Various changes proposed by Andreas
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html/
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Makefile
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Makefile
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@ -3,3 +3,9 @@ build: src/**.agda
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clean:
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find src -name "*.agdai" -type f -delete
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html:
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agda --html src/Cat.agda
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upload: html
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scp -r html/ remote11.chalmers.se:www/cat/doc/
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@ -1,21 +1,23 @@
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\chapter*{Abstract}
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The usual notion of propositional equality in intensional type-theory is
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restrictive. For instance it does not admit functional extensionality or
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univalence. This poses a severe limitation on both what is \emph{provable} and
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the \emph{re-usability} of proofs. Recent developments have, however, resulted
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in cubical type theory which permits a constructive proof of these two important
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notions. The programming language Agda has been extended with capabilities for
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working in such a cubical setting. This thesis will explore the usefulness of
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this extension in the context of category theory.
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The usual notion of propositional equality in intensional type-theory
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is restrictive. For instance it does not admit functional
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extensionality or univalence. This poses a severe limitation on both
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what is \emph{provable} and the \emph{re-usability} of proofs. Recent
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developments have, however, resulted in cubical type theory which
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permits a constructive proof of these two important notions. The
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programming language Agda has been extended with capabilities for
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working in such a cubical setting. This thesis will explore the
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usefulness of this extension in the context of category theory.
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The thesis will motivate and explain why propositional equality in cubical Agda
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is more expressive than in standard Agda. Alternative approaches to Cubical Agda
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will be presented and their pros and cons will be explained. It will emphasize
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why it is useful to have a constructive interpretation of univalence. As an
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example of this two formulations of monads will be presented: Namely monaeds in
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the monoidal form an monads in the Kleisli form.
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The thesis will motivate and explain why propositional equality in
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cubical Agda is more expressive than in standard Agda. Alternative
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approaches to Cubical Agda will be presented and their pros and cons
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will be explained. It will emphasize why it is useful to have a
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constructive interpretation of univalence. As an example of this two
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formulations of monads will be presented: Namely monads in the
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monoidal form and monads in the Kleisli form.
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Finally the thesis will explain the challenges that a developer will face when
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working with cubical Agda and give some techniques to overcome these
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difficulties. It will also try to suggest how furhter work can help allievate
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some of these challenges.
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Finally the thesis will explain the challenges that a developer will
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face when working with cubical Agda and give some techniques to
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overcome these difficulties. It will also try to suggest how further
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work can help alleviate some of these challenges.
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@ -9,7 +9,7 @@ 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}. It turns out to be useful to built the
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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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@ -1,13 +1,13 @@
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\chapter{Cubical Agda}
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\section{Propositional equality}
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Judgmental equality in Agda is a feature of the type-system. Its something that
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can be checked automatically by the type-checker: In the example from the
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Judgmental equality in Agda is a feature of the type system. Its something that
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can be checked automatically by the type checker: In the example from the
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introduction $n + 0$ can be judged to be equal to $n$ simply by expanding the
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definition of $+$.
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On the other hand, propositional equality is something defined within the
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language itself. Propositional equality cannot be derived automatically. The
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normal definition of judgmental equality is an inductive data-type. Cubical Agda
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normal definition of judgmental equality is an inductive data type. Cubical Agda
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discards this type in favor of a new primitives that has certain computational
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properties exclusive to it.
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@ -22,15 +22,13 @@ two points of $A$; $a_0, a_1 \tp A$ we can form the type:
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a_0 \equiv a_1 \tp \MCU
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\end{align}
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%
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In Agda this is defined as an inductive data-type with the single constructor
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In Agda this is defined as an inductive data type with the single constructor
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for any $a \tp A$:
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%
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\begin{align}
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\refl \tp a \equiv a
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\end{align}
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%
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For any $a \tp A$.
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There also exist a related notion of \emph{heterogeneous} equality which allows
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for equating points of different types. In this case given two types $A, B \tp
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\MCU$ and two points $a \tp A$, $b \tp B$ we can construct the type:
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@ -39,7 +37,7 @@ for equating points of different types. In this case given two types $A, B \tp
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a \cong b \tp \MCU
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\end{align}
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%
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This is likewise defined as an inductive data-type with a single constructors
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This is likewise defined as an inductive data type with a single constructors
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for any $a \tp A$:
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%
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\begin{align}
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@ -56,17 +54,17 @@ 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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$I$ 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 ``path-space'' of some path $p \tp
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$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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$p$, from the index-space to the path space:
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%
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$$
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p \tp \prod_{i \tp I} P\ i
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@ -80,16 +78,17 @@ endpoints. I.e.:
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p\ 1 & = a_1
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\end{align*}
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%
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The notion of ``homogeneous equalities'' is recovered when $P$ does not depend
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on its argument:
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The notion of \nomenindex{homogeneous equalities} is recovered when $P$ does not
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depend on its argument. That is for $A \tp \MCU$, $a_0, a_1 \tp A$ the
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homogenous equality between $a_0$ and $a_1$ is the type:
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%
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$$
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a_0 \equiv a_1 \defeq \Path\ (\lambda i \to A)\ a_0\ a_1
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$$
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%
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For $A \tp \MCU$, $a_0, a_1 \tp A$. I will generally prefer to use the notation
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I will generally prefer to use the notation
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$a \equiv b$ when talking about non-dependent paths and use the notation
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$\Path\ (\lambda i \to P\ i)\ a\ b$ when the path-space is of particular
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$\Path\ (\lambda i \to P\ i)\ a\ b$ when the path space is of particular
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interest.
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With this definition we can also recover reflexivity. That is, for any $A \tp
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@ -102,7 +101,7 @@ With this definition we can also recover reflexivity. That is, for any $A \tp
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\end{aligned}
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\end{equation}
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%
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Here the path-space is $P \defeq \lambda i \to A$ and it satsifies $P\ i = A$
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Here the path space is $P \defeq \lambda i \to A$ and it satsifies $P\ i = A$
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definitionally. So to inhabit it, is to give a path $I \to A$ which is
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judgmentally $a$ at either endpoint. This is satisfied by the constant path;
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i.e. the path that stays at $a$ at any index $i$.
