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16
thesis/main.tex
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@ -13,7 +13,7 @@
frame=lines,
autogobble
}
\usepackage[dvipsnames]{xcolor} % Coloured text etc.
\usepackage[dvipsnames]{xcolor} % Coloured text etc.
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{thmtools}
@ -36,15 +36,12 @@
\usepackage{makeidx}
\usepackage{graphicx}
\usepackage{fvextra}
\usepackage[style=ieee, sorting=ynt]{biblatex} %advanced citations
\usepackage[english=british]{csquotes} %biblatex recommended to load this
\usepackage[style=ieee, sorting=ynt]{biblatex} % advanced citations
\usepackage[english=british]{csquotes} % biblatex recommended to load this
\usepackage{etoolbox,xpatch}
\makeatletter
\AtBeginEnvironment{minted}{\dontdofcolorbox}
\def\dontdofcolorbox{\renewcommand\fcolorbox[4][]{##4}}
\xpatchcmd{\inputminted}{\minted@fvset}{\minted@fvset\dontdofcolorbox}{}{}
\xpatchcmd{\mintinline}{\minted@fvset}{\minted@fvset\dontdofcolorbox}{}{} % see https://tex.stackexchange.com/a/401250/
\AtBeginEnvironment{minted}{\dontdofcolorbox}\def\dontdofcolorbox{\renewcommand\fcolorbox[4][]{##4}}\xpatchcmd{\inputminted}{\minted@fvset}{\minted@fvset\dontdofcolorbox}{}{}\xpatchcmd{\mintinline}{\minted@fvset}{\minted@fvset\dontdofcolorbox}{}{} % see https://tex.stackexchange.com/a/401250/
\makeatother
\usepackage{scrhack}
@ -54,11 +51,11 @@
%\usepackage[justific=raggedright,totoc]{idxlayout}
\usepackage[type=CC, modifier=by-sa,version=4.0]{doclicense}
\addto\extrasenglish{%
\addto\extrasenglish{
\renewcommand{\chapterautorefname}{Section}
\renewcommand{\sectionautorefname}{Section}
\renewcommand{\subsectionautorefname}{Subsection}
}
}
\newcommand\chap[1]{%
\chapter*{#1}%
@ -111,6 +108,7 @@
% category C
\newcommand*{\C}{\ensuremath{\mathcal{C}}}
\newcommand*{\D}{\ensuremath{\mathcal{D}}}
% objects of category
\newcommand*{\obj}[1]{\ensuremath{\vert #1 \vert}}
% category of elgot algebras on #1

87
thesis/src/01_preliminaries.tex
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@ -114,13 +114,30 @@ The internal logic of cartesian closed categories is the simply typed $\lambda$-
Monads are widely known among programmers as a way of modelling effects in pure languages and are also central to this thesis. Let us recall the basic definitions\cite{Lane1971}\cite{moggi}.
