nLab
center

Contents

Contents

Idea

In general, the center (or centre) of an algebraic object AA is the collection of elements of AA which “commute with all elements of AA.” 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 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.

  • For a generator GG of a category 𝒞\mathcal{C} there is an embedding of Z(𝒞)Z(\mathcal{C}) into the monoid Hom(G,G)Hom(G,G) given by ηη G\eta\mapsto\eta _G. In particular, if Hom(G,G)Hom(G,G) or Z(Hom(G,G))Z(Hom(G,G)) is trivial, as happens e.g. for SetSet with G=*G=\ast, then so is Z(𝒞)Z(\mathcal{C}) (Hofmann 1975).

  • For Cauchy complete 𝒞\mathcal{C} the idempotent elements of Z(𝒞)Z(\mathcal{C}) correspond precisely to the quintessential localizations of 𝒞\mathcal{C} (Johnstone 1996).

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.

References

  • R.-E. Hoffmann, Über das Zentrum einer Kategorie , Math. Nachr. 68 (1975) pp.299-306.

  • P. Johnstone, Remarks on Quintessential and Persistent Localizations , TAC 2 no.8 (1996) pp.90-99. (pdf)

See also

Last revised on April 13, 2019 at 05:12:54. See the history of this page for a list of all contributions to it.