nLab Bénabou cosmos

Contents

Context

Enriched category theory

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Idea

A Bénabou cosmos is a monoidal category 𝒱\mathcal{V} with good properties that make it well-suited for 𝒱\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 SetSet-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 𝒱\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 𝒱\mathcal{V}-enriched category theory (e.g. concerning enriched presheaves).

Definition

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 𝒱\mathcal{V} being rich enough so that the theory of categories enriched in 𝒱\mathcal{V} develops to a large extent just as the theory of ordinary categories.

See also:

The indexed generalization:

  • Mike Shulman, Enriched indexed categories, Theory and Applications of Categories, 28 21 (2013) 616-695 (tac:28-21)

Created on October 11, 2021 at 03:57:58. See the history of this page for a list of all contributions to it.