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\begin{document}

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\section*{change of enriching 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{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{generalizations_and_enhancements}{Generalizations and enhancements}\dotfill \pageref*{generalizations_and_enhancements} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Given two [[monoidal categories]] $(V_1,\otimes_1, I_1)$ and $(V_2, \otimes_2, I_2)$ used for enrichment in [[enriched category theory]] (for instance two [[Bénabou cosmoi]]), a [[lax monoidal functor]] between them

\begin{displaymath}
F 
  \;\colon\;
  (V_1, \otimes_1, I_1)
    \longrightarrow
  (V_2, \otimes_2, I_2)
\end{displaymath}
canonically induces a [[2-functor]]

\begin{displaymath}
F_\ast \;\colon\; V_1 Cat \longrightarrow V_2 Cat
\end{displaymath}
between [[categories of V-enriched categories]] sending a $V_1$-[[enriched category]] $\mathcal{C}$ to the $V_2$-[[enriched category]] $F_\ast(\mathcal{C})$ such that

\begin{enumerate}%
\item $F_\ast(\mathcal{C})$ has the same [[objects]] as $\mathcal{C}$;


\item the [[hom-objects]] of $F_\ast(\mathcal{C})$ are the [[images]] of the hom-objects of $\mathcal{C}$ under $F$:

\begin{displaymath}
F_\ast(\mathcal{C})(x,y)
  \;\coloneqq\;
  F\big(\mathcal{C}(x,y)\big)
\end{displaymath}

\item the [[composition]], [[unit]], [[associator]] and [[unitor]] [[morphisms]] in $F_\ast(\mathcal{C})$ are the [[images]] of those of $\mathcal{C}$ composed with the structure morphisms of the [[lax monoidal functor|lax monoidal]]-[[structure]] on $F$.



\end{enumerate}
(\hyperlink{EilenbergKelly65}{Eilenberg-Kelly 65})

This operation $F_\ast$ is sometimes called \textbf{change of enriching category} or \textbf{change of enriching base} or just \textbf{change of base} (e.g. \hyperlink{Crutwell14}{Crutwell 14, chapter 4}, \hyperlink{Riehl14}{Riehl 14, lemma 3.4.3}). But notice that, despite some vague similarity, this is different from [[base change]] of [[slice categories]].

In the special case that $\mathcal{C}$ has a single object and is hence (if thought of as [[pointed object|pointed]] by that object) equivalently a [[monoid object]] in $V_1$, this statement reduces to the statement that lax monoidal functors preserve [[monoids]] (\href{monoidal+functor#MonoidsToMonoidsByLaxMonoidal}{this Prop.})

\hypertarget{generalizations_and_enhancements}{}\subsection*{{Generalizations and enhancements}}\label{generalizations_and_enhancements}

\begin{itemize}%
\item The operation of change of enriching category is functorial from [[MonCat]] to [[2Cat]]. In particular, any [[monoidal adjunction]] $V_1\rightleftarrows V_2$ gives rise to a [[2-adjunction]] $V_1 Cat\rightleftarrows V_2 Cat$ (and also for profunctors).


\item $V$-enriched categories can be defined more generally when $V$ is a [[multicategory]], and any functor $F:V_1\to V_2$ between multicategories induces a change of enrichment 2-functor. Note that a functor of multicategorise between the underlying multicategories of two monoidal categories corresponds to a \emph{lax} monoidal functor, as in the original version above. The multicategorical version also includes change of enrichment between [[closed categories]].


\item A different generalization is when $V$ is a [[bicategory]], yielding [[bicategory-enriched categories]]. Any lax functor of bicategories induces a similar functor between its 2-categories of enriched categories.


\item When $V_1$ and $V_2$ are cocomplete monoidal categories (or locally cocomplete bicategories), so that the bicategories $V_i$-[[Prof]] of $V_i$-categories and $V_i$-[[profunctors]] exist, change of enrichment also induces a [[lax functor]] $V_1 Prof \to V_2 Prof$. To make this version functorial on $MonCat_{cocomplete}$, we need to consider $V_i Prof$ as [[double categories]], yielding a functor from [[MonCat]] to [[DblCat]].


\item Finally, a generalization subsuming all of these is that for any [[virtual double category]] $V$ we can construct another virtual double category $V Prof$, and this construction is functorial.



\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{Underlying}\hypertarget{Underlying}{}
For any a monoidal category $V$, the functor $V(I,-): V \to Set$ is lax monoidal, hence induces a 2-functor from $V Cat$ to $Cat$. This assigns to any $V$-enriched category, $\mathcal{C}$, its [[underlying ordinary category]], usually denoted $\mathcal{C}_0$, defined by $\mathcal{C}_0(x,y) = V(I, hom(x,y))$.

