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\begin{document}

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\section*{topologically enriched category}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{enriched_category_theory}{}\paragraph*{{Enriched category theory}}\label{enriched_category_theory}

[[!include enriched category theory contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{topologically_enriched_presheaves}{Topologically enriched presheaves}\dotfill \pageref*{topologically_enriched_presheaves} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

In the following we say \emph{[[Top]]-[[enriched category]]} and \emph{[[Top]]-[[enriched functor]]} etc. for what often is referred to as ``[[topological category]]'' and ``[[topological functor]]'' etc. As discussed there, these latter terms are ambiguous.

\begin{defn}
\label{kTop}\hypertarget{kTop}{}
Write

\begin{displaymath}
Top_{cg} \hookrightarrow Top
\end{displaymath}
for the [[full subcategory]] of [[Top]] on the [[compactly generated topological spaces]]. Under forming [[Cartesian product]]

\begin{displaymath}
(-)\times (-)
  \;\colon\;
  Top_{cg} \times Top_{cg} \longrightarrow Top_{cg}
\end{displaymath}
and compactly generated [[mapping spaces]]

\begin{displaymath}
(-)^{(-)}
  \;\colon\;
  Top_{cg}^{op}\times Top_{cg}
  \longrightarrow
  Top_{cg}
\end{displaymath}
this is a [[cartesian closed category]] (see at \emph{[[convenient category of topological spaces]]}).

\end{defn}
\begin{defn}
\label{TopEnrichedCategory}\hypertarget{TopEnrichedCategory}{}
A \textbf{[[topologically enriched category]]} $\mathcal{C}$ is a $Top_{cg}$-[[enriched category]], hence:

\begin{enumerate}%
\item a [[class]] $Obj(\mathcal{C})$, called the \textbf{class of [[objects]]};


\item for each $a,b\in Obj(\mathcal{C})$ a [[compactly generated topological space]]

\begin{displaymath}
\mathcal{C}(a,b)\in Top_{cg}
  \,,
\end{displaymath}
called the \textbf{space of [[morphisms]]} or the \textbf{[[hom-space]]} between $a$ and $b$;


\item for each $a,b,c\in Obj(\mathcal{C})$ a [[continuous function]]

\begin{displaymath}
\circ_{a,b,c} \;\colon\; \mathcal{C}(a,b)\times \mathcal{C}(b,c) \longrightarrow \mathcal{C}(a,c)
\end{displaymath}
out of the cartesian product, called the \emph{[[composition]]} operation


\item for each $a \in Obj(\mathcal{C})$ a point $id_a\in \mathcal{C}(a,a)$, called the \emph{[[identity]]} morphism on $a$



\end{enumerate}
such that the composition is [[associativity|associative]] and [[unitality|unital]].

\end{defn}
\begin{remark}
\label{UnderlyingCategoryOfTopEnrichedCategory}\hypertarget{UnderlyingCategoryOfTopEnrichedCategory}{}
Given a [[topologically enriched category]] as in def. \ref{TopEnrichedCategory}, then forgetting the topology on the [[hom-spaces]] (along the [[forgetful functor]] $U \colon Top_k \to Set$) yields an ordinary [[locally small category]] with

\begin{displaymath}
Hom_{\mathcal{C}}(a,b) = U(\mathcal{C}(a,b))
  \,.
\end{displaymath}
It is in this sense that $\mathcal{C}$ is a category with [[extra structure]], and hence ``[[enriched category|enriched]]''.

\end{remark}
The archetypical example is the following:

\begin{example}
\label{TopkAsATopologicallyEnrichedCategory}\hypertarget{TopkAsATopologicallyEnrichedCategory}{}
The category $Top_{cg}$ from def. \ref{kTop} itself, being a [[cartesian closed category]], canonically obtains the structure of a [[topologically enriched category]], def. \ref{TopEnrichedCategory}, with [[hom-spaces]] given by compactly generated [[mapping spaces]]

\begin{displaymath}
Top_{cg}(X,Y)
  \coloneqq
  Y^X
\end{displaymath}
and with [[composition]]

\begin{displaymath}
Y^X \times Z^Y
  \longrightarrow
  Z^X
\end{displaymath}
given by the (product$\dashv$ mapping-space)-[[adjunct]] of the [[evaluation morphism]]

\begin{displaymath}
X \times Y^X \times Z^Y
  \overset{(ev, id)}{\longrightarrow}
  Y \times Z^Y
  \overset{ev}{\longrightarrow}
  Z
  \,.
\end{displaymath}
\end{example}
\begin{defn}
\label{TopologicallyEnrichedFunctor}\hypertarget{TopologicallyEnrichedFunctor}{}
A [[topologically enriched functor]] between two [[topologically enriched categories]]

\begin{displaymath}
F \;\colon\;  \mathcal{C}  \longrightarrow \mathcal{D}
\end{displaymath}
is a $Top_{cg}$-[[enriched functor]], hence:

\begin{enumerate}%
\item a [[function]]

\begin{displaymath}
F_0 \colon Obj(\mathcal{C}) \longrightarrow Obj(\mathcal{D})
\end{displaymath}
of [[objects]];


\item for each $a,b \in Obj(\mathcal{C})$ a [[continuous function]]

\begin{displaymath}
F_{a,b} \;\colon\; \mathcal{C}(a,b) \longrightarrow \mathcal{D}(F_0(a), F_0(b))
\end{displaymath}
of [[hom-spaces]]



\end{enumerate}
such that this preserves [[composition]] and [[identity]] morphisms in the evident sense.

