In general, the center (or centre) of an object is the collection of things which “commute with all elements of .” This has a number of specific incarnations.
The original example is the center of a group , which is defined to be the subgroup consisting of all elements such that for all elements the equality 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 is an abelian Lie subalgebra , consisting of all elements such that for all . There are generalizations for some other kinds of algebras.
The centre of a monoid can be horizontally categorified to the center of a category. Specifically, the center of a category is defined to be the commutative monoid of endo-natural-transformations of the identity functor of . It is straightforward to check that this reduces to the usual definition if 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 is an n-category, then its center is the monoidal -category of endo-transformations of its identity functor. One expects that in general, this center will actually admit a natural structure of braided monoidal -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 -category by regarding it as a one-object -category, according to the delooping hypothesis. It follows that the center of a monoidal -category should naturally be a braided monoidal -category. This is known to be true when (the center of a monoid is a commutative monoid) and also for and .
Note that a monoidal -category has two different centers: if we regard it as a one-object -category, then its center is a braided monoidal -category, but if we regard it merely as an -category, then its center is a braided monoidal -category. The latter construction makes no reference to the monoidal structure. Likewise, a braided monoidal -category has three different centers, depending on whether we regard it as an -category, a connected -category, or a 2-connected -category, and so on (a -tuply monoidal -category has different centers).
It seems that in applications, however, one is usually most interested in the sort of center of a monoidal -category obtained by regarding it as a one-object -category, thereby obtaining a braided monoidal -category. It is in this case, and seemingly this case only, that the center comes with a natural forgetful functor to , corresponding to the classical inclusion of the center of a monoid. (For , however, this functor will not be an inclusion; the objects of the center of are objects of equipped with additional structure.)
Moreover, one expects that if we perform this “canonical” operation on a k-tuply monoidal n-category (for ), the resulting braided monoidal -category will actually be -tuply monoidal. This is known to be true in the cases : 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 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 , not . 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 -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 -tuply monoidal -category is a -tuply monoidal -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 -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 -tuply monoidal -category for ! How many of these different centres that a -tuply monoidal -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 -tuply monoidal -category is -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. (’_‘)