# nLab Bénabou cosmos

Contents

### Context

#### Enriched category theory

enriched category theory

## Extra stuff, structure, property

### Homotopical enrichment

#### Monoidal categories

monoidal categories

# Contents

## Idea

A Bénabou cosmos is a monoidal category $\mathcal{V}$ with good properties that make it well-suited for $\mathcal{V}$-enriched category theory. (There are variants of this idea, see at cosmos for more).

The archetypical Bénabou cosmos is the category Set of all sets, equipped with its Cartesian product-monoidal structure; and $Set$-enriched category theory is the ordinary theory of locally small categories, whose hom-sets are sets.

The point of the notion of Bénabou cosmoi is to allow more general hom-objects than just sets, while imposing enough conditions on the category $\mathcal{V}$ which these form in order that most standard definitions and proofs of plain category theory (e.g. properties of categories of presheaves) generalize to $\mathcal{V}$-enriched category theory (e.g. concerning enriched presheaves).

## Definition

###### Definition

A Bénabou cosmos is (Street 74, p. 1) a

###### Remark

There is a generalization of Def. to the context of enriched indexed categories (Shulman 2013). While Def. is not “elementary” (as it involves infinitary (non-finite) limits and colimits), the indexed version is elementary, as the infinitary structure is folded into the indexing base category. The notion of Bénabou cosmoi is recovered as particular indexed cosmoi over Set.

## Examples

###### Example

(topoi are cosmoi)
Every Grothendieck topos is a Bénabou cosmos, where the symmetric monoidal structure is cartesian. Examples in this class include:

## References

Apparently there is no explicit written account by Jean Bénabou on the notion, but one finds it recounted in Street 74, p. 1:

to J. Benabou the word means “bicomplete symmetric monoidal category”, such categories $\mathcal{V}$ being rich enough so that the theory of categories enriched in $\mathcal{V}$ develops to a large extent just as the theory of ordinary categories.