Finish section on category of sets
This commit is contained in:
parent
f6cf051519
commit
129eef1150
|
|
@ -0,0 +1,72 @@
|
|||
Andrea Vezzosi <vezzosi@chalmers.se> Tue, Apr 24, 2018 at 2:02 PM
|
||||
To: Frederik Hanghøj Iversen <fhi.1990@gmail.com>
|
||||
Cc: Thierry Coquand <coquand@chalmers.se>
|
||||
On Tue, Apr 24, 2018 at 12:57 PM, Frederik Hanghøj Iversen
|
||||
<fhi.1990@gmail.com> wrote:
|
||||
> I've written the first few sections about my implementation. I was wondering
|
||||
> if you could have a quick look at it. You don't need to read it
|
||||
> word-for-word but I would like some indication from you if this is the sort
|
||||
> of thing you would like to see in the final report.
|
||||
|
||||
Yes! I would say this very much fits the bill of what the main part of
|
||||
the report should be, then you could have a discussion section where
|
||||
you might put some analysis of the pros and cons of cubical, design
|
||||
choices you made, and your experience overall.
|
||||
|
||||
I wonder if there should be some short introduction to Cubical Type
|
||||
Theory before this chapter, so you can introduce the Path type by
|
||||
itself and show some simple proof with it. e.g. how to get function
|
||||
extensionality.
|
||||
|
||||
You mention a few "combinators" like propPi and lemPropF, you might
|
||||
want to call them just lemmas, so it's clearer that these can be
|
||||
proven in --cubical.
|
||||
|
||||
>
|
||||
> I refer you specifically to "Chapter 2 - Implementation" on p. 6.
|
||||
>
|
||||
> In this chapter I plan to additionally include some text about the proof we
|
||||
> did that products are mere propositions and the proof about the two
|
||||
> equivalent notions of a monad.
|
||||
|
||||
I've read the chapter up until 2.3 and skimmed the rest for now, but I
|
||||
accumulated some editing suggestions I copy here.
|
||||
Remember to look for things like these when you proof-read the rest :)
|
||||
|
||||
|
||||
You should be careful to properly introduce things before you use
|
||||
them, like IsPreCategory (I'd prefer if it took the raw category as
|
||||
argument btw) and its fields isIdentity, isAssociative, .. come up a
|
||||
bit out of the blue from the end of page 8.
|
||||
Maybe the easiest is to show the definition of IsPreCategory.
|
||||
|
||||
Maybe give a type for propIsIdentity and mention the other prop* are similar.
|
||||
|
||||
Also the notation "isIdentity_a" to apply projections is a bit unusual
|
||||
so it needs to be introduced as well.
|
||||
To be fair it would be simpler to stick to function application
|
||||
(though I see that it would introduce more parentheses),
|
||||
|
||||
"The situation is a bit more complicated when we have a dependent
|
||||
type" could be more clear by being more specific:
|
||||
"The situation is a bit more complicated when the type of a field
|
||||
depends on a previous field"
|
||||
|
||||
Here too it might be more concrete if you also give the code for IsCategory.
|
||||
|
||||
In Path ( λ i → Univalent_{p i} ) isPreCategory_a isPreCategory_b
|
||||
I suggest parentheses around (p i), but also you should be consistent
|
||||
on whether you want to call the proof "p" or "p_{isPreCategory}",
|
||||
finally i'm guessing the two fields should be "isUnivalent" rather
|
||||
than "isPreCategory".
|
||||
|
||||
You can cite the book on the specific definition of isEquiv,
|
||||
"contractible fibers" in section 4.4, the grad lemma is also from
|
||||
somewhere but I don't remember off-hand.
|
||||
|
||||
You have not defined what you mean by _\~=_ and isomorphism.