\begin{definition}[Monad]
A monad on a category $\C$ is a triple $(F, \eta, \mu)$, where $F : \C \rightarrow \C$ is an endofunctor and $\eta : Id \rightarrow F, \mu : F^2 \rightarrow F$ are natural transformations, satisfying the following laws:
\begin{alignat*}{1}
&\mu_X \circ \mu_{FX} = \mu_X \circ F\mu_X \tag*{(M1)}\label{M1}\\
&\mu_X \circ \eta_{FX} = id_{FX} \tag*{(M2)}\label{M2}\\
&\mu_X \circ F\eta_X = id_{FX} \tag*{(M3)}\label{M3}
A monad $\mathbf{T}$ on a category $\C$ is a triple $(T, \eta, \mu)$, where $T : \C \rightarrow \C$ is an endofunctor and $\eta : Id \rightarrow T, \mu : TT \rightarrow T$ are natural transformations, satisfying the following laws:
\begin{alignat*}{2}
&\mu_X \circ \mu_{TX} &&= \mu_X \circ T\mu_X \tag*{(M1)}\label{M1}\\
&\mu_X \circ \eta_{TX} &&= id_{TX} \tag*{(M2)}\label{M2}\\
&\mu_X \circ T\eta_X &&= id_{TX} \tag*{(M3)}\label{M3}
\end{alignat*}
\todo[inline]{diagrams}
These laws are expressed by the following diagrams:
% with indices: % https://q.uiver.app/#q=WzAsOCxbMCwwLCJUVFRYIl0sWzIsMCwiVFRYIl0sWzAsMiwiVFRYIl0sWzIsMiwiVFgiXSxbNCwwLCJUWCJdLFs2LDAsIlRUWCJdLFs4LDAsIlRYIl0sWzYsMiwiVFgiXSxbMCwxLCJcXG11X3tUWH0iXSxbMCwyLCJUXFxtdV9YIiwyXSxbMSwzLCJcXG11X1giXSxbNSw3LCJcXG11X1giXSxbNCw1LCJcXGV0YV97VFh9Il0sWzYsNSwiVFxcZXRhX1giXSxbNCw3LCJpZF97VFh9IiwyXSxbNiw3LCJpZF97VFh9IiwyXSxbMiwzLCJcXG11X1giLDJdXQ==
% https://q.uiver.app/#q=WzAsOCxbMCwwLCJUVFRYIl0sWzIsMCwiVFRYIl0sWzAsMiwiVFRYIl0sWzIsMiwiVFgiXSxbNCwwLCJUWCJdLFs2LDAsIlRUWCJdLFs4LDAsIlRYIl0sWzYsMiwiVFgiXSxbMCwxLCJcXG11Il0sWzAsMiwiVFxcbXUiLDJdLFsxLDMsIlxcbXUiXSxbNSw3LCJcXG11Il0sWzQsNSwiXFxldGEiXSxbNiw1LCJUIl0sWzQsNywiaWQiLDJdLFs2LDcsImlkIiwyXSxbMiwzLCJcXG11IiwyXV0=
\[\begin{tikzcd}
TTTX && TTX && TX && TTX && TX \\
\\
TTX && TX &&&& TX
\arrow["\mu", from=1-1, to=1-3]
\arrow["T\mu"', from=1-1, to=3-1]
\arrow["\mu", from=1-3, to=3-3]
\arrow["\mu", from=1-7, to=3-7]
\arrow["\eta", from=1-5, to=1-7]
\arrow["T", from=1-9, to=1-7]
\arrow["id"', from=1-5, to=3-7]
\arrow["id"', from=1-9, to=3-7]
\arrow["\mu"', from=3-1, to=3-3]
\end{tikzcd}\]
\end{definition}
Morphisms between monads are natural transformations that respect the monad operations:
@ -146,19 +163,19 @@ Morphisms between monads are natural transformations that respect the monad oper
\arrow["{\mu^T}", from=2-7, to=3-6]
\end{tikzcd}\]
\end{definition}
\begin{definition}[The Category of Monads]
Monads on a category $\C$ together with monad morphisms yield a category that we call $\monads{\C}$. The identity morphism is the identity natural transformation that trivially respects the monad operations and composition of morphisms is composition of natural transformations.
Monads on a category $\C$ together with monad morphisms form a category that we call $\monads{\C}$. The identity morphism is the identity natural transformation that trivially respects the monad operations and composition of morphisms is composition of natural transformations.
