Remove some TODO-notes, add section on motifs
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@ -1,22 +1,24 @@
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\chapter{Conclusion}
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This thesis highlighted some issues with the standard inductive definition of
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propositional equality used in Agda. Functional extensionality and univalence
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are examples of two propositions not admissible in Intensional Type Theory
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(ITT). This has a big impact on what is provable and the reusability of proofs.
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This issue is overcome with an extension to Agda's type system called Cubical
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Agda. With Cubical Agda both functional extensionality and univalence are
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admissible. Cubical Agda is more expressive, but there are certain issues that
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arise that are not present in standard Agda. For one thing ITT and standard Agda
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enjoys Uniqueness of Identity Proofs (UIP). This is not the case in Cubical
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Agda. In stead there exists a hierarchy of types with increasing
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\nomen{homotopical structure}{homotopy levels}. It turns out to be useful to built the
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formalization with this hierarchy in mind as it can simplify proofs
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considerably. Another issue one must overcome in Cubical Agda is when a type has
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a field whose type depends on a previous field. In this case paths between such
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types will be heterogeneous paths. This problem is related to Cubical Agda not
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having the K-rule \TODO{Not mentioned anywhere in the report}. In practice it
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turns out to be considerably more difficult to work heterogeneous paths than
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with homogeneous paths. The thesis demonstrated some techniques to overcome
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This thesis highlighted some issues with the standard inductive
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definition of propositional equality used in Agda. Functional
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extensionality and univalence are examples of two propositions not
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admissible in Intensional Type Theory (ITT). This has a big impact on
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what is provable and the reusability of proofs. This issue is overcome
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with an extension to Agda's type system called Cubical Agda. With
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Cubical Agda both functional extensionality and univalence are
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admissible. Cubical Agda is more expressive, but there are certain
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issues that arise that are not present in standard Agda. For one thing
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ITT and standard Agda enjoys Uniqueness of Identity Proofs (UIP). This
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is not the case in Cubical Agda. In stead there exists a hierarchy of
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types with increasing \nomen{homotopical structure}{homotopy levels}.
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It turns out to be useful to built the formalization with this
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hierarchy in mind as it can simplify proofs considerably. Another
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issue one must overcome in Cubical Agda is when a type has a field
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whose type depends on a previous field. In this case paths between
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such types will be heterogeneous paths. This problem is related to
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Cubical Agda not having the K-rule. In practice it turns out to be
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considerably more difficult to work heterogeneous paths than with
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homogeneous paths. The thesis demonstrated some techniques to overcome
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these difficulties, such as based path-induction.
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This thesis formalized some of the core concepts from category theory including;
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@ -54,17 +54,18 @@ Judgmental equality in Cubical Agda is encapsulated with the type:
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\Path \tp (P \tp I → \MCU) → P\ 0 → P\ 1 → \MCU
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\end{equation}
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%
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$I$ is a special data type (\TODO{that also has special computational properties
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AFAIK}) called the index set. $I$ can be thought of simply as the interval on
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the real numbers from $0$ to $1$. $P$ is a family of types over the index set
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$I$. I will sometimes refer to $P$ as the \nomenindex{path space} of some path $p \tp
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\Path\ P\ a\ b$. By this token $P\ 0$ then corresponds to the type at the
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left-endpoint and $P\ 1$ as the type at the right-endpoint. The type is called
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$\Path$ because it is connected with paths in homotopy theory. The intuition
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behind this is that $\Path$ describes paths in $\MCU$ -- i.e.\ between types. For
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a path $p$ for the point $p\ i$ the index $i$ describes how far along the path
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one has moved. An inhabitant of $\Path\ P\ a_0\ a_1$ is a (dependent-) function,
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$p$, from the index-space to the path space:
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$I$ is a special data type called the index set. $I$ can be thought of
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simply as the interval on the real numbers from $0$ to $1$. $P$ is a
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family of types over the index set $I$. I will sometimes refer to $P$
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as the \nomenindex{path space} of some path $p \tp \Path\ P\ a\ b$. By
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this token $P\ 0$ then corresponds to the type at the left-endpoint
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and $P\ 1$ as the type at the right-endpoint. The type is called
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$\Path$ because it is connected with paths in homotopy theory. The
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intuition behind this is that $\Path$ describes paths in $\MCU$ --
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i.e.\ between types. For a path $p$ for the point $p\ i$ the index $i$
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describes how far along the path one has moved. An inhabitant of
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$\Path\ P\ a_0\ a_1$ is a (dependent-) function, $p$, from the
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index-space to the path space:
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%
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$$
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p \tp \prod_{i \tp I} P\ i
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@ -194,7 +195,7 @@ indeed both $\top$ and $\bot$ are propositions:
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The term $\varnothing$ is used here to denote an impossible pattern. It is a
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theorem that if a mere proposition $A$ is inhabited, then so is it contractible.