|
||||
|
||||
|
||||
Cheers,
|
||||
Andrea
|
||||
[Quoted text hidden]
|
||||
|
|
@ -189,7 +189,7 @@ $$
|
|||
and one heterogeneous:
|
||||
%
|
||||
$$
|
||||
Path\ (\lambda i \to Univalent_{p\ i})\ \isPreCategory_a\ \isPreCategory_b
|
||||
\Path\ (\lambda\; i \mto Univalent_{p\ i})\ \isPreCategory_a\ \isPreCategory_b
|
||||
$$
|
||||
%
|
||||
Which depends on the choice of $p_{\isPreCategory}$. The first of these we can
|
||||
|
|
@ -203,7 +203,7 @@ path between some two elements in $A$; $p : a_0 \equiv a_1$ then we can built a
|
|||
heterogeneous path between any two $b$'s at the endpoints:
|
||||
%
|
||||
$$
|
||||
Path\ (\lambda i \to B\ (p\ i))\ b0\ b1
|
||||
\Path\ (\lambda\; i \mto B\ (p\ i))\ b0\ b1
|
||||
$$
|
||||
%
|
||||
where $b_0 \tp B a_0$ and $b_1 \tp B\ a_1$. This is quite a mouthful, but the
|
||||
|
|
@ -216,7 +216,7 @@ applying using congruence of paths: $\congruence\ \mathit{isIdentity}\
|
|||
p_{\isPreCategory}$
|
||||
|
||||
When we have a proper category we can make precise the notion of ``identifying
|
||||
isomorphic types'' (TODO cite awodey here). That is, we can construct the
|
||||
isomorphic types'' \TODO{cite awodey here}. That is, we can construct the
|
||||
function:
|
||||
%
|
||||
$$
|
||||
|
|
@ -227,11 +227,11 @@ One application of this, and a perhaps somewhat surprising result, is that
|
|||
terminal objects are propositional. Intuitively; they do not have any
|
||||
interesting structure. The proof of this follows from the usual observation that
|
||||
any two terminal objects are isomorphic. The proof is omitted here, but the
|
||||
curious reader can check the implementation for the details. (TODO: The proof is
|
||||
a bit fun, should I include it?)
|
||||
curious reader can check the implementation for the details. \TODO{The proof is
|
||||
a bit fun, should I include it?}
|
||||
|
||||
\section{Equivalences}
|
||||
\label{equiv}
|
||||
\label{sec:equiv}
|
||||
The usual notion of a function $f : A \to B$ having an inverses is:
|
||||
%
|
||||
$$
|
||||
|
|
@ -253,7 +253,7 @@ what an equivalence $\isEquiv : (A \to B) \to \MCU$ must supply:
|
|||
%
|
||||
Having such an interface gives us both 1) a way to think rather abstractly about
|
||||
how to work with equivalences and 2) to use ad-hoc definitions of equivalences.
|
||||
The specific instantiation of $\isEquiv$ as defined in \cite{cubical} is:
|
||||
The specific instantiation of $\isEquiv$ as defined in \cite{cubical-agda} is:
|
||||
%
|
||||
$$
|
||||
isEquiv\ f \defeq \prod_{b : B} \isContr\ (\fiber\ f\ b)
|
||||
|
|
@ -269,7 +269,7 @@ once we have shown that this definition actually works as an equivalence.
|
|||
|
||||
The first function from the list of requirements we will call
|
||||
$\mathit{fromIsomorphism}$, this is known as $\mathit{gradLemma}$ in
|
||||
\cite{cubical} the second one we will refer to as $\mathit{toIsmorphism}$. It's
|
||||
\cite{cubical-agda} the second one we will refer to as $\mathit{toIsmorphism}$. It's
|
||||
implementation can be found in the sources. Likewise the proof that this
|
||||
equivalence is propositional can be found in my implementation.
|
||||
|
||||
|
|
@ -353,7 +353,7 @@ $$
|
|||
\isoToId \tp A \approxeq B \to A \equiv B
|
||||
$$
|
||||
%
|
||||
The next few theorems are variations on theorem 9.1.9 from \cite{HoTT-book}. Let
|
||||
The next few theorems are variations on theorem 9.1.9 from \cite{hott-2013}. Let
|
||||
an isomorphism $A \approxeq B$ in some category $\bC$ be given. Name the
|
||||
isomorphism $\iota \tp A \to B$ and its inverse $\widetilde{\iota} \tp B \to A$.
|
||||
Since $\bC$ is a category (and therefore univalent) the isomorphism induces a
|
||||
|
|
@ -425,7 +425,7 @@ univalence in a very simple category where the structure of the category is
|
|||
rather simple.
