nLab cosmos

Idea

In (enriched) category theory by a cosmos one means a “good place in which to do (enriched) category theory”. Typically what is meant is a base of enrichment $V$ (being by default a monoidal category or more generally a bicategory or similar) which carries enough extra structure for intended category-theoretic-constructions (such as enriched functor categories, etc.) to exist in $V$-enriched category theory.

The word is chosen by analogy with the notion of toposes which may be regarded as “a good places to do set theory”.

Definitions

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 $V$-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.

Street (1974) 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 VCat for non-cartesian $V$, since internal fibrations are poorly behaved there. The definition is given in Street (1980):

A fibrational cosmos is a 2-category $K$ such that

• finite limits exist in $K$;

• there is a 2-adjunction $P^* \colon K \leftrightarrows K^{\text{coop}} : P$;

• each object $A$ admits a discrete fibration from $P A$ satisfying two technical properties.

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

Street’s second definition

Street (1981) instead defines 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

• Bénabou cosmos

• base of enrichment

• 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.

References

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. Bénabou 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. in Category Seminar, Lecture Notes in Mathematics 420, Springer (1974) [doi:10.1007/BFb0063103]

• Ross Street, Cosmoi of internal categories, Transactions of the American Mathematical Society 258 2 (1980) 271-318 [doi:10.2307/1998059]

• Ross Street, Cauchy characterization of enriched categories, Rend. Sem. Mat. Fis. Milano 51 (1981): 217-233, Reprints in Theory and Applications of Categories, 4 (2004) 1-16 [tac:tr4, pdf]

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