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If it is not inhabited it is equivalent to the empty-type (or false
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proposition).\TODO{Cite!!}
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proposition).\TODO{Cite}
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I will refer to a type $A \tp \MCU$ as a \emph{mere} proposition if I want to
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stress that we have $\isProp\ A$.
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@ -242,11 +243,16 @@ Let $\left\Vert A \right\Vert = n$ denote that the level of $A$ is $n$.
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Proposition: For any homotopic level $n$ this is a mere proposition.
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%
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\section{A few lemmas}
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Rather than getting into the nitty-gritty details of Agda I venture to take a
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more ``combinator-based'' approach. That is, I will use theorems about paths
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already that have already been formalized. Specifically the results come from
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the Agda library \texttt{cubical} (\TODO{Cite}). I have used a handful of
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results from this library as well as contributed a few lemmas myself.\footnote{The module \texttt{Cat.Prelude} lists the upstream dependencies. As well my contribution to \texttt{cubical} can be found in the git logs \TODO{Cite}.}
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Rather than getting into the nitty-gritty details of Agda I venture to
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take a more ``combinator-based'' approach. That is, I will use
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theorems about paths already that have already been formalized.
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Specifically the results come from the Agda library \texttt{cubical}
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(\cite{cubical-demo}). I have used a handful of results from this
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library as well as contributed a few lemmas myself.\footnote{The
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module \texttt{Cat.Prelude} lists the upstream dependencies. As well
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my contribution to \texttt{cubical} can be found in the git logs
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which are available at
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\hrefsymb{https://github.com/Saizan/cubical-demo}{\texttt{https://github.com/Saizan/cubical-demo}}.}
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These theorems are all purely related to homotopy theory and cubical Agda and as
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such not specific to the formalization of Category Theory. I will present a few
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@ -97,10 +97,25 @@ a significant impact on the complexity of these kinds of proofs.
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\TODO{Universe levels.}
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\subsection{Motifs}
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An oft-used technique in this development is using based path
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induction to prove certain properties. One particular challenge that
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arises when doing so is that Agda is not able to automatically infer
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the family that one wants to do induction over. For instance in the
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proof $\var{sym}\ (\var{sym}\ p) ≡ p$ from \ref{eq:sym-invol} the
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family that we chose to do induction over was $D\ b'\ p' \defeq
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\var{sym}\ (\var{sym}\ p') ≡ p'$. However, if one interactively tries
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to give this hole, all the information that Agda can provide is that
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one must provide an element of $𝒰$. Agda could be more helpful in this
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context, perhaps even infer this family in some situations. In this
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very simple example this is of course not a big problem, but there are
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examples in the source code where this gets more involved.
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\section{Future work}
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\subsection{Agda \texttt{Prop}}
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Jesper Cockx' work extending the universe-level-laws for Agda and the
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\texttt{Prop}-type.
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\TODO{Do I want to include this?}
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\subsection{Compiling Cubical Agda}
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\label{sec:compiling-cubical-agda}
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@ -117,3 +132,8 @@ library exposes could provide one particularly interesting case-study.
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\subsection{Higher inductive types}
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This library has not explored the usefulness of higher inductive types
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in the context of Category Theory.