|
||||
|
||||
Let $\bC$ be some category, we then define the opposite category
|
||||
$\bC^{\matit{Op}}$. It has the same objects, but the type of arrows are flipped,
|
||||
$\bC^{\mathit{Op}}$. It has the same objects, but the type of arrows are flipped,
|
||||
that is to say an arrow from $A$ to $B$ in the opposite category corresponds to
|
||||
an arrow from $B$ to $A$ in the underlying category. The identity arrow is the
|
||||
same as the one in the underlying category (they have the same type). Function
|
||||
|
|
@ -442,7 +442,7 @@ Since $\rrr$ is reverse function composition this is just the symmetric version
|
|||
of associativity.
|
||||
%
|
||||
$$
|
||||
\matit{identity} \rrr f \equiv f \x f \rrr identity \equiv f
|
||||
\mathit{identity} \rrr f \equiv f \x f \rrr identity \equiv f
|
||||
$$
|
||||
%
|
||||
This is just the swapped version of identity.
|
||||
|
|
@ -452,7 +452,7 @@ arguments. Or in other words since $\Hom_{A\ B}$ is a set for all $A\ B \tp
|
|||
\Object$ then so is $\Hom_{B\ A}$.
|
||||
|
||||
Now, to show that this category is univalent is not as straight-forward. Lucliy
|
||||
section \ref{equiv} gave us some tools to work with equivalences. We saw that we
|
||||
section \ref{sec:equiv} gave us some tools to work with equivalences. We saw that we
|
||||
can prove this category univalent by giving an inverse to
|
||||
$\idToIso_{\mathit{Op}} \tp (A \equiv B) \to (A \approxeq_{\mathit{Op}} B)$.
|
||||
From the original category we have that $\idToIso \tp (A \equiv B) \to (A \cong
|
||||
|
|
@ -500,7 +500,7 @@ This finished the proof that the opposite category is in fact a category. Now,
|
|||
to prove that that opposite-of is an involution we must show:
|
||||
%
|
||||
$$
|
||||
\prod_{\bC \tp \mathit{Category}} \left(\bC^{\matit{Op}}\right)^{\matit{Op}} \equiv \bC
|
||||
\prod_{\bC \tp \mathit{Category}} \left(\bC^{\mathit{Op}}\right)^{\mathit{Op}} \equiv \bC
|
||||
$$
|
||||
%
|
||||
As we've seen the laws in $\left(\bC^{\mathit{Op}}\right)^{\mathit{Op}}$ get
|
||||
|
|
@ -543,20 +543,23 @@ $$
|
|||
%
|
||||
Which, as we saw in section \ref{univalence}, is sufficient to show that the
|
||||
category is univalent. The way that I have shown this is with a three-step
|
||||
process. For objects $(A, s_A)\; (B, s_B) \tp \Set$ I show that.
|
||||
process. For objects $(A, s_A)\; (B, s_B) \tp \Set$ I show the following chain
|
||||
of equivalences:
|
||||
%
|
||||
\begin{align*}
|
||||
((A, s_A) \equiv (B, s_B)) & \simeq (A \equiv B) \\
|
||||
(A \equiv B) & \simeq (\fst A \simeq \fst B) \\
|
||||
(A \simeq B) & \simeq ((A, s_A) \approxeq (B, s_B))
|
||||
((A, s_A) \equiv (B, s_B))
|
||||
& \simeq (A \equiv B) && \ref{eq:equivPropSig} \\
|
||||
& \simeq (A \simeq B) && \text{Univalence} \\
|
||||
& \simeq ((A, s_A) \approxeq (B, s_B)) && \text{\ref{eq:equivSig} and \ref{eq:equivIso}}
|
||||
\end{align*}
|
||||
|
||||
And since $\simeq$ is an equivalence relation we can chain these equivalences
|
||||
together. Step one will be proven with the following lemma:
|
||||
%
|
||||
$$
|
||||
\begin{align}
|
||||
\label{eq:equivPropSig}
|
||||
\left(\prod_{a \tp A} \isProp (P\ a)\right) \to \prod_{x\;y \tp \sum_{a \tp A} P\ a} (x \equiv y) \simeq (\fst\ x \equiv \fst\ y)
|
||||
$$
|
||||
\end{align}
|
||||
%
|
||||
The lemma states that for pairs whose second component are mere propositions
|
||||