\end{definition}
For programmers a second equivalent definition is more useful:
\begin{definition}[Kleisli Triple]
A kleisli triple on a category $\C$ is a triple $(F, \eta, (-)^*)$, where $F : \obj{C} \rightarrow \obj{C}$ is a mapping on objects, $(\eta_X : X \rightarrow FX)_{X\in\obj{C}}$ is a family of morphisms and for every morphism $f : X \rightarrow FY$ there exists a morphism $f^* : FX \rightarrow FY$ called the kleisli lifting, where the following laws hold:
\begin{alignat*}{1}
&\eta_X^* = id_{FX} \tag*{(K1)}\label{K1}\\
&\eta_X \circ f^* = f \tag*{(K2)}\label{K2}\\
&f^* \circ g* = (f \circ g^*)^* \tag*{(K3)}\label{K3}
A Kleisli triple on a category $\C$ is a triple $(F, \eta, (-)^*)$, where $F : \obj{C} \rightarrow \obj{C}$ is a mapping on objects, $(\eta_X : X \rightarrow FX)_{X\in\obj{C}}$ is a family of morphisms and for every morphism $f : X \rightarrow FY$ there exists a morphism $f^* : FX \rightarrow FY$ called the kleisli lifting, where the following laws hold:
\begin{alignat*}{2}
&\eta_X^* &&= id_{FX} \tag*{(K1)}\label{K1}\\
&\eta_X \circ f^* &&= f \tag*{(K2)}\label{K2}\\
&f^* \circ g* &&= (f \circ g^*)^* \tag*{(K3)}\label{K3}
\end{alignat*}
\end{definition}
@ -176,8 +193,8 @@ This results in the following:
\begin{itemize}
\item $\vert \C^T \vert = \obj{C}$
\item $\mathcal{C^T}(X, Y) = \C(X, TY)$
\item composition of $f : X \rightarrow TY$ and $g : Y \rightarrow TZ$ is defined as $f \circ_{\C^T} g = f^* \circ_{\C} g$.
\item the identity morphisms are the unit morphisms of $T$, $id_X = \eta_X : X \rightarrow TX$
\item Composition of $f : X \rightarrow TY$ and $g : Y \rightarrow TZ$ is defined as $f \circ_{\C^T} g = f^* \circ_{\C} g$.
\item The identity morphisms are the unit morphisms of $T$, $id_X = \eta_X : X \rightarrow TX$
\end{itemize}
The laws of categories then follow from the Kleisli triple laws, making this indeed a category.
\end{definition}
@ -219,29 +236,36 @@ When modelling partiality with a monad, one would expect the following two progr
\end{multicols}
where p and q are (partial) computations. This condition can be expressed categorically, but first we need another definition:
\begin{definition}[Strong Monad] A monad $M$ on a cartesian category $\C$ is called strong if there exists a natural transformation $\tau_{X,Y} : X \times MY \rightarrow M(X \times Y)$, satisfying the following conditions:
\begin{alignat*}{1}
&M\pi_2 \circ \tau_{1,X} = \pi_2 \tag*{(S1)}\label{S1}\\
&\tau_{X,Y} \circ (id_X \times \eta_Y) = \eta_{X\times Y} \tag*{(S2)}\label{S2}\\
&\tau_{X,Y} \circ (id_X \times \mu_Y) = \mu_{X\times Y} \circ M\tau_{X,Y} \circ \tau_{X,MY} \tag*{(S3)}\label{S3}\\
&M \alpha_{X,Y,Z} \circ \tau_{X \times Y, Z} = \tau_{X, Y\times Z} \circ (id_X \times \tau_{Y, Z}) \circ \alpha_{X,Y,MZ} \tag*{(S4)}\label{S4}
\begin{definition}[Strong Monad] A monad $(T, \eta, \mu)$ on a cartesian category $\C$ is called strong if there exists a natural transformation $\tau_{X,Y} : X \times TY \rightarrow T(X \times Y)$, satisfying the following conditions:
\begin{alignat*}{2}
&T\pi_2 \circ \tau_{1,X} &&= \pi_2 \tag*{(S1)}\label{S1}\\
&\tau_{X,Y} \circ (id_X \times \eta_Y) &&= \eta_{X\times Y} \tag*{(S2)}\label{S2}\\
&\tau_{X,Y} \circ (id_X \times \mu_Y) &&= \mu_{X\times Y} \circ T\tau_{X,Y} \circ \tau_{X,TY} \tag*{(S3)}\label{S3}\\
&M \alpha_{X,Y,Z} \circ \tau_{X \times Y, Z} &&= \tau_{X, Y\times Z} \circ (id_X \times \tau_{Y, Z}) \circ \alpha_{X,Y,TZ} \tag*{(S4)}\label{S4}
\end{alignat*}
where $\alpha_{X,Y,Z} : X \times (Y \times Z) \rightarrow (X \times Y) \times Z$ is the obvious associativity morphism.
where $\alpha_{X,Y,Z} = \langle \langle \pi_1 , \pi_1 \circ \pi_2 \rangle , \pi_2 \circ \pi_2 \rangle : X \times (Y \times Z) \rightarrow (X \times Y) \times Z$ is the associativity morphism on products.
