nLab cosmos

Idea

A cosmos is a “good place in which to do category theory,” including both ordinary category theory as well as enriched category theory.

The word is chosen by analogy with topos which can be regarded as “a good place to do set theory,” but there are notable differences between the two situations; a more direct categorification of a topos is, unsurprisingly, a 2-topos. In contrast, cosmoi also include enriched category theory, while toposes do not allow non-cartesian enrichment.

There are a number of different, inequivalent, definitions of “cosmos” in the literature.

Bénabou cosmoi

A Bénabou cosmos (see Street 74, p. 1) is a complete and cocomplete (hence bicomplete) closed symmetric monoidal category.

This is the ideal situation for for $V$-enriched category theory.

Indexed Bénabou cosmoi

Shulman 2013 introduced an indexed generalization of Bénabou cosmoi, appropriate for studying enriched indexed categories over a base category. Notably, while the definition of Bénabou cosmoi is not “elementary” (it involves infinitary (non-finite) limits and colimits), the indexed version is elementary; the infinitary structure is folded into the indexing base category. The notion of Bénabou cosmoi is recovered as particular cosmoi indexed over Set.

Street’s “fibrational cosmoi”

Ross Street has taken a different tack, defining a “cosmos” to be the collection of (enriched) categories and relevant structure for doing category theory, rather than the “base” category $V$ over which the enrichment occurs.

In his paper “Elementary cosmoi,” Street defined a (fibrational) cosmos to be a 2-category in which internal fibrations are well-behaved and representable by a structure of “presheaf objects” (later realized to be a special sort of Yoneda structure). Note that while this includes $Cat$, it does not include $V$-$Cat$ for non-cartesian $V$, since internal fibrations are poorly behaved there. The definition is given in the paper “Cosmoi of internal categories”: a fibrational cosmos is a 2-category $K$ such that

• finite limits exist in $K$;
• there is a 2-adjunction $P^* : K \leftrightarrows K^{\text{coop}} : P$;
• each object $A$ admits a discrete fibration from $PA$ satisfying two technical properties.

The objects $PA$ are the “presheaf objects” that represent fibrations.

Street’s second definition

In his paper “Cauchy characterization of enriched categories,” Street instead defined a cosmos to be a 2-category that “behaves like the 2-category $V$-$Mod$ of enriched categories and profunctors”. The precise definition: a cosmos is a 2-category (or bicategory) such that:

• Small (weak, or bi-) coproducts exist.

• Each monad admits a Kleisli construction (analogous to the exactness of a topos).

• It is locally small-cocomplete, i.e. its hom-categories have small colimits that are preserved by composition on each side.

• There exists a small “Cauchy generator”.

These hypotheses imply that it is equivalent to the bicategory of categories and profunctors enriched over some “base” bicategory. (Note the generalization from enrichment over a monoidal category to enrichment over a bicategory.)

Defined in this way, cosmoi are closed under dualization, parametrization and localization, suitably defined.

Related notions

• An infinity-cosmos is a “good place in which to do higher category theory” as axiomatized by Riehl and Verity in their work on the foundations of $(\infty,1)$- and $(\infty,n)$-category theory.

Bibliography

Apparently there is no written account by Jean Bénabou of his definition of cosmos. 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.

• Ross Street, Elementary cosmoi I. Category Seminar. Springer, Berlin, Heidelberg, 1974. (doi:10.1007%2FBFb0063103)

• Ross Street, Cosmoi of internal categories, Transactions of the American Mathematical Society 258.2 (1980): 271-318.

• Ross Street, Cauchy characterization of enriched categories, Rend. Sem. Mat. Fis. Milano 51 (1981): 217-233. (pdf)

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