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 hGh\in G 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.

Definition

Let CC be an object in a 2-category. The center of CC, Z(C)Z(C) is the monoid of endomorphisms of the identity morphism, id C:CCid_C : C \rightarrow C.

One can invoke the Eckmann-Hilton argument to prove that vertical and horizontal composition agree on Z(C)Z(C) and are commutative.

Of rings

The center Z(R)Z(R) of a ring RR is defined to be the multiplicative subset consisting of all elements rRr \in R such that for all elements sRs \in R, rs=srr \cdot s = s \cdot r is true. RR is a commutative ring if RR is isomorphic to Z(R)Z(R).

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 lLl\in L. There are generalizations for some other kinds of algebras.

Of (higher) categories

Of general categories

The notion of center of a monoid has a horizontal categorification to a notion of center of a category.

For CC a category, its center is defined to be the commutative monoid

Z(C)[C,C](Id C,Id C) Z(C) \;\coloneqq\; [C,C](Id_C,Id_C)

of endo-natural transformation of the identity functor Id C:CCId_C \,\colon\, C \to C, i.e. the endomorphism monoid of Id CId_C in the functor category [C,C][C,C].

It is straightforward to check that this reduces to the usual definition of the center of monoid DD in the case that C=B(A,)C = \mathbf{B}(A,\cdot) is the corresponding delooping.

  • 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}) [Hoffmann (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)]

Of abelian categories

If a category carries further structure this may be inherited by its center. Notably the center of an additive category is not just a commutative monoid but a commutative ring (the endomorphism ring of its identity functor).

For more on this see at center of an abelian category.

Of higher categories

The notion of center also has a vertical categorification: 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).

Of \infty-groups

See center of an ∞-group.

References

See also

On the notion of center of a category:

and of an enriched category:

Last revised on June 20, 2023 at 10:53:29. See the history of this page for a list of all contributions to it.