equiality is equivalent to equality of the first components. In this case the
|
||||
|
|
@ -564,21 +567,105 @@ type-family $P$ is $\isSet$ which itself is a proposition for any type $A \tp
|
|||
\Type$. Step two is univalence. Step three will be proven with the following
|
||||
lemma:
|
||||
%
|
||||
$$
|
||||
\begin{align}
|
||||
\label{eq:equivSig}
|
||||
\prod_{a \tp A} \left( P\ a \simeq Q\ a \right) \to \sum_{a \tp A} P\ a \simeq \sum_{a \tp A} Q\ a
|
||||
$$
|
||||
\end{align}
|
||||
%
|
||||
Which says that if two type-families are equivalent at all points, then pairs
|
||||
with identitical first components and these families as second components will
|
||||
also be equivalent. For our purposes $P \defeq \isEquiv\ A\ B$ and $Q \defeq
|
||||
\mathit{Isomorphism}$. So we must finally prove:
|
||||
%
|
||||
$$
|
||||
\begin{align}
|
||||
\label{eq:equivIso}
|
||||
\prod_{f \tp A \to B} \left( \isEquiv\ A\ B\ f \simeq \mathit{Isomorphism}\ f \right)
|
||||
$$
|
||||
\end{align}
|
||||
|
||||
First, lets proove \ref{eq:equivPropSig}: Let $propP \tp \prod_{a \tp A} \isProp (P\ a)$ and $x\;y \tp \sum_{a \tp A} P\ a$ be given. Because
|
||||
of $\mathit{fromIsomorphism}$ it suffices to give an isomorphism between
|
||||
$x \equiv y$ and $\fst\ x \equiv \fst\ y$:
|
||||
%
|
||||
\begin{align*}
|
||||
f & \defeq \congruence\ \fst \tp x \equiv y \to \fst\ x \equiv \fst\ y \\
|
||||
g & \defeq \mathit{lemSig}\ \mathit{propP}\ x\ y \tp \fst\ x \equiv \fst\ y \to x \equiv y
|
||||
\end{align*}
|
||||
%
|
||||
\TODO{Is it confusing that I use point-free style here?}
|
||||
Here $\mathit{lemSig}$ is a lemma that says that if the second component of a
|
||||
pair is a proposition, it suffices to give a path between it's first components
|
||||
to construct an equality of the two pairs:
|
||||
%
|
||||
\begin{align*}
|
||||
\mathit{lemSig} \tp \left( \prod_{x \tp A} \isProp\ (B\ x) \right) \to
|
||||
\prod_{u\; v \tp \sum_{a \tp A} B\ a}
|
||||
\left( \fst\ u \equiv \fst\ v \right) \to u \equiv v
|
||||
\end{align*}
|
||||
%
|
||||
The proof that these are indeed inverses has been omitted. \TODO{Do I really
|
||||
want to ommit it?}\QED
|
||||
|
||||
\subsection{Categories}
|
||||
Now to prove \ref{eq:equivSig}: Let $e \tp \prod_{a \tp A} \left( P\ a \simeq
|
||||
Q\ a \right)$ be given. To prove the equivalence, it suffices to give an
|
||||
isomorphism between $\sum_{a \tp A} P\ a$ and $\sum_{a \tp A} Q\ a$, but since
|
||||
they have identical first components it suffices to give an isomorphism between
|
||||
$P\ a$ and $Q\ a$ for all $a \tp A$. This is exactly what we can get from
|
||||
the equivalence $e$.\QED
|
||||
|
||||
Lastly we prove \ref{eq:equivIso}. Let $f \tp A \to B$ be given. For the maps we
|
||||
choose:
|
||||
%
|
||||
\begin{align*}
|
||||
\mathit{toIso}
|
||||
& \tp \isEquiv\ f \to \mathit{Isomorphism}\ f \\
|
||||
\mathit{fromIso}
|
||||
& \tp \mathit{Isomorphism}\ f \to \isEquiv\ f
|
||||
\end{align*}
|
||||
%
|
||||
As mentioned in section \ref{sec:equiv}. These maps are not in general inverses
|
||||
of each other. In stead, we will use the fact that $A$ and $B$ are sets. The first thing we must prove is:
|
||||
%
|
||||
\begin{align*}
|
||||
\mathit{fromIso} \comp \mathit{toIso} \equiv \identity_{\isEquiv\ f}
|
||||
\end{align*}
|
||||
%
|
||||
For this we can use the fact that being-an-equivalence is a mere proposition.
