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\section*{center}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{of_groups_and_monoids}{Of groups and monoids}\dotfill \pageref*{of_groups_and_monoids} \linebreak
\noindent\hyperlink{of_lie_algebras}{Of Lie algebras}\dotfill \pageref*{of_lie_algebras} \linebreak
\noindent\hyperlink{OfCategories}{Of categories and higher categories}\dotfill \pageref*{OfCategories} \linebreak
\noindent\hyperlink{of_abelian_categories}{Of abelian categories}\dotfill \pageref*{of_abelian_categories} \linebreak
\noindent\hyperlink{of_groups}{Of $\infty$-groups}\dotfill \pageref*{of_groups} \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 general, the \emph{center} (or \emph{centre}) of an [[algebra|algebraic]] [[object]] $A$ is the collection of [[elements]] of $A$ which ``commute with all elements of $A$.'' This has a number of specific incarnations.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{of_groups_and_monoids}{}\subsubsection*{{Of groups and monoids}}\label{of_groups_and_monoids}

The original example is the \textbf{center} $Z(G)$ of a [[group]] $G$, which is defined to be the [[subgroup]] consisting of all elements $g\in G$ such that for all elements $h\in H$ the equality $g h=h g$ holds. The center is an [[abelian group|abelian]] subgroup, but not every abelian subgroup is in the center. See also [[centralizer]].

This notion of center of a group can be generalized to the center of a [[monoid]] in an obvious way.

\hypertarget{of_lie_algebras}{}\subsubsection*{{Of Lie algebras}}\label{of_lie_algebras}

The \textbf{center of a [[Lie algebra]]} $L$ is an abelian Lie subalgebra $Z(L)$, consisting of all elements $z\in L$ such that $[l,z]=0$ for all $z\in L$. There are generalizations for some other kinds of algebras.

\hypertarget{OfCategories}{}\subsubsection*{{Of categories and higher categories}}\label{OfCategories}

The center of a monoid can be [[horizontal categorification|horizontally categorified]] to the center of a [[category]]. Specifically, the \emph{center of a category} $C$ is defined to be the commutative monoid $[C,C](Id_C,Id_C)$ of [[natural transformation|endo-natural-transformations]] of the [[identity functor]] of $C$. It is straightforward to check that this reduces to the usual definition if $C = \mathbf{B}(A,\times)$ is the [[delooping]] of a [[monoid]].

\begin{itemize}%
\item For a [[generator]] $G$ of a category $\mathcal{C}$ there is an embedding of $Z(\mathcal{C})$ into the monoid $Hom(G,G)$ given by $\eta\mapsto\eta _G$. In particular, if $Hom(G,G)$ or $Z(Hom(G,G))$ is trivial, as happens e.g. for $Set$ with $G=\ast$, then so is $Z(\mathcal{C})$ (Hofmann 1975).


\item For [[Cauchy completion|Cauchy complete]] $\mathcal{C}$ the idempotent elements of $Z(\mathcal{C})$ correspond precisely to the \emph{quintessential localizations} of $\mathcal{C}$ (Johnstone 1996).



\end{itemize}
The notion of center can also be [[vertical categorification|vertically categorified]]. It is easy to categorify the notion of center of a category as defined above: if $C$ is an [[n-category]], then its \emph{center} is the monoidal $(n-1)$-category $[C,C](Id_C,Id_C)$ of endo-transformations of its identity functor. One expects that in general, this center will actually admit a natural structure of \emph{braided} monoidal $(n-1)$-category, just as the center of a category is actually a commutative monoid, not merely a monoid.

For instance if $C = \mathbf{B}_\otimes \mathcal{C}$ is the [[delooping]] of a [[monoidal category]], then this center is called the \emph{[[Drinfeld center]]} of $(C, \otimes)$.

Generally, we can now obtain a notion of the center of a monoidal $n$-category by regarding it as a one-object $(n+1)$-category, according to the [[delooping hypothesis]]. It follows that the center of a monoidal $n$-category should naturally be a braided monoidal $n$-category. This is known to be true when $n=0$ (the center of a monoid is a commutative monoid) and also for $n=1$ and $n=2$.

Note that a monoidal $n$-category has two different centers: if we regard it as a one-object $(n+1)$-category, then its center is a braided monoidal $n$-category, but if we regard it merely as an $n$-category, then its center is a braided monoidal $(n-1)$-category. The latter construction makes no reference to the monoidal structure. Likewise, a braided monoidal $n$-category has three different centers, depending on whether we regard it as an $n$-category, a connected $(n+1)$-category, or a 2-connected $(n+2)$-category, and so on (a $k$-tuply monoidal $n$-category has $k+1$ different centers).

It seems that in applications, however, one is usually most interested in the sort of center of a monoidal $n$-category $C$ obtained by regarding it as a one-object $(n+1)$-category, thereby obtaining a braided monoidal $n$-category. It is in this case, and seemingly this case only, that the center comes with a natural forgetful functor to $C$, corresponding to the classical inclusion of the center of a monoid. (For $n\gt 0$, however, this functor will not be an inclusion; the objects of the center of $C$ are objects of $C$ equipped with additional [[stuff, structure, property|structure]].)

Moreover, one expects that if we perform this ``canonical'' operation on a [[k-tuply monoidal n-category]] (for $k\ge 1$), the resulting braided monoidal $n$-category will actually be $(k+1)$-tuply monoidal. This is known to be true in the cases $n\le 4$: the center of a braided monoidal category is symmetric monoidal, the center of a braided monoidal 2-category is sylleptic, and the center of a sylleptic monoidal 2-category is symmetric.

Finally, if we decategorify further, we find that the center of a [[set]] (i.e. a [[0-category]]) is a monoidal [[(-1)-category]], i.e. the [[truth value]] ``true.'' This is what we ought to expect, since when $C$ is a set, there is precisely one endo-transformation of its identity endofunction (namely, the identity).

An old query about the categorical notion of center is archived at $n$Forum \href{http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2715&Focus=22852#Comment_22852}{here}.

\hypertarget{of_abelian_categories}{}\paragraph*{{Of abelian categories}}\label{of_abelian_categories}

A special case is the [[center of an abelian category]] which has a special entry because of a number of special applications and properties.

\hypertarget{of_groups}{}\subsubsection*{{Of $\infty$-groups}}\label{of_groups}

See [[center of an ∞-group]].

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

\begin{itemize}%
\item [[diagonal matrix]]


\item [[central product of groups]]



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

\begin{itemize}%
\item R.-E. Hoffmann, \emph{\"U{}ber das Zentrum einer Kategorie} , Math. Nachr. \textbf{68} (1975) pp.299-306.


\item [[Peter Johnstone|P. Johnstone]], \emph{Remarks on Quintessential and Persistent Localizations} , TAC \textbf{2} no.8 (1996) pp.90-99. (\href{http://www.tac.mta.ca/tac/volumes/1996/n8/n8.pdf}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{}

\end{itemize}
[[!redirects centre]] [[!redirects centers]] [[!redirects centres]]

[[!redirects center of a group]]



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