# nLab center of a category

Contents

### Context

category theory

#### Categorical algebra

internalization and categorical algebra

universal algebra

categorical semantics

# Contents

## Idea

The notion of center of a monoid has a horizontal categorification to a notion of 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 transformation of the identity functor $Id_C \,\colon\, C \to C$, i.e. the endomorphism monoid of $Id_C$ in the functor category $[C,C]$.

If $\mathcal{C}$ carries extra structure this may be inhereted 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 at center of an additive category.

## Examples

###### Example

For $A$ an ordinary monoid and $\mathbf{B}A$ it 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)]