|
||||
For the other direction:
|
||||
%
|
||||
\begin{align*}
|
||||
\mathit{toIso} \comp \mathit{fromIso} \equiv \identity_{\mathit{Isomorphism}\ f}
|
||||
\end{align*}
|
||||
%
|
||||
We will show that $\mathit{Isomorphism}\ f$ is also a mere proposition. To this
|
||||
end, let $X\;Y \tp \mathit{Isomorphism}\ f$ be given. Name the maps $x\;y \tp B
|
||||
\to A$ respectively. Now, the proof that $X$ and $Y$ are the same is a pair of
|
||||
paths: $p \tp x \equiv y$ and $\Path\ (\lambda\; i \mto
|
||||
\mathit{AreInverses}\ f\ (p\ i))\ \mathcal{X}\ \mathcal{Y}$ where $\mathcal{X}$
|
||||
and $\mathcal{Y}$ denotes the witnesses that $x$ (respectively $y$) is an
|
||||
inverse to $f$. $p$ is inhabited by:
|
||||
%
|
||||
\begin{align*}
|
||||
x
|
||||
& \equiv x \comp \identity \\
|
||||
& \equiv x \comp (f \comp y)
|
||||
&& \text{$y$ is an inverse to $f$} \\
|
||||
& \equiv (x \comp f) \comp y \\
|
||||
& \equiv \identity \comp y
|
||||
&& \text{$x$ is an inverse to $f$} \\
|
||||
& \equiv y
|
||||
\end{align*}
|
||||
%
|
||||
For the other (dependent) path we can prove that being-an-inverse-of is a
|
||||
proposition and then use $\lemPropF$. So we prove the generalization:
|
||||
%
|
||||
\begin{align*}
|
||||
\prod_{g : B \to A} \isProp\ (\mathit{AreInverses}\ f\ g)
|
||||
\end{align*}
|
||||
%
|
||||
But $\mathit{AreInverses}\ f\ g$ is a pair of equations on arrows, so we use
|
||||
$\propSig$ and the fact that both $A$ and $B$ are sets to close this proof.
|
||||
|
||||
\subsection{Category of 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.
|
||||
|
||||
|
|
@ -589,8 +676,40 @@ 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.
|
||||
|
||||
|
||||
\section{Product}
|
||||
In the following I'll demonstrate a technique for using categories to prove
|
||||
properties. The goal in this section is to show that products are propositions:
|
||||
%
|
||||
$$
|
||||
\prod_{\bC \tp \Category} \prod_{A\;B \tp \Object} \isProp\ (\mathit{Product}\ \bC\ A\ B)
|
||||
$$
|
||||
%
|
||||
Where $\mathit{Product}\ \bC\ A\ B$ denotes the type of products of objects $A$
|
||||
and $B$ in the category $\bC$. I do this by constructing a category whose
|
||||
terminal objects are equivalent to products in $\bC$, and since terminal objects
|
||||
are propositional in a proper category and equivalences preservehomotopy level,
|
||||
then we know that products also are propositions. But before we get to that,
|
||||
let's recall the definition of products.
