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\begin{document}

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\section*{power}

\begin{quote}%
This entry is about the [[formal dual]] to [[tensoring]] in the generality of [[category theory]]. For the different concept of \emph{[[cotensor product]]} of [[comodules]] see there.


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

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

[[!include enriched category theory contents]]

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In a [[closed monoidal category|closed]] [[symmetric monoidal category]] $V$ the [[internal hom]] $[-,-] : V^{op} \times V \to V$ satisfies the [[natural isomorphism]]

\begin{displaymath}
[v_1,[v_2,v_3]] \simeq [v_2,[v_1,v_3]]
\end{displaymath}
for all [[object]]s $v_i \in V$ (\href{closed+monoidal+category#TensorHomIsoInternalizes}{prop.}). If we regard $V$ as a $V$-[[enriched category]] we write $V(v_1,v_2) := [v_1,v_2]$ and this reads

\begin{displaymath}
V(v_1,V(v_2,v_3)) \simeq V(v_2,V(v_1,v_3))
  \,.
\end{displaymath}
If we now pass more generally to any $V$-[[enriched category]] $C$ then we still have the enriched [[hom object]] functor $C(-,-) : C^{op} \times C \to V$. One says that $C$ is \emph{powered} over $V$ if it is in addition equipped also with a mixed operation $\pitchfork : V^{op} \times C \to C$ such that $\pitchfork(v,c)$ behaves as if it were a hom of the object $v \in V$ into the object $c \in C$ in that it satisfies the natural isomorphism

\begin{displaymath}
C(c_1,\pitchfork(v,c_2)) \simeq V(v,C(c_1,c_2))
  \,.
\end{displaymath}
\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{udefn}
Let $V$ be a [[closed monoidal category|closed]] [[monoidal category]]. In a $V$-[[enriched category]] $C$, the \textbf{power} of an object $y\in C$ by an object $v\in V$ is an object $\pitchfork(v,y) \in C$ with a [[natural isomorphism]]

\begin{displaymath}
C(x, \pitchfork(v,y)) \cong V(v, C(x,y))
\end{displaymath}
where $C(-,-)$ is the $V$-valued hom of $C$ and $V(-,-)$ is the [[internal hom]] of $V$.

We say that $C$ is \textbf{powered} or \textbf{cotensored} over $V$ if all such power objects exist.

\end{udefn}
\begin{uremark}
Powers are frequently called \emph{cotensors} and a $V$-category having all powers is called \emph{cotensored}, while the word ``power'' is reserved for the case $V=$ [[Set]]. However, there seems to be no good reason for making this distinction. Moreover, the word ``tensor'' is fairly overused, and unfortunate since a tensor (= a [[copower]]) is a [[colimit]], while a cotensor (= power) is a [[limit]].

\end{uremark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{itemize}%
\item Powers are a special sort of [[weighted limit]]s. Conversely, all weighted limits can be constructed from powers together with [[conical limit]]s. The dual colimit notion of a power is a [[copower]].

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

\begin{itemize}%
\item $V$ itself is always powered over itself, with $\pitchfork(v_1,v_2) := [v_1,v_2]$.


\item Every [[locally small category]] $C$ ($V = (Set,\times)$ ) with all [[product]]s is powered over [[Set]]: the powering operation

\begin{displaymath}
\pitchfork(S,c) := \prod_{s\in S} c
\end{displaymath}
of an object $c$ by a set $S$ forms the $|S|$-fold [[cartesian product]] of $c$ with itself, where $|S|$ is the [[cardinality]] of $S$.

The defining natural isomorphism

\begin{displaymath}
Hom_C(c_1,\pitchfork(S,c_2))\simeq Hom_{Set}(S,Hom_C(c_1,c_2))
\end{displaymath}
is effectively the definition of the product (see [[limit]]).



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

\begin{itemize}%
\item [[tensored and cotensored category]]


\item [[copower]], [[(∞,1)-copower]]


\item [[pullback-power]]



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

\begin{itemize}%
\item [[Max Kelly]], section 3.7 of \emph{Basic concepts of enriched category theory} (\href{http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html}{tac} ,\href{http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf}{pdf})


\item [[Francis Borceux]], Vol 2, Section 6.5 of \emph{[[Handbook of Categorical Algebra]]}, Cambridge University Press (1994)



\end{itemize}
[[!redirects cotensor]] [[!redirects cotensoring]]

[[!redirects powers]] [[!redirects cotensors]] [[!redirects cotensored category]]

[[!redirects powering]]



\end{document}