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@ -143,11 +142,12 @@ With this we can now prove the desired equality $f \equiv g$ from section
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%
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Paths have some other important properties, but they are not the focus of
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this thesis. \TODO{Refer the reader somewhere for more info.}
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%
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\section{Homotopy levels}
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In ITT all equality proofs are identical (in a closed context). This means that,
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in some sense, any two inhabitants of $a \equiv b$ are ``equally good'' -- they
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do not have any interesting structure. This is referred to as Uniqueness of
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Identity Proofs (UIP). Unfortunately it is not possible to have a type-theory
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Identity Proofs (UIP). Unfortunately it is not possible to have a type theory
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with both univalence and UIP. In stead we have a hierarchy of types with an
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increasing amount of homotopic structure. At the bottom of this hierarchy we
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have the set of contractible types:
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@ -162,7 +162,7 @@ have the set of contractible types:
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\end{equation}
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%
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The first component of $\isContr\ A$ is called ``the center of contraction''.
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Under the propositions-as-types interpretation of type-theory $\isContr\ A$ can
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Under the propositions-as-types interpretation of type theory $\isContr\ A$ can
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be thought of as ``the true proposition $A$''. And indeed $\top$ is
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contractible:
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\begin{equation*}
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@ -181,7 +181,7 @@ The next step in the hierarchy is the set of mere propositions:
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\end{aligned}
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\end{equation}
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%
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$\isProp\ A$ can be thought of as the set of true and false propositions. And
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One can think of $\isProp\ A$ as the set of true and false propositions. And
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indeed both $\top$ and $\bot$ are propositions:
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%
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\begin{align*}
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@ -189,9 +189,9 @@ indeed both $\top$ and $\bot$ are propositions:
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λ\varnothing\ \varnothing & \tp \isProp\ ⊥
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\end{align*}
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%
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$\varnothing$ is used here to denote an impossible pattern. It is a theorem that
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if a mere proposition $A$ is inhabited, then so is it contractible. If it is not
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inhabited it is equivalent to the empty-type (or false
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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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I will refer to a type $A \tp \MCU$ as a \emph{mere} proposition if I want to
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@ -225,8 +225,9 @@ As the reader may have guessed the next step in the hierarchy is the type:
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%
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And so it continues. In fact we can generalize this family of types by indexing
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them with a natural number. For historical reasons, though, the bottom of the
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hierarchy, the contractible types, is said to be a \nomen{-2-type}, propositions
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are \nomen{-1-types}, (homotopical) sets are \nomen{0-types} and so on\ldots
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hierarchy, the contractible types, is said to be a \nomen{-2-type}{homotopy
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levels}, propositions are \nomen{-1-types}{homotopy levels}, (homotopical)
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sets are \nomen{0-types}{homotopy levels} and so on\ldots
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Just as with paths, homotopical sets are not at the center of focus for this
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thesis. But I mention here some properties that will be relevant for this
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@ -253,8 +254,8 @@ of these theorems here, as they will be used later in chapter
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\subsection{Path induction}
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\label{sec:pathJ}
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The induction principle for paths intuitively gives us a way to reason about a
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type-family indexed by a path by only considering if said path is $\refl$ (the
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``base-case''). For \emph{based path induction}, that equality is \emph{based}
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type family indexed by a path by only considering if said path is $\refl$ (the
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\nomen{base case}{path induction}). For \emph{based path induction}, that equality is \emph{based}
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at some element $a \tp A$.
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Let a type $A \tp \MCU$ and an element of the type $a \tp A$ be given. $a$ is said to be the base of the induction. Given a family of types:
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@ -292,7 +293,7 @@ D & \tp \prod_{b' \tp A} \prod_{p \tp a ≡ b'} \MCU \\
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D\ b'\ p' & \defeq \var{sym}\ (\var{sym}\ p') ≡ p'
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\end{align*}
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%
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The base-case will then be:
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The base case will then be:
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%
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\begin{align*}
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d & \tp \var{sym}\ (\var{sym}\ \refl) ≡ \refl \\
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@ -326,7 +327,7 @@ over the family:
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T\ d'\ r' & \defeq \trans\ p\ (\trans\ q\ r') ≡ \trans\ (\trans\ p\ q)\ r'
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\end{align*}
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%
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So the base-case is proven with $t$ which is defined as:
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So the base case is proven with $t$ which is defined as:
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%
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\begin{align*}
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\trans\ p\ (\trans\ q\ \refl) & ≡
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@ -342,7 +343,7 @@ conclusion \ref{eq:cum-trans} is inhabited by the term:
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\pathJ\ T\ t\ d\ r
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\end{align*}
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%
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We shall see another application on path-induction in \ref{eq:pathJ-example}.
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We shall see another application on path induction in \ref{eq:pathJ-example}.
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\subsection{Paths over propositions}
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\label{sec:lemPropF}
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@ -357,7 +358,7 @@ a heterogeneous path between any two elements of $p_0 \tp P\ a_0$ and $p_1 \tp
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P\ a_1$:
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%
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$$
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\lemPropF\ \var{propP}\ p \defeq \Path\ (\lambda\; i \mto P\ (p\ i))\ p_0\ p_1
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\lemPropF\ \var{propP}\ p \tp \Path\ (\lambda\; i \mto P\ (p\ i))\ p_0\ p_1
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$$
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%
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This is quite a mouthful. So let me try to show how this is a very general and
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@ -57,8 +57,8 @@ composition and identity, laws; preservation of identity and composition) plus
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the extra condition that it is univalent - namely that you can get an equality
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of two objects from an isomorphism.
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I make no distinction between a pre-category and a real category (as in the
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[HoTT]-sense). A pre-category in my implementation would be a category sans the
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I make no distinction between a pre category and a real category (as in the
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[HoTT]-sense). A pre category in my implementation would be a category sans the
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witness to univalence.
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I also prove that being a category is a proposition. This gives rise to an
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@ -2,33 +2,44 @@
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\label{ch:implementation}
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This implementation formalizes the following concepts:
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%
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\begin{enumerate}[i.]