|
||||
|
||||
Given a category $\bC$ and two objects $A$ and $B$ in $bC$ we define the product
|
||||
of $A$ and $B$ to be an object $A \x B$ in $\bC$ and two arrows $\pi_1 \tp A \x
|
||||
B \to A$ and $\pi_2 \tp A \x B \to B$ called the projections of the product. The projections must satisfy the following property:
|
||||
|
||||
For all $X \tp Object$, $f \tp \Arrow\ X\ A$ and $g \tp \Arrow\ X\ B$ we have
|
||||
that there exists a unique arrow $\pi \tp \Arrow\ X\ (A \x B)$ satisfying
|
||||
%
|
||||
\begin{align}
|
||||
\label{eq:umpProduct}
|
||||
%% \prod_{X \tp Object} \prod_{f \tp \Arrow\ X\ A} \prod_{g \tp \Arrow\ X\ B}\\
|
||||
%% \uexists_{f \x g \tp \Arrow\ X\ (A \x B)}
|
||||
\pi_1 \lll \pi \equiv f \x \pi_2 \lll \pi \equiv g
|
||||
%% ump : ∀ {X : Object} (f : ℂ [ X , A ]) (g : ℂ [ X , B ])
|
||||
%% → ∃![ f×g ] (ℂ [ fst ∘ f×g ] ≡ f P.× ℂ [ snd ∘ f×g ] ≡ g)
|
||||
\end{align*}
|
||||
$
|
||||
$\pi$ is called the product (arrow) of $f$ and $g$.
|
||||
|
||||
|
||||
\section{Monads}
|
||||
|
||||
%% \subsubsection{Functors}
|
||||
|
|
|
|||
|
|
@ -73,12 +73,13 @@ the (left) identity law of the underlying category to proove $\idFun \comp g
|
|||
%
|
||||
\subsection{Equality of isomorphic types}
|
||||
%
|
||||
Let $\top$ denote the unit type -- a type with a single constructor. In the
|
||||
propositions-as-types interpretation of type theory $\top$ is the proposition
|
||||
that is always true. The type $A \x \top$ and $A$ has an element for each $a :
|
||||
A$. So in a sense they are the same. The second element of the pair does not add
|
||||
any ``interesting information''. It can be useful to identify such types. In
|
||||
fact, it is quite commonplace in mathematics. Say we look at a set $\{x \mid
|
||||
Let $\top$ denote the unit type -- a type with a single constructor. In
|
||||
the propositions-as-types interpretation of type theory $\top$ is the
|
||||
proposition that is always true. The type $A \x \top$ and $A$ has an element for
|
||||
each $a : A$. So in a sense they are the same. The second element of the pair
|
||||
does not add any ``interesting information''. It can be useful to identify such
|
||||
types. In fact, it is quite commonplace in mathematics. Say we look at a set
|
||||
$\{x \mid
|
||||
\phi\ x \land \psi\ x\}$ and somehow conclude that $\psi\ x \equiv \top$ for all
|
||||
$x$. A mathematician would immediately conclude $\{x \mid \phi\ x \land
|
||||
\psi\ x\} \equiv \{x \mid \phi\ x\}$ without thinking twice. Unfortunately such
|
||||
|
|
@ -92,8 +93,9 @@ types. The principle of univalence says that:
|
|||
%
|
||||
$$\mathit{univalence} \tp (A \simeq B) \simeq (A \equiv B)$$
|
||||
%
|
||||
In particular this allows us to construct an equality from an equivalence $\mathit{ua} \tp
|
||||
(A \simeq B) \to (A \equiv B)$ and vice-versa.
|
||||
In particular this allows us to construct an equality from an equivalence
|
||||
($\mathit{ua} \tp (A \simeq B) \to (A \equiv B)$) and vice-versa.
|
||||
|
||||
\section{Formalizing Category Theory}
|
||||
%
|
||||
The above examples serve to illustrate the limitation of Agda. One case where
|
||||
|
|
@ -115,20 +117,21 @@ Inspiration:
|
|||
\end{verbatim}
|
||||
The idea of formalizing Category Theory in proof assistants is not new. There
|
||||
are a multitude of these available online. Just as first reference see this
|
||||
question on Math Overflow: \cite{mo-formalizations}. Notably these two implementations of category theory in Agda:
|
||||
question on Math Overflow: \cite{mo-formalizations}. Notably these
|
||||
implementations of category theory in Agda:
|
||||
\begin{itemize}
|
||||
\item
|
||||
\url{https://github.com/copumpkin/categories} - setoid interpretation
|
||||
\url{https://github.com/copumpkin/categories} -- setoid interpretation
|
||||
\item
|
||||
\url{https://github.com/pcapriotti/agda-categories} - homotopic setting with postulates
|
||||
\url{https://github.com/pcapriotti/agda-categories} -- homotopic setting with postulates
|
||||
\item
|
||||
\url{https://github.com/pcapriotti/agda-categories} - homotopic setting in coq
|
||||
\url{https://github.com/pcapriotti/agda-categories} -- homotopic setting in coq
|
||||
\item
|
||||
\url{https://github.com/mortberg/cubicaltt} - homotopic setting in \texttt{cubicaltt}
|
||||
\url{https://github.com/mortberg/cubicaltt} -- homotopic setting in \texttt{cubicaltt}
|
||||
\end{itemize}
|
||||
The contribution of this
|
||||
thesis is to explore how working in a cubical setting will make it possible to
|
||||
prove more things and to reuse proofs.
