nLab
center

Contents

Idea

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

Definitions

Of groups and monoids

The original example is the center Z(G)Z(G) of a group GG, which is defined to be the subgroup consisting of all elements gGg\in G such that for all elements hHh\in H the equality gh=hgg h=h g holds. The center is an 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.

Of Lie algebras

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

Of categories and higher categories

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

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 CC is an n-category, then its center is the monoidal (n1)(n-1)-category [C,C](Id C,Id C)[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 braided monoidal (n1)(n-1)-category, just as the center of a category is actually a commutative monoid, not merely a monoid.

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

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

Note that a monoidal nn-category has two different centers: if we regard it as a one-object (n+1)(n+1)-category, then its center is a braided monoidal nn-category, but if we regard it merely as an nn-category, then its center is a braided monoidal (n1)(n-1)-category. The latter construction makes no reference to the monoidal structure. Likewise, a braided monoidal nn-category has three different centers, depending on whether we regard it as an nn-category, a connected (n+1)(n+1)-category, or a 2-connected (n+2)(n+2)-category, and so on (a kk-tuply monoidal nn-category has k+1k+1 different centers).

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

Moreover, one expects that if we perform this “canonical” operation on a k-tuply monoidal n-category (for k1k\ge 1), the resulting braided monoidal nn-category will actually be (k+1)(k+1)-tuply monoidal. This is known to be true in the cases n4n\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 CC 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 nnForum here.

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.

Of \infty-groups

See center of an ∞-group.

Revised on August 5, 2014 06:46:32 by Thomas Holder (89.204.153.193)