nLab center




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.


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.


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.


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.