nLab center

Idea

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

Definitions

Of groups and monoids

The original example is the center $Z(G)$ of a group $G$, which is defined to be the subgroup consisting of all elements $g\in G$ such that for all elements $h\in G$ the equality $g h=h g$ 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 $C$ be an object in a 2-category. The center of $C$, $Z(C)$ is the monoid of endomorphisms of the identity morphism, $id_C : C \rightarrow C$.

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

Of rings

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

Of Lie algebras

The center of a Lie algebra $L$ is an abelian Lie subalgebra $Z(L)$, consisting of all elements $z\in L$ such that $[l,z]=0$ for all $l\in L$. 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 $C$ a category, its center is defined to be the commutative monoid

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

of endo-natural transformation of the identity functor $Id_C \,\colon\, C \to C$, i.e. the endomorphism monoid of $Id_C$ in the functor category $[C,C]$.

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

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

• For Cauchy complete $\mathcal{C}$ the idempotent elements of $Z(\mathcal{C})$ correspond precisely to the quintessential localizations of $\mathcal{C}$ [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 $C$ is an n-category, then its center is the monoidal $(n-1)$-category $[C,C](Id_C,Id_C)$ of endo-transformations of its identity functor. One expects that in general, this center will actually admit a natural structure of braided monoidal $(n-1)$-category, just as the center of a category is actually a commutative monoid, not merely a monoid.

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

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

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

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

Moreover, one expects that if we perform this “canonical” operation on a k-tuply monoidal n-category (for $k\ge 1$), the resulting braided monoidal $n$-category will actually be $(k+1)$-tuply monoidal. This is known to be true in the cases $n\le 4$: 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 $C$ is a set, there is precisely one endo-transformation of its identity endofunction (namely, the identity).

Of $\infty$-groups

See center of an ∞-group.

Related concepts

• center of a category

  center of an additive category

  Drinfeld center

• central product of groups

References

See also

• Wikipedia, Center (group theory)

On the notion of center of a category:

• Rudolf-E. Hoffmann, Über das Zentrum einer Kategorie, Math. Nachr. 68 (1975) 299-306 [doi:10.1002/mana.19750680122]

• Peter Johnstone, Remarks on Quintessential and Persistent Localizations, TAC 2 8 (1996) 90-99 [tac:2-08, pdf]

and of an enriched category:

• Harald Lindner. Center and trace, Archiv der Mathematik 35 (1980): 476-496. (doi:10.1007/BF01235372)