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\item Categories
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\item Functors
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\item Products
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\item Exponentials
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\item Cartesian closed categories
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\item Natural transformations
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\item Yoneda embedding
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\item Monads
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\item Categories
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\begin{enumerate}[i.]
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\item Opposite category
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\item Category of sets
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\item ``Pair category''
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\end{enumerate}
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\end{enumerate}
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\newcommand{\sourcebasepath}{http://web.student.chalmers.se/~hanghj/cat/doc/html/}
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\newcommand{\sourcelink}[1]{\href{\sourcebasepath#1.html}{\texttt{#1}}}
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\begin{center}
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\begin{tabular}{ l l }
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Name & Link \\
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\hline
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Categories & \sourcelink{Cat.Category} \\
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Functors & \sourcelink{Cat.Category.Functor} \\
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Products & \sourcelink{Cat.Category.Products} \\
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Exponentials & \sourcelink{Cat.Category.Exponentials} \\
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Cartesian closed categories & \sourcelink{Cat.Category.CartesianClosed} \\
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Natural transformations & \sourcelink{Cat.Category.NaturalTransformation} \\
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Yoneda embedding & \sourcelink{Cat.Category.Yoneda} \\
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Monads & \sourcelink{Cat.Category.Monads} \\
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%% Categories & \null \\
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%%
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Opposite category &
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\href{\sourcebasepath Cat.Category.html#22744}{\texttt{Cat.Category.Opposite}} \\
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Category of sets & \sourcelink{Cat.Categories.Sets} \\
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Span category &
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\href{\sourcebasepath Cat.Category.Product.html#2919}{\texttt{Cat.Category.Product.Span}} \\
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%%
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\end{tabular}
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\end{center}
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%
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Furthermore the following items have been partly formalized:
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%
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\begin{enumerate}[i.]
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\item The (higher) category of categories.
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\item Category of relations
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\item Category of functors and natural transformations -- only as a precategory
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\item Free category
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\item Monoidal objects
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\item Monoidal categories
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\end{enumerate}
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\begin{center}
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\begin{tabular}{ l l }
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Name & Link \\
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\hline
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Category of categories & \sourcelink{Cat.Categories.Cat} \\
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Category of relations & \sourcelink{Cat.Categories.Rel} \\
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Category of functors & \sourcelink{Cat.Categories.Fun} \\
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Free category & \sourcelink{Cat.Categories.Free} \\
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Monoids & \sourcelink{Cat.Category.Monoid} \\
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\end{tabular}
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\end{center}
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%
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As well as a range of various results about these. E.g. I have shown that the
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category of sets has products. In the following I aim to demonstrate some of the
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@ -70,22 +81,22 @@ constituents of a category and can be found in typical mathematical expositions
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on the topic. We, however, impose one further requirement on what it means to be
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a category, namely that the type of arrows form a set.
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Such categories are called \nomen{1-categories}. It is possible to relax this
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requirement. This would lead to the notion of higher categories (\cite[p.
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Such categories are called \nomen{1-categories}{1-category}. It is possible to relax
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this requirement. This would lead to the notion of higher categories (\cite[p.
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307]{hott-2013}). For the purpose of this project, however, this report will
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restrict itself to 1-categories. Making based on higher categories would be a
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very natural possible extension of this work.
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restrict itself to 1-categories\index{1-category}. Generalizing this work to
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higher categories would be a very natural possible extension of this work.
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Raw categories satisfying all of the above requirements are called a
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\nomen{pre}-categories. As a further requirement to be a proper category we
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\nomenindex{pre categories}. As a further requirement to be a proper category we
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require it to be univalent. Before we can define this, I must introduce two more
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definitions: If we let $p$ be a witness to the identity law, which formally is:
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%
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\begin{equation}
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\label{eq:identity}
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\var{IsIdentity} \defeq
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\prod_{A\ B \tp \Object} \prod_{f \tp A \to B}
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\id \comp f \equiv f \x f \comp \id \equiv f
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\prod_{A\ B \tp \Object} \prod_{f \tp \Arrow\ A\ B}
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\id \lll f \equiv f \x f \lll \id \equiv f
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\end{equation}
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%
|
||||
Then we can construct the identity isomorphism $\idIso \tp \identity,
|
||||
|
@ -190,8 +201,8 @@ This example illustrates nicely how we can use these combinators to reason about
|
|||
the other homotopic levels. These combinators are however not applicable in
|
||||
situations where we want to reason about other types - e.g. types we have
|
||||
defined ourselves. For instance, after we have proven that all the projections
|
||||
of pre-categories are propositions, then we would like to bundle this up to show
|
||||
that the type of pre-categories is also a proposition. Formally:
|
||||
of pre categories are propositions, then we would like to bundle this up to show
|
||||
that the type of pre categories is also a proposition. Formally:
|
||||
%
|
||||
\begin{equation}
|
||||
\label{eq:propIsPreCategory}
|
||||
|
@ -264,7 +275,7 @@ a priori that equality proofs are unique.
|
|||
|
||||
The situation is a bit more complicated when we have a dependent type. For
|
||||
instance, when we want to show that $\IsCategory$ is a mere proposition.
|
||||
$\IsCategory$ is a record with two fields, a witness to being a pre-category and
|
||||
$\IsCategory$ is a record with two fields, a witness to being a pre category and
|
||||
the univalence condition. Recall that the univalence condition is indexed by the
|
||||
identity-proof. So to follow the same recipe as above, let $a\ b \tp
|
||||
\IsCategory$ be given, to show them equal, we now need to give two paths. One homogeneous:
|
||||
|
@ -423,7 +434,7 @@ equalities and isomorphisms (on arrows). It is worthwhile to dwell on this for a
|
|||
few seconds. This type looks very similar to univalence for types and is
|
||||
therefore perhaps a bit more intuitive to grasp the implications of. Of course
|
||||
univalence for types (which is a proposition -- i.e. provable) does not imply
|
||||
univalence of all pre-category since morphisms in a category are not regular
|
||||
univalence of all pre category since morphisms in a category are not regular
|
||||
functions -- in stead they can be thought of as a generalization hereof. The univalence criterion therefore is simply a way of restricting arrows
|
||||
to behave similarly to maps.