\end{example}
\begin{example}
\label{FreeCats}\hypertarget{FreeCats}{}
If $V$ is a [[cocomplete category|cocomplete]] monoidal category, the functor $V(I,-)$ above has a [[left adjoint]] that takes a set $X$ to the [[copower]] of $X$ copies of $I$. The resulting 2-functor $Cat \to V Cat$ takes an ordinary category to the ``free'' $V$-category it generates. Such $V$-categories are used, for instance, in subsuming ``conical'' [[limits]] under enriched [[weighted limits]].

By functoriality, the adjunction between these two functors gives rise to a 2-adjunction $Cat \rightleftarrows V Cat$.

\end{example}
\begin{example}
\label{GenUnderlying}\hypertarget{GenUnderlying}{}
More generally, if an adjunction $F:V_1 \rightleftarrows V_2:U$ can be regarded as a [[free-forgetful adjunction]], then the corresponding adjunction between enriched categories can also be so regarded. For instance, if $V_1 = Top$ and $V_2 = G Top$ for some [[topological group]] $G$, then the right adjoint $V_2 Cat \to V_1 Cat$ takes the ``underlying topologically enriched category'' of a category enriched in $G$-spaces (its morphisms are the fixed-point spaces of the original $G$-space-enriched category; we think of these as the ``$G$-equivariant maps'' while the original spaces consisted of not-necessarily-equivariant maps acted on by conjugation). Similarly, any category enriched in [[spectra]] has an underlying category enriched in spaces (whose hom-spaces are the 0-spaces of the original hom-spectra), any [[dg-category]] has an underlying [[Ab-category]] (whose morphisms are the degree-0 ones in the original category), and so on.

\end{example}
\begin{example}
\label{RealSing}\hypertarget{RealSing}{}
The [[geometric realization]] and [[total singular complex]] adjunction $Real:SSet \rightleftarrows Top : Sing$ between [[simplicial sets]] and [[topological spaces]] induces another adjunction $SSet Cat \rightleftarrows Top Cat$ between [[simplicially enriched categories]] and [[topologically enriched categories]].

\end{example}
\begin{example}
\label{HomotopyCat}\hypertarget{HomotopyCat}{}
If $V=$[[SSet]] or [[Top]], then the set of connected components $\pi_0:V\to Set$ is lax monoidal. The resulting change of enrichment functor takes a $V$-category $C$ to its ``naive homotopy category'' $h C(x,y) = \pi_0(C(x,y))$ obtained by ``identifying homotopic morphisms''.

Similarly, the [[fundamental groupoid]] functor $\Pi_1:V\to Gpd$ is lax monoidal, so any $V$-category has an underlying [[(2,1)-category]].

\end{example}
\begin{example}
\label{EnrichedHomotopyCat}\hypertarget{EnrichedHomotopyCat}{}
An enhancement of the last example is that if $V$ is any [[monoidal model category]], then its [[homotopy category]] $Ho(V)$ comes with a lax monoidal functor $\gamma : V \to Ho(V)$. Thus any $V$-category $C$ has an underlying $Ho(V)$-enriched ``homotopy category'' $h C$. Usually the underlying ordinary category of $h C$ is the naive homotopy category from the previous example.

\end{example}
\begin{example}
\label{Tau1}\hypertarget{Tau1}{}
The nerve functor $N:Cat \to SSet$ preserves products and has a left adjoint $\tau_1$ that also preserves products. Thus, we have a change of enrichment adjunction in which any [[2-category]] can be regarded as a [[simplicially enriched category]], and similarly any simplicially enriched category has a ``homotopy 2-category''. The latter plays an important role in the theory of [[quasi-categories]].

\end{example}
\begin{example}
\label{PolyMorphisms}\hypertarget{PolyMorphisms}{}
\textbf{([[poly-morphisms]])}

For $F =P \;\colon\; Set \to Set$ the [[power set]]-[[functor]], the change of base functor $P_\ast \;\colon\;
 Cat \to Cat$ sends plain [[categories]] to plain categories. For $C$ any category, the morphisms of $C^{poly} \coloneqq P\ast(C)$ are called the \emph{[[poly-morphisms]]} of $C$ in \href{poly-morphism#Mochizuki12}{Mochizuki 12, section 0}.

\end{example}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Samuel Eilenberg]], [[Max Kelly]], \emph{Closed categories}. Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965).


\item [[Geoff Cruttwell]], chapter 4 of \emph{Normed spaces and the Change of Base for Enriched Categories}, 2014 (\href{http://pages.cpsc.ucalgary.ca/~gscruttw/publications/thesis4.pdf}{pdf})


\item [[Emily Riehl]], lemma 3.4.3 in \emph{[[Categorical Homotopy Theory]]}, Cambridge University Press 2014



\end{itemize}
[[!redirects changes of enriching category]] [[!redirects changes of enriching categories]] [[!redirects change of enriching categories]]

[[!redirects change of enriching base]] [[!redirects changes of enriching base]] [[!redirects changes of enriching bases]]

[[!redirects change of base in enriched category theory]] [[!redirects changes of base in enriched category theory]] [[!redirects changes of bases in enriched category theory]]



\end{document}