\end{definition}
Now we can express the above condition:
\begin{definition}[Commutative Monad]
A strong monad $M$ is called commutative if the (right) strength $\tau$ commutes with the induced (left) strength $\hat{\tau}_{X,Y} = Mswap \circ \tau_{Y,X} \circ swap$. Concretely if the following equation holds:
\[
\tau_{X,Y}^* \circ \hat{\tau}_{X, MY} = \hat{\tau}_{X,Y}^* \circ \tau_{X, MY}
\]
A strong monad $\mathbf{T}$ is called commutative if the (right) strength $\tau$ commutes with the induced (left) strength $\sigma_{X,Y} = Mswap \circ \tau_{Y,X} \circ swap$. Concretely if the following equation holds:
% https://q.uiver.app/#q=WzAsNCxbMCwyLCJUKFggXFx0aW1lcyBUWSkiXSxbMiwwLCJUKFRYIFxcdGltZXMgWSkiXSxbMiwyLCJUKFggXFx0aW1lcyBZKSJdLFswLDAsIlRYIFxcdGltZXMgVFkiXSxbMywxLCJcXHRhdSJdLFszLDAsIlxcc2lnbWEiLDJdLFswLDIsIlxcdGF1XioiLDJdLFsxLDIsIlxcc2lnbWFeKiJdXQ==
\[\begin{tikzcd}
{TX \times TY} && {T(TX \times Y)} \\
\\
{T(X \times TY)} && {T(X \times Y)}
\arrow["\tau", from=1-1, to=1-3]
\arrow["\sigma"', from=1-1, to=3-1]
\arrow["{\tau^*}"', from=3-1, to=3-3]
\arrow["{\sigma^*}", from=1-3, to=3-3]
\end{tikzcd}\]
\end{definition}
\section{Free Objects}
Free objects are constructions capturing the essence of structures in a minimal way, we will rely on free structures in chapter~\ref{chp:iteration} to define a monad in a general setting. We recall the definition to establish some notation:
\begin{definition}[Free Object]
Let $\C, \mathcal{D}$ be categories and $U : C \rightarrow D$ a forgetful functor (whose construction usually is obvious). A free object on some object $X \in \mathcal{D}$ is an object $FX \in \C$ together with a morphism $\eta : X \rightarrow UFX$ such that the following universal property holds for any $Y \in \obj{C}$ and $f : X \rightarrow UY$:
Let $\C, \D$ be categories and $U : \C \rightarrow \D$ be a forgetful functor (whose construction usually is obvious). A free object on some object $X \in \obj{\D}$ is an object $FX \in \obj{\C}$ together with a morphism $\eta : X \rightarrow UFX$ such that the following universal property holds for any $Y \in \obj{\C}$ and $f : X \rightarrow UY$:
% https://q.uiver.app/#q=WzAsMyxbMCwwLCJYIl0sWzEsMCwiVVkiXSxbMCwxLCJGWCJdLFswLDEsImYiXSxbMCwyLCJcXGV0YSIsMl0sWzIsMSwiXFxleGlzdHMhXFxsbGJyYWNrZXQgZiBcXHJyYnJhY2tldCIsMl1d
\[\begin{tikzcd}
X & UY \\
@ -250,4 +274,11 @@ Let $\C, \mathcal{D}$ be categories and $U : C \rightarrow D$ a forgetful functo
\arrow["\eta"', from=1-1, to=2-1]
\arrow["{\exists!\llbracket f \rrbracket}"', from=2-1, to=1-2]
\end{tikzcd}\]
\end{definition}
\end{definition}
\todo[inline]{adjunction}
\todo[inline]{monad}
\section{Coalgebras}
\todo[inline]{introduce (terminal) coalgebras, proof lambeks lemma}

21
thesis/src/02_agda-categories.tex
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@ -1,12 +1,12 @@
\chapter{Category Theory in Agda}
There are many formalizations of category theory in proof assistants like Coq or Agda. The benefits are clear, having a usable formalization allows one to reason about categorical notions in a typechecked environment that makes errors less likely.