|
||||
|
||||
|
@ -541,7 +552,7 @@ over-bar. So e.g. $\idToIso$ is a function in the underlying category and the
|
|||
corresponding thing is denoted $\wideoverbar{\idToIso}$ in the opposite
|
||||
category.
|
||||
|
||||
Showing that this forms a pre-category is rather straightforward.
|
||||
Showing that this forms a pre category is rather straightforward.
|
||||
%
|
||||
$$
|
||||
h \rrr (g \rrr f) \equiv h \rrr g \rrr f
|
||||
|
@ -826,13 +837,12 @@ that there exists a unique arrow $\pi \tp \Arrow\ X\ (A \x B)$ satisfying
|
|||
%
|
||||
$\pi$ is called the product (arrow) of $f$ and $g$.
|
||||
|
||||
\subsection{Pair category}
|
||||
\subsection{Span category}
|
||||
|
||||
\newcommand\pairA{\mathcal{A}}
|
||||
\newcommand\pairB{\mathcal{B}}
|
||||
Given a base category $\bC$ and two objects in this category $\pairA$ and
|
||||
$\pairB$ we can construct the ``pair category'': \TODO{This is a working title,
|
||||
it is nice to have a name for this thing to refer back to}
|
||||
$\pairB$ we can construct the \nomenindex{span category}:
|
||||
|
||||
The type of objects in this category will be an object in the underlying
|
||||
category, $X$, and two arrows (also from the underlying category)
|
||||
|
@ -964,7 +974,7 @@ $$
|
|||
This we get from \ref{eq:productEqPrinc} and the fact that homotopical structure
|
||||
is cumulative.
|
||||
|
||||
This finishes the proof that this is a valid pre-category.
|
||||
This finishes the proof that this is a valid pre category.
|
||||
|
||||
\subsubsection{Univalence}
|
||||
To prove that this is a proper category it must be shown that it is univalent.
|
||||
|
@ -1151,9 +1161,10 @@ gory details.
|
|||
%
|
||||
\subsection{Propositionality of products}
|
||||
%
|
||||
Now that we have constructed the ``pair category'' I will demonstrate how to use
|
||||
this to prove that products are propositional. I will do this by showing that
|
||||
terminal objects in this category are equivalent to products:
|
||||
Now that we have constructed the span category\index{span category} I will
|
||||
demonstrate how to use this to prove that products are propositional. I will
|
||||
do this by showing that terminal objects in this category are equivalent to
|
||||
products:
|
||||
%
|
||||
\begin{align}
|
||||
\var{Terminal} ≃ \var{Product}\ ℂ\ \mathcal{A}\ \mathcal{B}
|
||||
|
@ -1178,7 +1189,7 @@ indeed a product. That is, for an object $Y$ and two arrows $y_𝒜 \tp
|
|||
%
|
||||
Since $X, x_𝒜, x_ℬ$ is a terminal object there is a \emph{unique} arrow from
|
||||
this object to any other object, so also $Y, y_𝒜, y_ℬ$ in particular (which is
|
||||
also an object in the pair category). The arrow we will play the role of $f$ and
|
||||
also an object in the span category). The arrow we will play the role of $f$ and
|
||||
it immediately satisfies \ref{eq:pairCondRev}. Any other arrow satisfying these
|
||||
conditions will be equal since $f$ is unique.
|
||||
|
||||
|
@ -1262,14 +1273,15 @@ Denote the arrow-map of $\EndoR$ as $\fmap$, then this data must satisfy the
|
|||
following laws:
|
||||
%
|
||||
\begin{align}
|
||||
\label{eq:monad-monoidal-laws}
|
||||
\begin{split}
|
||||
\label{eq:monad-monoidal-laws-0}
|
||||
\join \lll \fmap\ \join
|
||||
& ≡ \join \lll \join\ \fmap \\
|
||||
\join \lll \pure\ \fmap & ≡ \identity \\
|
||||
& ≡ \join \lll \join \\
|
||||
\label{eq:monad-monoidal-laws-1}
|
||||
\join \lll \pure\ & ≡ \identity \\
|
||||
\label{eq:monad-monoidal-laws-2}
|
||||
\join \lll \fmap\ \pure & ≡ \identity
|
||||
\end{split}
|
||||
\end{align}
|
||||
\newcommand\monoidallaws{\ref{eq:monad-monoidal-laws-0}, \ref{eq:monad-monoidal-laws-1} and \ref{eq:monad-monoidal-laws-2}}%
|
||||
%
|
||||
The implicit arguments to the arrows above have been left out and the objects
|
||||
they range over are universally quantified.
|
||||
|
@ -1279,13 +1291,13 @@ they range over are universally quantified.
|
|||
The Kleisli-formulation consists of the following data:
|
||||
%
|
||||
\begin{align}
|
||||
\label{eq:monad-kleisli-data}
|
||||
\begin{split}
|
||||
\EndoR & \tp \Object → \Object \\
|
||||
\pure & \tp % \prod_{X \tp Object}
|
||||
\label{eq:monad-kleisli-data}
|
||||
\EndoR & \tp \Object → \Object \\
|
||||
\pure & \tp % \prod_{X \tp Object}
|
||||
\Arrow\ X\ (\EndoR\ X) \\
|
||||
\bind & \tp % \prod_{X\;Y \tp Object} → \Arrow\ X\ (\EndoR\ Y)
|
||||
\Arrow\ (\EndoR\ X)\ (\EndoR\ Y)
|
||||
\Arrow\ (\EndoR\ X)\ (\EndoR\ Y)
|
||||
\end{split}
|
||||
\end{align}
|
||||
%
|
||||
|
@ -1298,17 +1310,14 @@ is a regular maps on objects and a pair of arrows.
