With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
In enriched category theory, a Bénabou cosmos is a monoidal category $\mathcal{V}$ with good properties that make it a base of enrichment 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 enriched functor categories and specifically of categories of presheaves) generalize to $\mathcal{V}$-enriched category theory (e.g. concerning enriched presheaves).
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.
(topoi are cosmoi)
Every Grothendieck topos is a Bénabou cosmos, where the symmetric monoidal structure is cartesian. Examples in this class include:
The archetypical such example is the category Set of set. $Set$-enriched categories are ordinary locally small categories.
The category sSet of simplicial sets is the presheaf topos over the simplex category, whose cartesian monoidal structure is discussed at products of simplicial sets. $sSet$-enriched categories are often called simplicial categories (though see there for ambiguity in this terminology). They serve as foundation for much of homotopy theory, see at simplicial homotopy theory and simplicial model categories for more.
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.
See also:
The indexed generalization:
Last revised on August 23, 2023 at 11:03:40. See the history of this page for a list of all contributions to it.