Also ideally such a development will bring researchers together and enable them to work in a unified setting.
In this thesis we will work with the dependently typed programming language Agda~\cite{agda} and the agda-categories~\cite{agda-categories} library by Jason Hu and Jacques Carette that gives us a good foundation of categorical definitions to work with.
There are many formalizations of category theory in proof assistants like Coq or Agda. The benefits of such a formalization are clear: having a usable formalization allows one to reason about categorical notions in a typechecked environment that makes errors less likely.
Ideally such a development will bring researchers together and enable them to work in a unified setting that enables efficient communication of ideas and concepts.
In this thesis we will work with the dependently typed programming language Agda~\cite{agda} and the agda-categories~\cite{agda-categories} library that serves as an extensive foundation of categorical definitions.
This chapter shall serve as a quick introduction to the library and the formalization of this thesis.
\section{Setoid Enriched Categories}
The usual textbook definition of a category hides some design decisions that have to be made when implementing it in type theory. One would usually see something like this:
The usual textbook definition of a category glosses over some design decisions that have to be made when implementing it in type theory. One would usually see something like this:
\begin{definition}[Category]
A category consists of
\begin{itemize}
@ -18,10 +18,17 @@ A category consists of
where the composition is associative and the identity morphisms are identities with respect to the composition.
\end{definition}
Here a \textit{collection} usually is something that behaves set-like, but prevents size issues (there is no collection of all collections, preventing Russel's Paradox), it is not instantly clear how to translate this to a proof assistant. Furthermore in mathematical textbooks equality between morphisms is usually taken for granted, i.e. there is some global notion of equality that is clear to everyone. In Agda we need to be more thorough, there is no global notion of equality that 'just works', e.g. the standard notion of propositional equality has trouble dealing with functions and needs extra axioms like functional extensionality.
Here \textit{collection} refers to something that behaves set-like, which is not a set and is needed to prevent size issues (there is no set of all sets, otherwise we would obtain Russel's paradox, but there is a collection of all sets), it is not immediately clear how to translate this to type theory.
Furthermore in mathematical textbooks equality between morphisms is usually taken for granted, i.e. there is some global notion of equality that is clear to everyone. In type theory we need to be more thorough as there is no global notion of equality, eligible for all purposes, e.g. the standard notion of propositional equality has issues when dealing with functions in that it requires extra axioms like functional extensionality.
The definition of category that we will work with can be seen in listing~\ref{lst:category} (unnecessary information has been stripped). The key differences to the definition above are firstly that instead of talking about collections, Agda's infinite \Verb{Set} hierarchy is utilized to prevent size issues, i.e. the whole category is parametrized by 3 universe levels, one for objects, one for morphisms and one for equalities. The consequence is that the category does not contain a set of all morphisms, instead it contains a set of morphisms for any pair of objects. Furthermore the sets of morphisms are equipped with an equivalence relation \Verb{_≈_}, making them setoids. This solves the aforementioned issue of how to implement equality between morphisms, it is just added to the definition of a category. This kind of category is also called a \textit{setoid-enriched category}.
Because of this proofs like \Verb{∘-resp-≈} are needed throughout the library to make sure that operations on morphisms respect the equivalence relation. Lastly, the authors also add symmetric proofs like \Verb{sym-assoc} to definitions, in this case to guarantee that the opposite category of the opposite category is equal to the original category, and a similar reason for requiring \Verb{identity²}, we won't address the need for these proofs and just accept the requirement as given for the rest of the thesis.