|
|||
This data must satisfy:
|
||||
%
|
||||
\begin{align}
|
||||
\label{eq:monad-monoidal-laws}
|
||||
\begin{split}
|
||||
\bind\ \pure & ≡ \identity_{\EndoR\ X}
|
||||
\\
|
||||
% \prod_{f \tp \Arrow\ X\ (\EndoR\ Y)}
|
||||
\pure \fish f & ≡ f
|
||||
\\
|
||||
% \prod_{\substack{g \tp \Arrow\ Y\ (\EndoR\ Z)\\f \tp \Arrow\ X\ (\EndoR\ Y)}}
|
||||
\label{eq:monad-kleisli-laws-0}
|
||||
\bind\ \pure & ≡ \identity_{\EndoR\ X} \\
|
||||
\label{eq:monad-kleisli-laws-1}
|
||||
\pure \fish f & ≡ f \\
|
||||
\label{eq:monad-kleisli-laws-2}
|
||||
(\bind\ f) \rrr (\bind\ g) & ≡ \bind\ (f \fish g)
|
||||
\end{split}
|
||||
\end{align}
|
||||
\newcommand\kleislilaws{\ref{eq:monad-kleisli-laws-0}, \ref{eq:monad-kleisli-laws-1} and \ref{eq:monad-kleisli-laws-2}}%
|
||||
%
|
||||
Here likewise the arrows $f \tp \Arrow\ X\ (\EndoR\ Y)$ and $g \tp
|
||||
\Arrow\ Y\ (\EndoR\ Z)$ are universally quantified (as well as the objects they
|
||||
|
@ -1317,10 +1326,6 @@ f \rrr (\bind\ g)$ . (\TODO{Better way to typeset $\fish$?})
|
|||
|
||||
\subsection{Equivalence of formulations}
|
||||
%
|
||||
In my implementation I proceed to show how the one formulation gives rise to
|
||||
the other and vice-versa. For the present purpose I will briefly sketch some
|
||||
parts of this construction:
|
||||
|
||||
The notation I have chosen here in the report
|
||||
overloads e.g. $\pure$ to both refer to a natural transformation and an arrow.
|
||||
This is of course not a coincidence as the arrow in the Kleisli formulation
|
||||
|
@ -1329,15 +1334,141 @@ called $\pure$.
|
|||
|
||||
In the monoidal formulation we can define $\bind$:
|
||||
%
|
||||
\newcommand\joinX{\wideoverbar{\join}}%
|
||||
\newcommand\bindX{\wideoverbar{\bind}}%
|
||||
\newcommand\EndoRX{\wideoverbar{\EndoR}}%
|
||||
\newcommand\pureX{\wideoverbar{\pure}}%
|
||||
\newcommand\fmapX{\wideoverbar{\fmap}}%
|
||||
\begin{align}
|
||||
\bind \defeq \join \lll \fmap\ f
|
||||
\bind\ f \defeq \joinX \lll \fmap\ f
|
||||
\end{align}
|
||||
%
|
||||
And likewise in the Kleisli formulation we can define $\join$:
|
||||
%
|
||||
\begin{align}
|
||||
\join \defeq \bind\ \identity
|
||||
\join \defeq \bindX\ \identity
|
||||
\end{align}
|
||||
%
|
||||
It now remains to show that we can prove the various laws given this choice. I
|
||||
refer the reader to my implementation for the details.
|
||||
Here $\joinX$ corresponds to the arrow from the natural
|
||||
transformation $\join$. $\bindX$ on the other hand corresponds to a
|
||||
natural transformation constructed from $\bind$. It now remains to show that
|
||||
this construction indeed gives rise to a monad. This will be done in two steps.
|
||||
First we will assume that we have a monad in the monoidal form; $(\EndoR, \pure,
|
||||
\join)$ and then show that $\EndoR, \pure, \bind$ is indeed a monad in the
|
||||
Kleisli form. In the second part we will show the other direction.
|
||||
|
||||
\subsubsection{Monoidal to Kleisli}
|
||||
Let $(\EndoR, \pure, \join)$ be given as in \ref{eq:monad-monoidal-data}
|
||||
satisfying the laws \monoidallaws. For the data of the Kleisli
|
||||
formulation we pick:
|
||||
%
|
||||
\begin{align}
|
||||
\begin{split}
|
||||
\EndoR & \defeq \EndoRX \\
|
||||
\pure & \defeq \pureX \\
|
||||
\bind\ f & \tp \joinX \lll \fmapX\ f
|
||||
\end{split}
|
||||
\end{align}
|
||||
%
|
||||
$\EndoRX$ is the object map of the endo-functor $\EndoR$,
|
||||
$\pureX$ and $\joinX$ are the arrows from the natural
|
||||
transformations $\pure$ and $\join$ respectively. $\fmapX$ is the
|
||||
arrow map of the endo-functor $\EndoR$. It now just remains to verify
|
||||
the laws \kleislilaws. For \ref{eq:monad-kleisli-laws-0}:
|
||||
%
|
||||
\begin{align*}
|
||||
\bind\ \pure & ≡
|
||||
\join \lll (\fmap\ \pure) && \text{By definition} \\
|
||||
& ≡ \identity && \text{By \ref{eq:monad-monoidal-laws-2}}
|
||||
\end{align*}
|
||||
%
|
||||
For \ref{eq:monad-kleisli-laws-1}:
|
||||
%
|
||||
\begin{align*}
|
||||
\pure \fish f
|
||||
& \equiv %%%
|
||||
\pure \ggg \bind\ f && \text{By definition} \\ & ≡
|
||||
\bind\ f \lll \pure && \text{By definition} \\ & ≡
|
||||
\joinX \lll \fmapX\ f \lll \pureX && \text{By definition} \\ & ≡
|
||||
\joinX \lll \pureX \lll f && \text{$\pure$ is a natural transformation} \\ & ≡
|
||||
\identity \lll f && \text{By \ref{eq:monad-monoidal-laws-1}} \\ & ≡
|
||||
f && \text{Left identity}
|
||||
\end{align*}
|
||||
%
|
||||
For \ref{eq:monad-kleisli-laws-2}:
|
||||
\begin{align*}
|
||||
\bind\ g \rrr \bind\ f & ≡
|
||||
\bind\ f \lll \bind\ g
|
||||
\\ & ≡
|
||||
%% %%%%