A [[homomorphism]] of topologically enriched functors

\begin{displaymath}
\eta \;\colon\; F \Rightarrow G
\end{displaymath}
is a $Top_{cg}$-[[enriched natural transformation]]: for each $c \in Obj(\mathcal{C})$ a choice of morphism $\eta_c \in \mathcal{D}(F(c),G(c))$ such that for each pair of objects $c,d \in \mathcal{C}$ the two continuous functions

\begin{displaymath}
\eta_d \circ F(-) \;\colon\; \mathcal{C}(c,d) \longrightarrow \mathcal{D}(F(c), G(d))
\end{displaymath}
and

\begin{displaymath}
G(-) \circ \eta_c \;\colon\; \mathcal{C}(c,d) \longrightarrow \mathcal{D}(F(c), G(d))
\end{displaymath}
agree.

We write $[\mathcal{C}, \mathcal{D}]$ for the resulting category of topologically enriched functors. This itself naturally obtains the structure of [[topologically enriched category]], see at \emph{[[enriched functor category]]}.

\end{defn}
\hypertarget{topologically_enriched_presheaves}{}\subsection*{{Topologically enriched presheaves}}\label{topologically_enriched_presheaves}

\begin{example}
\label{TopologicallyEnrichedFunctorsToTopk}\hypertarget{TopologicallyEnrichedFunctorsToTopk}{}
For $\mathcal{C}$ any topologically enriched category, def. \ref{TopEnrichedCategory} then a topologically enriched functor

\begin{displaymath}
F \;\colon\; \mathcal{C} \longrightarrow Top_{cg}
\end{displaymath}
to the archetical topologically enriched category from example \ref{TopkAsATopologicallyEnrichedCategory} may be thought of as a topologically [[enriched presheaf|enriched]] [[copresheaf]], at least if $\mathcal{C}$ is [[small category|small]] (in that its [[class]] of objects is a proper [[set]]).

Hence the category of topologically enriched functors

\begin{displaymath}
[\mathcal{C}, Top_{cg}]
\end{displaymath}
according to def. \ref{TopologicallyEnrichedFunctor} may be thought of as the ([[copresheaf|co-]])[[presheaf category]] over $\mathcal{C}$ in the realm of topological enriched categories.

A functor $F \in [\mathcal{C}, Top_{cg}]$ is equivalently

\begin{itemize}%
\item a [[compactly generated topological space]] $F_a\in Top_{cg}$ for each object $a \in Obj(\mathcal{C})$;


\item a [[continuous function]]

\begin{displaymath}
F_a \times \mathcal{C}(a,b) \longrightarrow F_b
\end{displaymath}
for all pairs of objects $a,b \in Obj(\mathcal{C})$



\end{itemize}
such that composition is respected, in the evident sense.

For every object $c \in \mathcal{C}$, there is a topologically enriched [[representable functor]], denoted $y(c) or \mathcal{C}(c,-)$ which sends objects to

\begin{displaymath}
y(c)(d) = \mathcal{C}(c,d) \in Top_{cg}
\end{displaymath}
and whose action on morphisms is, under the above identification, just the [[composition]] operation in $\mathcal{C}$.

\end{example}
There is a full blown $Top_{cg}$-[[enriched Yoneda lemma]]. The following records a slightly simplified version.

\begin{prop}
\label{TopologicallyEnrichedYonedaLemma}\hypertarget{TopologicallyEnrichedYonedaLemma}{}
\textbf{(topologically enriched Yoneda-lemma)}

Let $\mathcal{C}$ be a [[topologically enriched category]], def. \ref{TopEnrichedCategory}, write $[\mathcal{C}, Top_{cg}]$ for its category of topologically enriched (co-)presheaves, and for $c\in Obj(\mathcal{C})$ write $y(c) = \mathcal{C}(c,-) \in [\mathcal{C}, Top_k]$ for the topologically enriched functor that it represents, all according to example \ref{TopologicallyEnrichedFunctorsToTopk}. Recall also the $Top_{cg}$-tensored functors $F \cdot X$ from that example.

For $c\in Obj(\mathcal{C})$, $X \in Top$ and $F \in [\mathcal{C}, Top_{cg}]$, there is a [[natural bijection]] between

\begin{enumerate}%
\item morphisms $y(c) \cdot X \longrightarrow F$ in $[\mathcal{C}, Top_{cg}]$;


\item morphisms $X \longrightarrow F(c)$ in $Top_{cg}$.