The definition of category that we will work with can be seen in listing~\ref{lst:category} (unnecessary information has been stripped).
The key differences to the definition above are firstly that instead of talking about collections, Agda's infinite \Verb{Set} hierarchy is utilized to prevent size issues.
The notion of category is thus parametrized by 3 universe levels, one for objects, one for morphisms and one for equalities.
The consequence is that the category does not contain a type of all morphisms, instead it contains a type of morphisms for any pair of objects.
Furthermore the types of morphisms are equipped with an equivalence relation \Verb{_≈_}, making them setoids.
This addresses the aforementioned issue of how to implement equality between morphisms: the notion of equality is just added to the definition of a category. This kind of category is also called a \textit{setoid-enriched category}.
Because of using a custom equality relation, proofs like \Verb{∘-resp-≈} are needed throughout the library to make sure that operations on morphisms respect the equivalence relation.
Lastly, the authors also include symmetric proofs like \Verb{sym-assoc} to definitions, in this case to guarantee that the opposite category of the opposite category is equal to the original category, and a similar reason for requiring \Verb{identity²}, we won't address the need for these proofs and just accept the requirement as given for the rest of the thesis.
\begin{listing}[H]
\begin{minted}{agda}

18
thesis/src/03_partiality-monads.tex
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@ -330,25 +330,25 @@ To show that $\mathbf{D}$ is commutative we will use another proof principle pre
$\mathbf{D}$ is commutative.
\end{theorem}
\begin{proof}
Using corollary~\ref{cor:solution} it suffices to show that both $\tau^* \circ \hat{\tau}$ and $\hat{\tau}^* \circ \tau$ are solutions of some guarded morphism $g$.
Using corollary~\ref{cor:solution} it suffices to show that both $\tau^* \circ \sigma$ and $\sigma^* \circ \tau$ are solutions of some guarded morphism $g$.
Let $w = (dstr + dstr) \circ dstl \circ (out \times out)$ and
\[g = out^{-1} \circ [ i_1 + D i_1 \circ \hat{\tau} , i_2 \circ [ D i_1 \circ \tau , later \circ now \circ i_2 ] ] \circ w\]
$g$ is trivially guarded by $[ id + D i_1 \circ \hat{\tau} , i_2 \circ [ D i_1 \circ \tau , later \circ now \circ i_2 ] ] \circ w$.
\[g = out^{-1} \circ [ i_1 + D i_1 \circ \sigma , i_2 \circ [ D i_1 \circ \tau , later \circ now \circ i_2 ] ] \circ w\]
$g$ is trivially guarded by $[ id + D i_1 \circ \sigma , i_2 \circ [ D i_1 \circ \tau , later \circ now \circ i_2 ] ] \circ w$.
We are left to show that $\tau^* \circ \hat{\tau}$ and $\hat{\tau}^* \circ \tau$ are solutions of $g$, i.e.:
We are left to show that $\tau^* \circ \sigma$ and $\sigma^* \circ \tau$ are solutions of $g$, i.e.:
\[\tau^* \circ \hat{\tau} = out^{-1} \circ [ id + \hat{\tau} , i_2 \circ [ \tau , later \circ \tau^* \circ \hat{\tau} ] ] \circ w = [ now , \tau^* \circ \hat{\tau}]^* \circ g \]
\[\hat{\tau}^* \circ \tau = out^{-1} \circ [ id + \hat{\tau} , i_2 \circ [ \tau , later \circ \hat{\tau}^* \circ \tau ] ] \circ w = [ now , \hat{\tau}^* \circ \tau]^* \circ g \]