|
||||
|
||||
The contribution of this thesis is to explore how working in a cubical setting
|
||||
will make it possible to prove more things and to reuse proofs.
|
||||
|
||||
There are alternative approaches to working in a cubical setting where one can
|
||||
still have univalence and functional extensionality. One option is to postulate
|
||||
|
|
|
|||
|
|
@ -59,3 +59,9 @@
|
|||
\newcommand\rrr{\ggg}
|
||||
\newcommand\fst{\mathit{fst}}
|
||||
\newcommand\snd{\mathit{snd}}
|
||||
\newcommand\Path{\mathit{Path}}
|
||||
\newcommand\Category{\mathit{Category}}
|
||||
\newcommand\TODO[1]{TODO: \emph{#1}}
|
||||
\newcommand*{\QED}{\hfill\ensuremath{\square}}%
|
||||
\newcommand\uexists{\exists!}
|
||||
\newcommand\Arrow{\mathit{Arrow}}
|
||||
|
|
|
|||
|
|
@ -26,8 +26,8 @@
|
|||
\bibliographystyle{plainnat}
|
||||
\nocite{cubical-demo}
|
||||
\nocite{coquand-2013}
|
||||
%% \bibliography{refs}
|
||||
%% \begin{appendices}
|
||||
\bibliography{refs}
|
||||
\begin{appendices}
|
||||
%% \input{planning.tex}
|
||||
%% \input{halftime.tex}
|
||||
\end{appendices}
|
||||
|
|
|
|||
|
|
@ -10,7 +10,7 @@ open import Cubical.GradLemma hiding (isoToPath)
|
|||
open import Cat.Prelude using
|
||||
( lemPropF ; setPi ; lemSig ; propSet
|
||||
; Preorder ; equalityIsEquivalence ; propSig ; id-coe
|
||||
; Setoid )
|
||||
; Setoid ; _$_ ; propPi )
|
||||
|
||||
import Cubical.Univalence as U
|
||||
|
||||
|
|
@ -133,7 +133,7 @@ module _ {ℓa ℓb ℓ : Level} (A : Set ℓa) (B : Set ℓb) where
|
|||
-- | The other inverse law does not hold in general, it does hold, however,
|
||||
-- | if `A` and `B` are sets.
|
||||
module _ (sA : isSet A) (sB : isSet B) where
|
||||
module _ {f : A → B} (iso : Isomorphism f) where
|
||||
module _ {f : A → B} 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)
|
||||
|
|
@ -146,22 +146,18 @@ module _ {ℓa ℓb ℓ : Level} (A : Set ℓa) (B : Set ℓb) where
|
|||
y ∎
|
||||
|
||||
propInv : ∀ g → isProp (AreInverses f g)
|
||||
propInv g t u i = a i , b i
|
||||
propInv g t u = λ i → a i , b i
|
||||
where
|
||||
a : (fst t) ≡ (fst u)
|
||||
a i = h
|
||||
a i = funExt hh
|
||||
where
|
||||
hh : ∀ a → (g ∘ f) a ≡ a
|
||||
hh a = sA ((g ∘ f) a) a (λ i → (fst t) i a) (λ i → (fst u) i a) i
|
||||
h : g ∘ f ≡ idFun A
|
||||
h i a = hh a i
|
||||
b : (snd t) ≡ (snd u)
|
||||
b i = h
|
||||
b i = funExt hh
|
||||
where
|
||||
hh : ∀ b → (f ∘ g) b ≡ b
|
||||
hh b = sB _ _ (λ i → snd t i b) (λ i → snd u 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
|
||||
|
