nLab
center

In general, the center (or centre) of an object $X$ is the collection of things which “commute with all elements of $X$.” This has a number of specific incarnations.

The original example is the 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 $gh=hg$ holds. The center is an abelian subgroup, but not every abelian subgroup is in the center. See also centralizer.

The center of a group can be generalized to the center of a monoid in an obvious way.

The 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.

Centers of categories and higher categories

The centre of a monoid can be horizontally categorified to the center of a category. Specifically, the center of a category $C$ is defined to be the commutative monoid $[C,C]({\mathrm{Id}}_C,{\mathrm{Id}}_C)$ of endo-natural-transformations of the identity functor of $C$. It is straightforward to check that this reduces to the usual definition if $C$ is a monoid, considered as a one-object category.

The notion of center can also be 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 center is the monoidal $(n-1)$-category $[C,C]({\mathrm{Id}}_C,{\mathrm{Id}}_C)$ of endo-transformations of its identity functor. One expects that in general, this center will actually admit a natural structure of braided monoidal $(n-1)$-category, just as the center of a category is actually a commutative monoid, not merely a monoid.

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 $n>0$, however, this functor will not be an inclusion; the objects of the center of $C$ are objects of $C$ equipped with additional 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).

Mike Shulman: It seems to me that the monoid of endofunctions of a set would be the decategorification of $[C,C]$, not $[C,C]({\mathrm{Id}}_C,{\mathrm{Id}}_C)$. The center of a set should be the endotransformations of the identity endofunction (of which there is only one, the identity). Moreover, since the center of a category is a commutative monoid, and the center of a bicategory is a braided monoidal category (horizontally categorifying the center of a monoidal category), the center construction acts like a knights-move on the periodic table; thus it makes sense that the center of a set should be a symmetric monoidal $(-1)$-category, i.e. “True.”

Toby: I didn't look closely enough at your centre of a category then! What you say here contradicts what you wrote below —that the centre of a $k$-tuply monoidal $n$-category is a $(k+1)$-tuply monoidal $n$-category, which is my understanding— and contradicts what John Baez writes in Section 1.1 (page 5) of HDA1.

Mike Shulman: I expanded it a lot; let me know if this is any better. It’s even more confusing than I realized at first.

Toby: I understand it, but it still doesn't actually include the centre of a set (or more generally of an $n$-category) that I learnt about from HDA1. Now, maybe that's not a very useful concept … except that it fits in so well with the centre of a $k$-tuply monoidal $n$-category for $k>0$! How many of these $k+1$ different centres that a $k$-tuply monoidal $n$-category has are used?

Mike Shulman: Hmm, that appears to be a different notion of center than the one I was used to. (I didn’t make this one up, but I don’t remember where I learned it; has anyone else seen it?) Perhaps that one is better; it also has the advantage that it gives automatically that the center of a $k$-tuply monoidal $n$-category is $(k+1)$-tuply monoidal.

Toby: I'll try to get John's attention.

Mike Shulman: The HDA1 definition also leaves me wondering: why make only that particular choice of what level to stop at? Is there anything interesting to say about other choices?

Toby: H'm, John is ‘too busy’ until September 22. (’_‘)