RawCategory ℂ public
|
||||
field
|
||||
isAssociative : IsAssociative
|
||||
isIdentity : IsIdentity identity
|
||||
arrowsAreSets : ArrowsAreSets
|
||||
open Univalence isIdentity public
|
||||
|
||||
record PreCategory : Set (lsuc (ℓa ⊔ ℓb)) where
|
||||
field
|
||||
isPreCategory : IsPreCategory
|
||||
open IsPreCategory isPreCategory public
|
||||
|
||||
record IsCategory : Set (lsuc (ℓa ⊔ ℓb)) where
|
||||
field
|
||||
isPreCategory : IsPreCategory
|
||||
open IsPreCategory isPreCategory public
|
||||
field
|
||||
univalent : Univalent
|
||||
\end{Verbatim}
|
||||
\caption{Excerpt from module \texttt{Cat.Category}. Records with inhabitants for
|
||||
the laws defined in the previous listings.}
|
||||
\end{figure}
|
||||
\clearpage
|
||||
\begin{figure}[h]
|
||||
\begin{Verbatim}
|
||||
module Opposite {ℓa ℓb : Level} where
|
||||
module _ (ℂ : Category ℓa ℓb) where
|
||||
private
|
||||
module _ where
|
||||
module ℂ = Category ℂ
|
||||
opRaw : RawCategory ℓa ℓb
|
||||
RawCategory.Object opRaw = ℂ.Object
|
||||
RawCategory.Arrow opRaw = flip ℂ.Arrow
|
||||
RawCategory.identity opRaw = ℂ.identity
|
||||
RawCategory._<<<_ opRaw = ℂ._>>>_
|
||||
|
||||
open RawCategory opRaw
|
||||
|
||||
isPreCategory : IsPreCategory opRaw
|
||||
IsPreCategory.isAssociative isPreCategory = sym ℂ.isAssociative
|
||||
IsPreCategory.isIdentity isPreCategory = swap ℂ.isIdentity
|
||||
IsPreCategory.arrowsAreSets isPreCategory = ℂ.arrowsAreSets
|
||||
|
||||
open IsPreCategory isPreCategory
|
||||
|
||||
----------------------------
|
||||
-- NEXT LISTING GOES HERE --
|
||||
----------------------------
|
||||
|
||||
isCategory : IsCategory opRaw
|
||||
IsCategory.isPreCategory isCategory = isPreCategory
|
||||
IsCategory.univalent isCategory
|
||||
= univalenceFromIsomorphism (isoToId* , inv)
|
||||
|
||||
opposite : Category ℓa ℓb
|
||||
Category.raw opposite = opRaw
|
||||
Category.isCategory opposite = isCategory
|
||||
|
||||
module _ {ℂ : Category ℓa ℓb} where
|
||||
open Category ℂ
|
||||
private
|
||||
rawInv : Category.raw (opposite (opposite ℂ)) ≡ raw
|
||||
RawCategory.Object (rawInv _) = Object
|
||||
RawCategory.Arrow (rawInv _) = Arrow
|
||||
RawCategory.identity (rawInv _) = identity
|
||||
RawCategory._<<<_ (rawInv _) = _<<<_
|
||||
|
||||
oppositeIsInvolution : opposite (opposite ℂ) ≡ ℂ
|
||||
oppositeIsInvolution = Category≡ rawInv
|
||||
\end{Verbatim}
|
||||
\caption{Excerpt from module \texttt{Cat.Category}. Showing that the opposite
|
||||
category is a category. Part of this listing has been cut out and placed in
|
||||
the next listing.}
|
||||
\end{figure}
|
||||
\clearpage
|
||||
For reasons of formatting the following listing is continued from the above with
|
||||
fewer levels of indentation.