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\subsection{Initiality conjecture}
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A fellow student here at Chalmers, Andreas Källberg, is currently
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working on proving the initiality conjecture\TODO{Citation}. He will
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be using this library to do so.
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@ -277,16 +277,19 @@ $i$ becomes the triple:
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\end{aligned}
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\end{equation}
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%
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I have found this to be a general pattern when proving things in homotopy type
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theory, namely that you have to wrap and unwrap equalities at different levels.
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It is worth noting that proving this theorem with the regular inductive equality
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type would already not be possible, since we at least need extensionality (the
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projections are all $\prod$-types). Assuming we had functional extensionality
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available to us as an axiom, we would use functional extensionality \TODO{in
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reverse?} to retrieve the equalities in $a$ and $b$, pattern-match on them to
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see that they are both $\refl$ and then close the proof with $\refl$.
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Of course this theorem is not so interesting in the setting of ITT since we know
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a priori that equality proofs are unique.
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I have found this to be a general pattern when proving things in
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homotopy type theory, namely that you have to wrap and unwrap
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equalities at different levels. It is worth noting that proving this
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theorem with the regular inductive equality type would already not be
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possible, since we at least need functional
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extensionality\index{functional extensionality} (the projections are
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all $\prod$-types). Assuming we had functional extensionality
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available to us as an axiom, we would use functional extensionality
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\TODO{in reverse?} to retrieve the equalities in $a$ and $b$,
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pattern-match on them to see that they are both $\refl$ and then close
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the proof with $\refl$. Of course this theorem is not so interesting
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in the setting of ITT since we know a priori that equality proofs are
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unique.
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The situation is a bit more complicated when we have a dependent type.
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For instance, when we want to show that $\IsCategory$ is a mere
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@ -326,8 +329,8 @@ $$
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And this finishes the proof that being-a-category is a mere proposition
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(\ref{eq:propIsPreCategory}).
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When we have a proper category we can make precise the notion of ``identifying
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isomorphic types'' \TODO{cite Awodey here}. That is, we can construct the
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When we have a proper category we can make precise the notion of
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``identifying isomorphic types''. That is, we can construct the
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function:
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%
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$$
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@ -345,7 +348,11 @@ terminal objects are propositional:
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It follows from the usual observation that any two terminal objects are
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isomorphic - and since categories are univalent, so are they equal. The proof is
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omitted here, but the curious reader can check the implementation for the
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details. \TODO{The proof is a bit fun, should I include it?}
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details. It is in the module:
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%
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\begin{center}
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\sourcelink{Cat.Category}
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\end{center}
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\section{Equivalences}
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\label{sec:equiv}
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@ -733,7 +740,7 @@ $x \equiv y$ and $\fst\ x \equiv \fst\ y$:
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\end{aligned}
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\end{equation*}
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%
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\TODO{Is it confusing that I use point-free style here?} Here $\var{lemSig}$ is
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\TODO{Is it confusing that I use point-free style here?}Here $\var{lemSig}$ is
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a lemma that says that if the second component of a pair is a proposition, it
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suffices to give a path between its first components to construct an equality of
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the two pairs:
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@ -744,15 +751,18 @@ the two pairs:
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\left( \fst\ u \equiv \fst\ v \right) \to u \equiv v
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\end{align*}
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%
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The proof that these are indeed inverses has been omitted. \TODO{Do I really
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want to omit it?}\QED
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The proof that these are indeed inverses has been omitted. The details
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can be found in the module:
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\begin{center}
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\sourcelink{Cat.Categories.Sets}
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\end{center}
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Now to prove \ref{eq:equivSig}: Let $e \tp \prod_{a \tp A} \left( P\ a \simeq
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Q\ a \right)$ be given. To prove the equivalence, it suffices to give an
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isomorphism between $\sum_{a \tp A} P\ a$ and $\sum_{a \tp A} Q\ a$, but since
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they have identical first components it suffices to give an isomorphism between
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$P\ a$ and $Q\ a$ for all $a \tp A$. This is exactly what we can get from
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the equivalence $e$.\QED
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Now to prove \ref{eq:equivSig}: Let $e \tp \prod_{a \tp A} \left( P\ a
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\simeq Q\ a \right)$ be given. To prove the equivalence, it suffices
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to give an isomorphism between $\sum_{a \tp A} P\ a$ and $\sum_{a \tp
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A} Q\ a$, but since they have identical first components it suffices
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to give an isomorphism between $P\ a$ and $Q\ a$ for all $a \tp A$.