|
@ -169,11 +165,12 @@ module _ {ℓa ℓb ℓ : Level} (A : Set ℓa) (B : Set ℓb) where
|
|||
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 ∎
|
||||
module _ (iso : Isomorphism f) where
|
||||
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
|
||||
|
|
@ -419,7 +416,7 @@ module _ {ℓa ℓb : Level} {A : Set ℓa} {P : A → Set ℓb} where
|
|||
equivPropSig pA p q = fromIsomorphism _ _ iso
|
||||
where
|
||||
f : ∀ {p q} → p ≡ q → fst p ≡ fst q
|
||||
f e i = fst (e i)
|
||||
f = cong fst
|
||||
g : ∀ {p q} → fst p ≡ fst q → p ≡ q
|
||||
g {p} {q} = lemSig pA p q
|
||||
ve-re : (e : p ≡ q) → (g ∘ f) e ≡ e
|
||||
|
|
@ -507,31 +504,26 @@ module _ {ℓa ℓb : Level} {A : Set ℓa} {P : A → Set ℓb} where
|
|||
→ ((a : A) → P a ≃ Q a) → Σ A P ≃ Σ A Q
|
||||
equivSig {Q = Q} eA = res
|
||||
where
|
||||
P≅Q : ∀ {a} → P a ≅ Q a
|
||||
P≅Q {a} = toIsomorphism _ _ (eA a)
|
||||
f : Σ A P → Σ A Q
|
||||
f (a , pA) = a , fst (eA a) pA
|
||||
f (a , pA) = a , fst P≅Q pA
|
||||
g : Σ A Q → Σ A P
|
||||
g (a , qA) = a , g' qA
|
||||
where
|
||||
k : Isomorphism _
|
||||
k = toIso _ _ (snd (eA a))
|
||||
open Σ k renaming (fst to g')
|
||||
g (a , qA) = a , fst (snd P≅Q) qA
|
||||
ve-re : (x : Σ A P) → (g ∘ f) x ≡ x
|
||||
ve-re x i = fst x , eq i
|
||||
ve-re (a , pA) i = a , eq i
|
||||
where
|
||||
eq : snd ((g ∘ f) x) ≡ snd x
|
||||
eq : snd ((g ∘ f) (a , pA)) ≡ pA
|
||||
eq = begin
|
||||
snd ((g ∘ f) x) ≡⟨⟩
|
||||
snd ((g ∘ f) (a , pA)) ≡⟨⟩
|
||||
snd (g (f (a , pA))) ≡⟨⟩
|
||||
g' (fst (eA a) pA) ≡⟨ lem ⟩
|
||||
pA ∎
|
||||
where
|
||||
open Σ x renaming (fst to a ; snd to pA)
|
||||
k : Isomorphism _
|
||||
k = toIso _ _ (snd (eA a))
|
||||
open Σ k renaming (fst to g' ; snd to inv)
|
||||
open Σ (snd P≅Q) renaming (fst to g' ; snd to inv)
|
||||
-- anti-funExt
|
||||
lem : (g' ∘ (fst (eA a))) pA ≡ pA
|
||||
lem i = fst inv i pA
|
||||
lem = cong (_$ pA) (fst (snd (snd P≅Q)))
|
||||
re-ve : (x : Σ A Q) → (f ∘ g) x ≡ x
|
||||
re-ve x i = fst x , eq i
|
||||
where
|
||||
|
|
|
|||
|
|
@ -62,10 +62,8 @@ module _ (ℓ : Level) where
|
|||
syntax ∃!-syntax (λ x → B) = ∃![ x ] B
|
||||
|
||||
module _ {ℓa ℓb} {A : Set ℓa} {P : A → Set ℓb} (f g : ∃! P) where
|
||||
open Σ (snd f) renaming (snd to u)
|
||||
|
||||
∃-unique : fst f ≡ fst g
|
||||
∃-unique = u (fst (snd g))
|
||||
∃-unique = (snd (snd f)) (fst (snd g))
|
||||
|
||||
module _ {ℓa ℓb : Level} {A : Set ℓa} {B : A → Set ℓb} {a b : Σ A B}
|
||||
(fst≡ : (λ _ → A) [ fst a ≡ fst b ])
|
||||
|
|
|
|||
Loading…
Reference in New Issue