|
||||
%
|
||||
\begin{figure}[h]
|
||||
\begin{Verbatim}
|
||||
module _ {A B : ℂ.Object} where
|
||||
open Σ (toIso _ _ (ℂ.univalent {A} {B}))
|
||||
renaming (fst to idToIso* ; snd to inv*)
|
||||
open AreInverses {f = ℂ.idToIso A B} {idToIso*} inv*
|
||||
|
||||
shuffle : A ≊ B → A ℂ.≊ B
|
||||
shuffle (f , g , inv) = g , f , inv
|
||||
|
||||
shuffle~ : A ℂ.≊ B → A ≊ B
|
||||
shuffle~ (f , g , inv) = g , f , inv
|
||||
|
||||
isoToId* : A ≊ B → A ≡ B
|
||||
isoToId* = idToIso* ∘ shuffle
|
||||
|
||||
inv : AreInverses (idToIso A B) isoToId*
|
||||
inv =
|
||||
( funExt (λ x → begin
|
||||
(isoToId* ∘ idToIso A B) x
|
||||
≡⟨⟩
|
||||
(idToIso* ∘ shuffle ∘ idToIso A B) x
|
||||
≡⟨ cong (λ φ → φ x)
|
||||
(cong (λ φ → idToIso* ∘ shuffle ∘ φ) (funExt (isoEq refl))) ⟩
|
||||
(idToIso* ∘ shuffle ∘ shuffle~ ∘ ℂ.idToIso A B) x
|
||||
≡⟨⟩
|
||||
(idToIso* ∘ ℂ.idToIso A B) x
|
||||
≡⟨ (λ i → verso-recto i x) ⟩
|
||||
x ∎)
|
||||
, funExt (λ x → begin
|
||||
(idToIso A B ∘ idToIso* ∘ shuffle) x
|
||||
≡⟨ cong (λ φ → φ x)
|
||||
(cong (λ φ → φ ∘ idToIso* ∘ shuffle) (funExt (isoEq refl))) ⟩
|
||||
(shuffle~ ∘ ℂ.idToIso A B ∘ idToIso* ∘ shuffle) x
|
||||
≡⟨ cong (λ φ → φ x)
|
||||
(cong (λ φ → shuffle~ ∘ φ ∘ shuffle) recto-verso) ⟩
|
||||
(shuffle~ ∘ shuffle) x
|
||||
≡⟨⟩
|
||||
x ∎)
|
||||
)
|
||||
\end{Verbatim}
|
||||
\caption{Excerpt from module \texttt{Cat.Category}. Proving univalence for the opposite category.}
|
||||
\end{figure}
|
||||
\clearpage
|
||||
\section{Products}
|
||||
\label{sec:app-products}
|
||||
\begin{figure}[h]
|
||||
\begin{Verbatim}
|
||||
module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||
open Category ℂ
|
||||
|
||||
module _ (A B : Object) where
|
||||
record RawProduct : Set (ℓa ⊔ ℓb) where
|
||||
no-eta-equality
|
||||
field
|
||||
object : Object
|
||||
fst : ℂ [ object , A ]
|
||||
snd : ℂ [ object , B ]
|
||||
|
||||
record IsProduct (raw : RawProduct) : Set (ℓa ⊔ ℓb) where
|
||||
open RawProduct raw public
|
||||
field
|
||||
ump : ∀ {X : Object} (f : ℂ [ X , A ]) (g : ℂ [ X , B ])
|
||||
→ ∃![ f×g ] (ℂ [ fst ∘ f×g ] ≡ f P.× ℂ [ snd ∘ f×g ] ≡ g)
|
||||
|
||||
record Product : Set (ℓa ⊔ ℓb) where
|
||||
field
|
||||
raw : RawProduct
|
||||
isProduct : IsProduct raw
|
||||
|
||||
open IsProduct isProduct public
|
||||
|
||||
\end{Verbatim}
|
||||
\caption{Excerpt from module \texttt{Cat.Category.Product}. Definition of products.}
|
||||
\end{figure}
|
||||
%
|
||||
\begin{figure}[h]
|
||||
\begin{Verbatim}
|
||||
module _{ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
|
||||
(let module ℂ = Category ℂ) {A* B* : ℂ.Object} where
|
||||
|
||||
module _ where
|
||||
raw : RawCategory _ _
|
||||
raw = record
|
||||
{ Object = Σ[ X ∈ ℂ.Object ] ℂ.Arrow X A* × ℂ.Arrow X B*
|
||||
; Arrow = λ{ (A , a0 , a1) (B , b0 , b1)
|
||||
→ Σ[ f ∈ ℂ.Arrow A B ]
|
||||
ℂ [ b0 ∘ f ] ≡ a0
|
||||
× ℂ [ b1 ∘ f ] ≡ a1
|
||||
}
|
||||
; identity = λ{ {X , f , g}
|
||||
→ ℂ.identity {X} , ℂ.rightIdentity , ℂ.rightIdentity
|
||||
}
|
||||
; _<<<_ = λ { {_ , a0 , a1} {_ , b0 , b1} {_ , c0 , c1}
|
||||
(f , f0 , f1) (g , g0 , g1)
|
||||
→ (f ℂ.<<< g)
|
||||
, (begin
|
||||
ℂ [ c0 ∘ ℂ [ f ∘ g ] ] ≡⟨ ℂ.isAssociative ⟩
|
||||
ℂ [ ℂ [ c0 ∘ f ] ∘ g ] ≡⟨ cong (λ φ → ℂ [ φ ∘ g ]) f0 ⟩
|
||||
ℂ [ b0 ∘ g ] ≡⟨ g0 ⟩
|
||||
a0 ∎
|
||||
)
|
||||
, (begin
|
||||
ℂ [ c1 ∘ ℂ [ f ∘ g ] ] ≡⟨ ℂ.isAssociative ⟩
|
||||
ℂ [ ℂ [ c1 ∘ f ] ∘ g ] ≡⟨ cong (λ φ → ℂ [ φ ∘ g ]) f1 ⟩