|
||||
\joinX \lll \fmapX\ g \lll \joinX \lll \fmapX\ f
|
||||
&& \text{\dots} \\ & ≡
|
||||
\joinX \lll \joinX \lll \fmapX^2\ g \lll \fmapX\ f
|
||||
&& \text{$\join$ is a natural transformation} \\ & ≡
|
||||
\joinX\ \lll \fmapX\ \joinX \lll \fmapX^2\ g \lll \fmapX\ f
|
||||
&& \text{By \ref{eq:monad-monoidal-laws-0}} \\ & ≡
|
||||
\joinX\ \lll \fmapX\ \joinX\ \lll \fmapX\ (\fmapX\ g) \lll \fmapX\ f
|
||||
&& \text{} \\ & ≡
|
||||
\joinX \lll \fmapX\ (\joinX \lll \fmapX\ g \lll f)
|
||||
&& \text{Distributive law for functors} \\ & \equiv
|
||||
%%%%
|
||||
\bind\ (g \fish f)
|
||||
\end{align*}
|
||||
\subsubsection{Kleisli to Monoidal}
|
||||
For the other direction we are given $(\EndoR, \pure, \bind)$ as in
|
||||
\ref{eq:monad-kleisli-data} satisfying the laws \kleislilaws. For the data of
|
||||
the monoidal formulation we pick:
|
||||
%
|
||||
\begin{align}
|
||||
\begin{split}
|
||||
\EndoR & \defeq \EndoRX \\
|
||||
\pure & \defeq \pureX \\
|
||||
\join & \defeq \bind\ \identity
|
||||
\end{split}
|
||||
\end{align}
|
||||
%
|
||||
Where $\EndoRX \defeq (\bind\ (\pure \lll f), \EndoR)$ and $\pureX \defeq
|
||||
\bind\ \identity$. We must now show the laws \monoidallaws, but we must also
|
||||
verify that our choice of $\EndoRX$ actually is a functor. I will ommit this
|
||||
here. In stead we shall see how these two mappings are indeed inverses.
|
||||
|
||||
\subsubsection{Equivalence}
|
||||
To prove that the two formulations are equivalent we must demonstrate that the
|
||||
two mappings sketched above are indeed inverses of each other. If we name the
|
||||
first mapping $\toKleisli$ and it's proposed inverse $\toMonoidal$
|
||||
then we must show:
|
||||
%
|
||||
\begin{align}
|
||||
\label{eq:monad-forwards}
|
||||
\toKleisli \comp \toMonoidal & ≡ \identity \\
|
||||
\label{eq:monad-backwards}
|
||||
\toMonoidal \comp \toKleisli & ≡ \identity
|
||||
\end{align}
|
||||
%
|
||||
For \ref{eq:monad-forwards} let $(\EndoR, \pure, \join)$ be a monad in the
|
||||
monoidal form. In my formulation the proof that being-a-monad is a proposition
|
||||
can be found. With this result in place we get an equality principle for
|
||||
kleisli-monads that say that to equate two such monads it suffices to equate
|
||||
their data-part. So it suffices to equate the data-parts of the
|
||||
\ref{eq:monad-forwards}. Such a proof is a triple equation the three projections
|
||||
of \ref{eq:monad-kleisli-data}. The first two hold definitionally -- essentially
|
||||
one just wraps and unwraps the morphism in a functor. For the last equation a
|
||||
little more work is required:
|
||||
%
|
||||
\begin{align*}
|
||||
\join \lll \fmap\ f & ≡
|
||||
\fmap\ f \rrr \join \\ & ≡
|
||||
\bind\ (f \rrr \pure) \rrr \bind\ \identity
|
||||
&& \text{By definition of $\fmap$ and $\join$} \\ & ≡
|
||||
\bind\ (f \rrr \pure \fish \identity)
|
||||
&& \text{By \ref{eq:monad-kleisli-laws-2}} \\ & ≡
|
||||
\bind\ (f \rrr \identity)
|
||||
&& \text{By \ref{eq:monad-kleisli-laws-1}} \\ & ≡
|
||||
\bind\ f
|
||||
\end{align*}
|
||||
%
|
||||
For the other direction we can likewise verify that the maps $\EndoR$, $\bind$,
|
||||
$\join$, and $\fmap$ are equal. The equality principle for functors gives us
|
||||
that this is enough to show that the the functor $\EndoR$ we construct is
|
||||
identical. Similarly for the natural transformations we have that the naturality
|
||||
condition is a proposition so the paths between the maps are sufficient.
|
||||
|
|
|
@ -9,19 +9,21 @@ limitations inherent in ITT and -- by extension -- Agda.
|
|||
Consider the functions:
|
||||
%
|
||||
\begin{multicols}{2}
|
||||
\noindent
|
||||
\begin{equation*}
|
||||
f \defeq (n \tp \bN) \mto (0 + n \tp \bN)
|
||||
\end{equation*}
|
||||
\begin{equation*}
|
||||
g \defeq (n \tp \bN) \mto (n + 0 \tp \bN)
|
||||
\end{equation*}
|
||||
\end{multicols}
|
||||
\noindent%
|
||||
\begin{equation*}%
|
||||
f \defeq \lambda\ (n \tp \bN) \to (0 + n \tp \bN)
|
||||
\end{equation*}%
|
||||
\begin{equation*}%
|
||||
g \defeq \lambda\ (n \tp \bN) \to (n + 0 \tp \bN)
|
||||
\end{equation*}%
|
||||
\end{multicols}%
|
||||
%
|
||||
$n + 0$ is \nomen{definitionally} equal to $n$, which we write as $n + 0 = n$.
|
||||
This is also called \nomen{judgmental} equality. We call it definitional
|
||||
equality because the \emph{equality} arises from the \emph{definition} of $+$
|
||||
which is:
|
||||
The term $n + 0$ is
|
||||
\nomenindex{definitionally} equal to $n$, which we
|
||||
write as $n + 0 = n$. This is also called
|
||||
\nomenindex{judgmental equality}.