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This is exactly what we can get from the equivalence $e$.\QED
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Lastly we prove \ref{eq:equivIso}. Let $f \tp A \to B$ be given. For the maps we
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choose:
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@ -1050,7 +1060,10 @@ $b_0, b_1$ corresponds to a pair of paths between $a_0,b_0$ and $a_1,b_1$ (check
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the implementation for the details).
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\emph{Proposition} \ref{eq:univ-1} is isomorphic to \ref{eq:univ-2}:
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\TODO{Super complicated}
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This proof of this has been omitted but can be found in the module:
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\begin{center}
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\sourcelink{Cat.Categories.Span}
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\end{center}
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\emph{Proposition} \ref{eq:univ-2} is isomorphic to \ref{eq:univ-3}: For this I
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will show two corollaries of \ref{eq:coeCod}: For an isomorphism $(\iota,
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g & \equiv f \lll \inv{\iota}
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\end{align}
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%
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Proof: \TODO{\ldots}
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The proof is omitted but can be found in the module:
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\begin{center}
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\sourcelink{Cat.Category}
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\end{center}
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Now we can prove the equivalence in the following way: Given $(f, \inv{f},
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\var{inv}_f) \tp X \cong Y$ and two heterogeneous paths
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@ -207,14 +207,13 @@ implementations of category theory in Agda:
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%
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The contribution of this thesis is to explore how working in a cubical setting
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will make it possible to prove more things and to reuse proofs and to try and
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compare some aspects of this formalization with the existing ones.\TODO{How can
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I live up to this?}
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compare some aspects of this formalization with the existing ones.
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There are alternative approaches to working in a cubical setting where
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one can still have univalence and functional extensionality. One
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option is to postulate these as axioms. This approach, however, has
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other shortcomings, e.g. you lose \nomenindex{canonicity}
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(\TODO{Pageno!} \cite{huber-2016}).
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(\cite[p. 3]{huber-2016}).
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Another approach is to use the \emph{setoid interpretation} of type
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theory (\cite{hofmann-1995,huber-2016}). With this approach one works
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@ -227,16 +226,17 @@ relation a priori. So this must be verified manually by the developer.
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Furthermore, functions between different setoids must be shown to be
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setoid homomorphism, that is; they preserve the relation.
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This approach has other drawbacks; it does not satisfy
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all propositional equalities of type theory (\TODO{Citation needed}), is
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cumbersome to work with in practice (\cite[p. 4]{huber-2016}) and makes
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equational proofs less reusable since equational proofs $a \sim_{X} b$ are
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inherently `local' to the extensional set $(X , \sim)$.
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This approach has other drawbacks; it does not satisfy all
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propositional equalities of type theory a priori. That is, the
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developer must manually show that e.g.\ the relation is a congruence.
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Equational proofs $a \sim_{X} b$ are in some sense `local' to the
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extensional set $(X , \sim)$. To e.g.\ prove that $x ∼ y → f\ x ∼
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f\ y$ for some function $f \tp A → B$ between two extensional sets $A$
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and $B$ it must be shown that $f$ is a groupoid homomorphism. This
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makes it very cumbersome to work with in practice (\cite[p.
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4]{huber-2016}).
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\section{Conventions}
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\TODO{Talk a bit about terminology. Find a good place to stuff this little
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section.}
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In the remainder of this paper I will use the term
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\nomenindex{Type} to describe --
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well, types. Thereby diverging from the notation in Agda where the keyword
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