nLab center of a category

Idea

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

Definition

For $\mathcal{C}$ a category, its center $Z(\mathcal{C})$ is defined to be the commutative monoid

$$Z(\mathcal{C}) \;\coloneqq\; [\mathcal{C},\mathcal{C}]\big( Id_\mathcal{C} ,\, Id_\mathcal{C} \big)$$

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

If $\mathcal{C}$ carries extra structure, then 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 the identity functor. For more on this, see center of an additive category.

Examples

Example

For $A$ an ordinary monoid and $\mathbf{B}A$ its delooping category, the ordinary center of $A$ is naturally identified with the category-theoretic center of $\mathbf{B}A$.

Proposition

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)]

Proposition

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

[Johnstone (1996)]

Related concepts

• center, center of a monoid

• center of an additive category

• Drinfeld center

References

• 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]

See also:

• Wikipedia, Center (category theory)