|
||||
We call it definitional equality because the \emph{equality} arises
|
||||
from the \emph{definition} of $+$ which is:
|
||||
%
|
||||
\begin{align*}
|
||||
+ & \tp \bN \to \bN \to \bN \\
|
||||
|
@ -29,11 +31,12 @@ which is:
|
|||
n + (\suc{m}) & \defeq \suc{(n + m)}
|
||||
\end{align*}
|
||||
%
|
||||
Note that $0 + n$ is \emph{not} definitionally equal to $n$. $0 + n$ is in
|
||||
normal form. I.e.; there is no rule for $+$ whose left-hand-side matches this
|
||||
expression. We \emph{do}, however, have that they are \nomen{propositionally}
|
||||
equal, which we write as $n + 0 \equiv n$. Propositional equality means that
|
||||
there is a proof that exhibits this relation. Since equality is a transitive
|
||||
Note that $0 + n$ is \emph{not} definitionally equal to $n$. $0 + n$
|
||||
is in normal form. I.e.; there is no rule for $+$ whose left hand side
|
||||
matches this expression. We \emph{do}, however, have that they are
|
||||
\nomen{propositionally}{propositional equality} equal, which we write
|
||||
as $n + 0 \equiv n$. Propositional equality means that there is a
|
||||
proof that exhibits this relation. Since equality is a transitive
|
||||
relation we have that $n + 0 \equiv 0 + n$.
|
||||
|
||||
Unfortunately we don't have $f \equiv g$.\footnote{Actually showing this is
|
||||
|
@ -43,7 +46,8 @@ not true.} There is no way to construct a proof asserting the obvious
|
|||
equivalence of $f$ and $g$ -- even though we can prove them equal for all
|
||||
points. This is exactly the notion of equality of functions that we are
|
||||
interested in; that they are equal for all inputs. We call this
|
||||
\nomen{point-wise equality}, where the \emph{points} of a function refers
|
||||
|
||||
\nomenindex{point-wise equality}, where the \emph{points} of a function refers
|
||||
to its arguments.
|
||||
|
||||
In the context of category theory functional extensionality is e.g. needed to
|
||||
|
@ -51,14 +55,14 @@ show that representable functors are indeed functors. The representable functor
|
|||
for a category $\bC$ and a fixed object in $A \in \bC$ is defined to be:
|
||||
%
|
||||
\begin{align*}
|
||||
\fmap \defeq X \mto \Hom_{\bC}(A, X)
|
||||
\fmap \defeq \lambda\ X \to \Hom_{\bC}(A, X)
|
||||
\end{align*}
|
||||
%
|
||||
The proof obligation that this satisfies the identity law of functors
|
||||
($\fmap\ \idFun \equiv \idFun$) thus becomes:
|
||||
%
|
||||
\begin{align*}
|
||||
\Hom(A, \idFun_{\bX}) = (g \mto \idFun \comp g) \equiv \idFun_{\Sets}
|
||||
\Hom(A, \idFun_{\bX}) = (\lambda\ g \to \idFun \comp g) \equiv \idFun_{\Sets}
|
||||
\end{align*}
|
||||
%
|
||||
One needs functional extensionality to ``go under'' the function arrow and apply
|
||||
|
@ -67,20 +71,23 @@ the (left) identity law of the underlying category to prove $\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 have the same shape (Greek; \nomen{isomorphic}). 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 an identification can not
|
||||
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 \tp A$. So in a sense they have the same shape
|
||||
(Greek;
|
||||
\nomenindex{isomorphic}). 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 an identification can not
|
||||
be performed in ITT.
|
||||
|
||||
More specifically what we are interested in is a way of identifying
|
||||
\nomen{equivalent} types. I will return to the definition of equivalence later
|
||||
|
||||
\nomenindex{equivalent} types. I will return to the definition of equivalence later
|
||||
in section \S\ref{sec:equiv}, but for now it is sufficient to think of an
|
||||
equivalence as a one-to-one correspondence. We write $A \simeq B$ to assert that
|
||||
$A$ and $B$ are equivalent types. The principle of univalence says that:
|
||||
|
@ -88,7 +95,7 @@ $A$ and $B$ are equivalent 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.
|
||||
($\mathit{ua} \tp (A \simeq B) \to (A \equiv B)$) and vice versa.
|
||||
|
||||
\section{Formalizing Category Theory}
|
||||
%
|
||||
|
@ -127,7 +134,7 @@ implementations of category theory in Agda:
|
|||
\item
|
||||
\url{https://github.com/mortberg/cubicaltt}
|
||||
|
||||
A formalization in CubicalTT - a language designed for cubical-type-theory.
|
||||
A formalization in CubicalTT - a language designed for cubical type theory.
|
||||
Formalizes many different things, but only a few concepts from category
|
||||
theory.
|
||||
|
||||
|
@ -141,19 +148,27 @@ compare some aspects of this formalization with the existing ones.\TODO{How can
|
|||
There are alternative approaches to working in a cubical setting where one can
|
||||
still have univalence and functional extensionality. One option is to postulate
|
||||
these as axioms. This approach, however, has other shortcomings, e.g.; you lose
|
||||
\nomen{canonicity} (\TODO{Pageno!} \cite{huber-2016}). Canonicity means that any
|
||||
well-typed term evaluates to a \emph{canonical} form. For example for a closed
|
||||
|
||||
\nomenindex{canonicity} (\TODO{Pageno!} \cite{huber-2016}). Canonicity means that any
|
||||
well typed term evaluates to a \emph{canonical} form. For example for a closed
|
||||
term $e \tp \bN$ it will be the case that $e$ reduces to $n$ applications of
|
||||
$\mathit{suc}$ to $0$ for some $n$; $e = \mathit{suc}^n\ 0$. Without canonicity
|
||||
terms in the language can get ``stuck'' -- meaning that they do not reduce to a
|
||||
canonical form.
