Use darkorange for all bordercolors
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@ -8,6 +8,7 @@ This implementation formalizes the following concepts:
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\begin{tabular}{ l l }
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\begin{tabular}{ l l }
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Name & Link \\
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Name & Link \\
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\hline
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\hline
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Equivalences & \sourcelink{Cat.Equivalence} \\
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Categories & \sourcelink{Cat.Category} \\
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Categories & \sourcelink{Cat.Category} \\
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Functors & \sourcelink{Cat.Category.Functor} \\
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Functors & \sourcelink{Cat.Category.Functor} \\
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Products & \sourcelink{Cat.Category.Product} \\
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Products & \sourcelink{Cat.Category.Product} \\
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@ -19,12 +20,9 @@ Monads & \sourcelink{Cat.Category.Monad} \\
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Kleisli Monads & \sourcelink{Cat.Category.Monad.Kleisli} \\
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Kleisli Monads & \sourcelink{Cat.Category.Monad.Kleisli} \\
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Monoidal Monads & \sourcelink{Cat.Category.Monad.Monoidal} \\
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Monoidal Monads & \sourcelink{Cat.Category.Monad.Monoidal} \\
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Voevodsky's construction & \sourcelink{Cat.Category.Monad.Voevodsky} \\
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Voevodsky's construction & \sourcelink{Cat.Category.Monad.Voevodsky} \\
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%% Categories & \null \\
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%%
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Opposite category & \sourcelink{Cat.Categories.Opposite} \\
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Opposite category & \sourcelink{Cat.Categories.Opposite} \\
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Category of sets & \sourcelink{Cat.Categories.Sets} \\
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Category of sets & \sourcelink{Cat.Categories.Sets} \\
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Span category & \sourcelink{Cat.Categories.Span} \\
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Span category & \sourcelink{Cat.Categories.Span} \\
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%%
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\end{tabular}
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\end{tabular}
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\end{center}
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\end{center}
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%
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%
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@ -42,12 +40,16 @@ Monoids & \sourcelink{Cat.Category.Monoid} \\
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\end{tabular}
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\end{tabular}
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\end{center}
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\end{center}
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%
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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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As well as a range of various results about these. E.g. I have shown
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category of sets has products. In the following I aim to demonstrate some of the
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that the category of sets has products. In the following I aim to
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techniques employed in this formalization and in the interest of brevity I will
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demonstrate some of the techniques employed in this formalization and
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not detail all the things I have formalized. In stead, I have selected a parts
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in the interest of brevity I will not detail all the things I have
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of this formalization that highlight some interesting proof techniques relevant
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formalized. In stead, I have selected parts of this formalization that
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to doing proofs in Cubical Agda.
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highlight some interesting proof techniques relevant to doing proofs
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in Cubical Agda. This chapter will focus on the definition of
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\emph{categories}, \emph{equivalences}, the \emph{opposite category},
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the \emph{category of sets}, \emph{products}, the \emph{span category}
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and the two formulations of \emph{monads}.
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One such technique that is pervasive to this formalization is the idea of
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One such technique that is pervasive to this formalization is the idea of
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distinguishing types with more or less homotopical structure. To do this I have
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distinguishing types with more or less homotopical structure. To do this I have
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@ -792,16 +794,19 @@ proposition and then use $\lemPropF$. So we prove the generalization:
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But $\var{AreInverses}\ f\ g$ is a pair of equations on arrows, so we use
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But $\var{AreInverses}\ f\ g$ is a pair of equations on arrows, so we use
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$\propSig$ and the fact that both $A$ and $B$ are sets to close this proof.
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$\propSig$ and the fact that both $A$ and $B$ are sets to close this proof.
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\subsection{Category of categories}
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%% \subsection{Category of categories}
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Note that this category does in fact not exist. In stead I provide the
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definition of the ``raw'' category as well as some of the laws.
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Furthermore I provide some helpful lemmas about this raw category. For instance
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%% Note that this category does in fact not exist. In stead I provide
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I have shown what would be the exponential object in such a category.
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%% the definition of the ``raw'' category as well as some of the laws.
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These lemmas can be used to provide the actual exponential object in a context
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%% Furthermore I provide some helpful lemmas about this raw category.
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where we have a witness to this being a category. This is useful if this library
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%% For instance I have shown what would be the exponential object in
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is later extended to talk about higher categories.
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%% such a category.
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%% These lemmas can be used to provide the actual exponential object
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%% in a context where we have a witness to this being a category. This
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%% is useful if this library is later extended to talk about higher
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%% categories.
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\section{Products}
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\section{Products}
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\label{sec:products}
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\label{sec:products}
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@ -3,13 +3,15 @@
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\usepackage{natbib}
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\usepackage{natbib}
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\bibliographystyle{plain}
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\bibliographystyle{plain}
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\usepackage{xcolor}
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\usepackage[
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\usepackage[
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hidelinks,
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%% hidelinks,
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pdfusetitle,
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pdfusetitle,
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pdfsubject={category theory},
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pdfsubject={category theory},
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pdfkeywords={type theory, homotopy theory, category theory, agda}]
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pdfkeywords={type theory, homotopy theory, category theory, agda}]
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{hyperref}
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{hyperref}
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\definecolor{darkorange}{HTML}{ff8c00}
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\hypersetup{allbordercolors={darkorange}}
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\usepackage{graphicx}
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\usepackage{graphicx}
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\usepackage{parskip}
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\usepackage{parskip}
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