|
||||
ℂ [ b1 ∘ g ] ≡⟨ g1 ⟩
|
||||
a1 ∎
|
||||
)
|
||||
}
|
||||
}
|
||||
\end{Verbatim}
|
||||
\caption{Excerpt from module \texttt{Cat.Category.Product}. Definition of ``pair category''.}
|
||||
\end{figure}
|
||||
\clearpage
|
||||
\section{Monads}
|
||||
\label{sec:app-monads}
|
||||
\begin{figure}[h]
|
||||
\begin{Verbatim}
|
||||
module Cat.Category.Monad.Kleisli
|
||||
{ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||
|
||||
record RawMonad : Set ℓ where
|
||||
field
|
||||
omap : Object → Object
|
||||
pure : {X : Object} → ℂ [ X , omap X ]
|
||||
bind : {X Y : Object} → ℂ [ X , omap Y ] → ℂ [ omap X , omap Y ]
|
||||
|
||||
fmap : ∀ {A B} → ℂ [ A , B ] → ℂ [ omap A , omap B ]
|
||||
fmap f = bind (pure <<< f)
|
||||
|
||||
_>=>_ : {A B C : Object}
|
||||
→ ℂ [ A , omap B ] → ℂ [ B , omap C ] → ℂ [ A , omap C ]
|
||||
f >=> g = f >>> (bind g)
|
||||
|
||||
join : {A : Object} → ℂ [ omap (omap A) , omap A ]
|
||||
join = bind identity
|
||||
|
||||
IsIdentity = {X : Object}
|
||||
→ bind pure ≡ identity {omap X}
|
||||
IsNatural = {X Y : Object} (f : ℂ [ X , omap Y ])
|
||||
→ pure >=> f ≡ f
|
||||
IsDistributive = {X Y Z : Object}
|
||||
(g : ℂ [ Y , omap Z ]) (f : ℂ [ X , omap Y ])
|
||||
→ (bind f) >>> (bind g) ≡ bind (f >=> g)
|
||||
|
||||
record IsMonad (raw : RawMonad) : Set ℓ where
|
||||
open RawMonad raw public
|
||||
field
|
||||
isIdentity : IsIdentity
|
||||
isNatural : IsNatural
|
||||
isDistributive : IsDistributive
|
||||
\end{Verbatim}
|
||||
\caption{Excerpt from module \texttt{Cat.Category.Monad.Kleisli}. Definition of
|
||||
Kleisli monads.}
|
||||
\end{figure}
|
||||
%
|
||||
\begin{figure}[h]
|
||||
\begin{Verbatim}
|
||||
module Cat.Category.Monad.Monoidal
|
||||
{ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
|
||||
|
||||
private
|
||||
ℓ = ℓa ⊔ ℓb
|
||||
|
||||
open Category ℂ using (Object ; Arrow ; identity ; _<<<_)
|
||||
|
||||
record RawMonad : Set ℓ where
|
||||
field
|
||||
R : EndoFunctor ℂ
|
||||
pureNT : NaturalTransformation Functors.identity R
|
||||
joinNT : NaturalTransformation F[ R ∘ R ] R
|
||||
|
||||
Romap = Functor.omap R
|
||||
fmap = Functor.fmap R
|
||||
|
||||
pureT : Transformation Functors.identity R
|
||||
pureT = fst pureNT
|
||||
|
||||
pure : {X : Object} → ℂ [ X , Romap X ]
|
||||
pure = pureT _
|
||||
|
||||
pureN : Natural Functors.identity R pureT
|
||||
pureN = snd pureNT
|
||||
|
||||
joinT : Transformation F[ R ∘ R ] R
|
||||
joinT = fst joinNT
|
||||
join : {X : Object} → ℂ [ Romap (Romap X) , Romap X ]
|
||||
join = joinT _
|
||||
joinN : Natural F[ R ∘ R ] R joinT
|
||||
joinN = snd joinNT
|
||||
|
||||
bind : {X Y : Object} → ℂ [ X , Romap Y ] → ℂ [ Romap X , Romap Y ]
|
||||
bind {X} {Y} f = join <<< fmap f
|
||||
|
||||
IsAssociative : Set _
|
||||
IsAssociative = {X : Object}
|
||||
→ joinT X <<< fmap join ≡ join <<< join
|
||||
IsInverse : Set _
|
||||
IsInverse = {X : Object}
|
||||
→ join <<< pure ≡ identity {Romap X}
|
||||
× join <<< fmap pure ≡ identity {Romap X}
|
||||
|
||||
record IsMonad (raw : RawMonad) : Set ℓ where
|
||||
open RawMonad raw public
|
||||
field
|
||||
isAssociative : IsAssociative
|
||||
isInverse : IsInverse
|
||||
\end{Verbatim}
|
||||
\caption{Excerpt from module \texttt{Cat.Category.Monad.Monoidal}. Definition of
|
||||
Monoidal monads.}
|
||||
\end{figure}
|
||||
Loading…
Reference in a new issue