|
||||
|
||||
Another approach is to use the \emph{setoid interpretation} of type theory
|
||||
(\cite{hofmann-1995,huber-2016}). With this approach one works with
|
||||
\nomen{extensional sets} $(X, \sim)$, that is a type $X \tp \MCU$ and an
|
||||
equivalence relation $\sim \tp X \to X \to \MCU$ on that type. Under the setoid
|
||||
interpretation the equivalence relation serve as a sort of ``local''
|
||||
propositional equality. This approach has other drawbacks; it does not satisfy
|
||||
Another approach is to use the \emph{setoid interpretation} of type
|
||||
theory (\cite{hofmann-1995,huber-2016}). With this approach one works
|
||||
with
|
||||
\nomenindex{extensional sets} $(X, \sim)$, that is a type $X \tp \MCU$
|
||||
and an equivalence relation $\sim\ \tp X \to X \to \MCU$ on that type.
|
||||
Under the setoid interpretation the equivalence relation serve as a
|
||||
sort of ``local'' propositional equality. Since the developer gets to
|
||||
pick this relation it is not guaranteed to be a congruence relation
|
||||
apriori. So this must be verified manually by the developer.
|
||||
Furthermore, functions between different setoids must be shown to be
|
||||
setoid homomorphism, that is; they preserve the relation.
|
||||
|
||||
This approach has other drawbacks; it does not satisfy
|
||||
all propositional equalities of type theory (\TODO{Citation needed}), is
|
||||
cumbersome to work with in practice (\cite[p. 4]{huber-2016}) and makes
|
||||
equational proofs less reusable since equational proofs $a \sim_{X} b$ are
|
||||
|
@ -163,14 +178,21 @@ inherently `local' to the extensional set $(X , \sim)$.
|
|||
\TODO{Talk a bit about terminology. Find a good place to stuff this little
|
||||
section.}
|
||||
|
||||
In the remainder of this paper I will use the term \nomen{Type} to describe --
|
||||
In the remainder of this paper I will use the term
|
||||
\nomenindex{Type} to describe --
|
||||
well, types. Thereby diverging from the notation in Agda where the keyword
|
||||
\texttt{Set} refers to types. \nomen{Set} on the other hand shall refer to the
|
||||
\texttt{Set} refers to types.
|
||||
\nomenindex{Set} on the other hand shall refer to the
|
||||
homotopical notion of a set. I will also leave all universe levels implicit.
|
||||
|
||||
And I use the term \nomen{arrow} to refer to morphisms in a category, whereas
|
||||
the terms morphism, map or function shall be reserved for talking about
|
||||
type-theoretic functions; i.e. functions in Agda.
|
||||
And I use the term
|
||||
\nomenindex{arrow} to refer to morphisms in a category,
|
||||
whereas the terms
|
||||
\nomenindex{morphism},
|
||||
\nomenindex{map} or
|
||||
\nomenindex{function}
|
||||
shall be reserved for talking about type theoretic functions; i.e.
|
||||
functions in Agda.
|
||||
|
||||
$\defeq$ will be used for introducing definitions. $=$ will be used to for
|
||||
judgmental equality and $\equiv$ will be used for propositional equality.
|
||||
|
@ -181,12 +203,16 @@ All this is summarized in the following table:
|
|||
\begin{tabular}{ c c c }
|
||||
Name & Agda & Notation \\
|
||||
\hline
|
||||
\nomen{Type} & \texttt{Set} & $\Type$ \\
|
||||
\nomen{Set} & \texttt{Σ Set IsSet} & $\Set$ \\
|
||||
|
||||
\varindex{Type} & \texttt{Set} & $\Type$ \\
|
||||
|
||||
\varindex{Set} & \texttt{Σ Set IsSet} & $\Set$ \\
|
||||
Function, morphism, map & \texttt{A → B} & $A → B$ \\
|
||||
Dependent- ditto & \texttt{(a : A) → B} & $∏_{a \tp A} B$ \\
|
||||
\nomen{Arrow} & \texttt{Arrow A B} & $\Arrow\ A\ B$ \\
|
||||
\nomen{Object} & \texttt{C.Object} & $̱ℂ.Object$ \\
|
||||
|
||||
\varindex{Arrow} & \texttt{Arrow A B} & $\Arrow\ A\ B$ \\
|
||||
|
||||
\varindex{Object} & \texttt{C.Object} & $̱ℂ.Object$ \\
|
||||
Definition & \texttt{=} & $̱\defeq$ \\
|
||||
Judgmental equality & \null & $̱=$ \\
|
||||
Propositional equality & \null & $̱\equiv$
|
||||
|
|
|
@ -17,7 +17,6 @@
|
|||
\newcommand{\UU}{\ensuremath{\mathcal{U}}\xspace}
|
||||
\let\type\UU
|
||||
\newcommand{\MCU}{\UU}
|
||||
\newcommand{\nomen}[1]{\emph{#1}}
|
||||
\newcommand{\todo}[1]{\textit{#1}}
|
||||
\newcommand{\comp}{\circ}
|
||||
\newcommand{\x}{\times}
|
||||
|
@ -33,7 +32,9 @@
|
|||
}
|
||||
\makeatother
|
||||
\newcommand{\var}[1]{\ensuremath{\mathit{#1}}}
|
||||
\newcommand{\varindex}[1]{\ensuremath{\mathit{#1}}\index{#1}}
|
||||
\newcommand{\varindex}[1]{\ensuremath{\var{#1}}\index{$\var{#1}$}}
|
||||
\newcommand{\nomen}[2]{\emph{#1}\index{#2}}
|
||||
\newcommand{\nomenindex}[1]{\nomen{#1}{#1}}
|
||||
|
||||
\newcommand{\Hom}{\varindex{Hom}}
|
||||
\newcommand{\fmap}{\varindex{fmap}}
|
||||
|
@ -93,3 +94,5 @@
|
|||
\newcommand\funExt{\varindex{funExt}}
|
||||
\newcommand{\suc}[1]{\varindex{suc}\ #1}
|
||||
\newcommand{\trans}{\varindex{trans}}
|
||||
\newcommand{\toKleisli}{\varindex{toKleisli}}
|
||||
\newcommand{\toMonoidal}{\varindex{toMonoidal}}
|
||||
|
|
|
@ -113,3 +113,4 @@
|
|||
}
|
||||
|
||||
\makeatother
|
||||
\usepackage{xspace}
|
||||
|
|
Loading…
Reference in a new issue