In general, the center (or centre) of an algebraic 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.
This notion of 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 center 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 the delooping of a monoid.
For a generator of a category there is an embedding of into the monoid given by . In particular, if or is trivial, as happens e.g. for with , then so is (Hofmann 1975).
For Cauchy complete the idempotent elements of correspond precisely to the quintessential localizations of (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 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.
Generally, 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).
An old query about the categorical notion of center is archived at Forum here.
A special case is the center of an abelian category which has a special entry because of a number of special applications and properties.
See center of an ∞-group.
R.-E. Hoffmann, Über das Zentrum einer Kategorie , Math. Nachr. 68 (1975) pp.299-306.