\end{enumerate}
\end{prop}
\begin{proof}
Given a morphism $\eta \colon y(c) \cdot X \longrightarrow F$ consider its component

\begin{displaymath}
\eta_c \;\colon\; \mathcal{C}(c,c)\times X \longrightarrow F(c)
\end{displaymath}
and restrict that to the identity morphism $id_c \in \mathcal{C}(c,c)$ in the first argument

\begin{displaymath}
\eta_c(id_c,-) \;\colon\; X \longrightarrow F(c)
  \,.
\end{displaymath}
We claim that just this $\eta_c(id_c,-)$ already uniquely determines all components

\begin{displaymath}
\eta_d \;\colon\; \mathcal{C}(c,d)\times X \longrightarrow F(d)
\end{displaymath}
of $\eta$, for all $d \in Obj(\mathcal{C})$: By definition of the transformation $\eta$ (def. \ref{TopologicallyEnrichedFunctor}), the two functions

\begin{displaymath}
F(-) \circ \eta_c
  \;\colon\;
  \mathcal{C}(c,d)
  \longrightarrow
  F(d)^{\mathcal{C}(c,c) \times X}
\end{displaymath}
and

\begin{displaymath}
\eta_d  \circ \mathcal{C}(c,-) \times X
  \;\colon\;
  \mathcal{C}(c,d)
  \longrightarrow
  F(d)^{\mathcal{C}(c,c) \times X}
\end{displaymath}
agree. This means that they may be thought of jointly as a function with values in commuting squares in $Top$ of this form:

\begin{displaymath}
f
  \;\;\;\;
  \mapsto
  \;\;\;\;
  \itexarray{
    \mathcal{C}(c,c) \times X &\overset{\eta_c}{\longrightarrow}& F(c)
    \\
    {}^{\mathllap{\mathcal{C}(c,f)}}\downarrow && \downarrow^{\mathrlap{F(f)}}
    \\
    \mathcal{C}(c,d) \times X &\underset{\eta_d}{\longrightarrow}& F(d)
  }
\end{displaymath}
For any $f \in \mathcal{C}(c,d)$, consider the restriction of

\begin{displaymath}
\eta_d \circ \mathcal{C}(c,f) \in F(d)^{\mathcal{C}(c,c) \times X}
\end{displaymath}
to $id_c \in \mathcal{C}(c,c)$, hence restricting the above commuting squares to

\begin{displaymath}
f
  \;\;\;\;
  \mapsto
  \;\;\;\;
  \itexarray{
    \{id_c\} \times X &\overset{\eta_c}{\longrightarrow}& F(c)
    \\
    {}^{\mathllap{\mathcal{C}(c,f)}}\downarrow && \downarrow^{\mathrlap{F}(f)}
    \\
    \{f\} \times X &\underset{\eta_d}{\longrightarrow}& F(d)
  }
\end{displaymath}
This shows that $\eta_d$ is fixed to be the function

\begin{displaymath}
\eta_d(f,x) = F(f)\circ \eta_c(id_c,x)
\end{displaymath}
and this is a continuous function since all the operations it is built from are continuous.

Conversely, given a continuous function $\alpha \colon X \longrightarrow F(c)$, define for each $d$ the function

\begin{displaymath}
\eta_d \colon (f,x) \mapsto F(f) \circ \alpha
  \,.
\end{displaymath}
Running the above analysis backwards shows that this determines a transformation $\eta \colon y(c)\times X \to F$.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
With $Top_{cg}$ equipped with the [[classical model structure on topological spaces]], which is a [[locally presentable (∞,1)-category|presentation]] for the archetypical [[(∞,1)-category]] [[∞Grpd]] of [[∞-groupoids]], then the topological functor category

\begin{displaymath}
[\mathcal{C},Top_{cg}]
\end{displaymath}
(def. \ref{TopologicallyEnrichedFunctor}, def. \ref{TopkAsATopologicallyEnrichedCategory}) is a model for the [[(∞,1)-category of (∞,1)-presheaves]] on $\mathcal{C}^{op}$. This is made precise by the \emph{[[model structure on enriched functors]]}, $[\mathcal{C},Top_{Quillen}]_{proj}$. See at \emph{\href{classical+model+structure+on+topological+spaces#ModelStructureOnTopEnrichedFunctors}{classical model structure on topological spaces -- Model structure on functors}} for details.

\end{remark}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[simplicially enriched category]]


\item [[model structure on enriched functors]]


\item [[(infinity,1)-category]]



\end{itemize}
[[!redirects topologically enriched categories]]

[[!redirects Top-enriched category]] [[!redirects Top-enriched categories]]

[[!redirects topologically enriched functor]] [[!redirects topologically enriched functors]]

[[!redirects Top-enriched functor]] [[!redirects Top-enriched functors]]

[[!redirects pointed topologically enriched category]] [[!redirects pointed topologically enriched categories]]

[[!redirects pointed topologically enriched functor]] [[!redirects pointed topologically enriched functors]]



\end{document}