\[\tau^* \circ \sigma = out^{-1} \circ [ id + \sigma , i_2 \circ [ \tau , later \circ \tau^* \circ \sigma ] ] \circ w = [ now , \tau^* \circ \sigma]^* \circ g \]
\[\sigma^* \circ \tau = out^{-1} \circ [ id + \sigma , i_2 \circ [ \tau , later \circ \sigma^* \circ \tau ] ] \circ w = [ now , \sigma^* \circ \tau]^* \circ g \]
The first step in both equations can be proven by monicity of $out$ and then using \ref{D3} and the dual diagram for $\hat{\tau}$ which is a direct consequence of \ref{D3}:
The first step in both equations can be proven by monicity of $out$ and then using \ref{D3} and the dual diagram for $\sigma$ which is a direct consequence of \ref{D3}:
% https://q.uiver.app/#q=WzAsNSxbMCwwLCJYIFxcdGltZXMgRFkiXSxbMCwxLCJEKFggXFx0aW1lcyBZKSJdLFs0LDAsIlggXFx0aW1lcyBZICsgWCBcXHRpbWVzIERZIl0sWzQsMSwiWCBcXHRpbWVzIFkgK0QoWCBcXHRpbWVzIFkpIl0sWzIsMCwiWCBcXHRpbWVzIChZICsgRFkpIl0sWzAsMSwiXFxoYXR7XFx0YXV9IiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzEsMywib3V0Il0sWzIsMywiaWQgKyBcXGhhdHtcXHRhdX0iLDJdLFswLDQsImlkIFxcdGltZXMgb3V0Il0sWzQsMiwiZHN0bCJdXQ==
\[\begin{tikzcd}[ampersand replacement=\&]
{X \times DY} \&\& {X \times (Y + DY)} \&\& {X \times Y + X \times DY} \\
{D(X \times Y)} \&\&\&\& {X \times Y +D(X \times Y)}
\arrow["{\hat{\tau}}", dashed, from=1-1, to=2-1]
\arrow["{\sigma}", dashed, from=1-1, to=2-1]
\arrow["out", from=2-1, to=2-5]
\arrow["{id + \hat{\tau}}"', from=1-5, to=2-5]
\arrow["{id + \sigma}"', from=1-5, to=2-5]
\arrow["{id \times out}", from=1-1, to=1-3]
\arrow["dstl", from=1-3, to=1-5]
\end{tikzcd}\]

12
thesis/src/04_iteration.tex
View file

@ -282,15 +282,15 @@ Of course there is also a symmetric version of this:
& {K(X \times KY)} && {K(X \times Y)} \\
{KX \times Y}
\arrow["\tau", from=1-2, to=1-4]
\arrow["{\hat{\tau}}"', from=1-2, to=3-2]
\arrow["{\hat{\tau}^*}", from=1-4, to=3-4]
\arrow["{\sigma}"', from=1-2, to=3-2]
\arrow["{\sigma^*}", from=1-4, to=3-4]
\arrow["{\tau^*}"', from=3-2, to=3-4]
\arrow["{id \times \eta}", curve={height=-12pt}, from=4-1, to=1-2]
\arrow["{\hat{\tau}}"', curve={height=12pt}, from=4-1, to=3-4]
\arrow["{\sigma}"', curve={height=12pt}, from=4-1, to=3-4]
\end{tikzcd}\]
The proofs for \ref{sharpr1} and the proof that $\hat{\tau}^* \circ \tau$ is right iteration preserving are straightforward and can be looked up in the formalization.
The proof that $\tau^* \circ \hat{\tau}$ is right iteration preserving is non-trivial, so we will look at it in more detail:
Let $Z \in \obj{\C}, h : Z \rightarrow KY + Z$ and let us introduce a definition for brevity: $\psi = \tau^* \circ \hat{\tau}$. We now use remark~\ref{rem:proofbyleftstability} to show that $\psi$ is right iteration preserving:
The proofs for \ref{sharpr1} and the proof that $\sigma^* \circ \tau$ is right iteration preserving are straightforward and can be looked up in the formalization.
The proof that $\tau^* \circ \sigma$ is right iteration preserving is non-trivial, so we will look at it in more detail:
Let $Z \in \obj{\C}, h : Z \rightarrow KY + Z$ and let us introduce a definition for brevity: $\psi = \tau^* \circ \sigma$. We now use remark~\ref{rem:proofbyleftstability} to show that $\psi$ is right iteration preserving:
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\[\begin{tikzcd}