Given two monoidal categories and used for enrichment in enriched category theory (for instance two Bénabou cosmoi), a lax monoidal functor between them
canonically induces a 2-functor
between categories of V-enriched categories sending a -enriched category to the -enriched category such that
has the same objects as ;
the hom-objects of are the images of the hom-objects of under :
the composition, unit, associator and unitor morphisms in are the images of those of composed with the structure morphisms of the lax monoidal-structure on .
This operation is sometimes called change of enriching category or change of enriching base or just change of base (e.g. Crutwell 14, chapter 4, Riehl 14, lemma 3.4.3). But notice that, despite some vague similarity, this is different from base change of slice categories.
In the special case that has a single object and is hence (if thought of as pointed by that object) equivalently a monoid object in , this statement reduces to the statement that lax monoidal functors preserve monoids (this Prop.)
The operation of change of enriching category is functorial from MonCat to 2Cat. In particular, any monoidal adjunction gives rise to a 2-adjunction (and also for profunctors).
-enriched categories can be defined more generally when is a multicategory, and any functor between multicategories induces a change of enrichment 2-functor. Note that a functor of multicategorise between the underlying multicategories of two monoidal categories corresponds to a lax monoidal functor, as in the original version above. The multicategorical version also includes change of enrichment between closed categories.
A different generalization is when is a bicategory, yielding bicategory-enriched categories. Any lax functor of bicategories induces a similar functor between its 2-categories of enriched categories.
When and are cocomplete monoidal categories (or locally cocomplete bicategories), so that the bicategories -Prof of -categories and -profunctors exist, change of enrichment also induces a lax functor . To make this version functorial on , we need to consider as double categories, yielding a functor from MonCat to DblCat?.
Finally, a generalization subsuming all of these is that for any virtual double category we can construct another virtual double category , and this construction is functorial.
For any a monoidal category , the functor is lax monoidal, hence induces a 2-functor from to . This assigns to any -enriched category, , its underlying ordinary category, usually denoted , defined by .
If is a cocomplete monoidal category, the functor above has a left adjoint that takes a set to the copower of copies of . The resulting 2-functor takes an ordinary category to the “free” -category it generates. Such -categories are used, for instance, in subsuming “conical” limits under enriched weighted limits.
By functoriality, the adjunction between these two functors gives rise to a 2-adjunction .
More generally, if an adjunction can be regarded as a free-forgetful adjunction, then the corresponding adjunction between enriched categories can also be so regarded. For instance, if and for some topological group , then the right adjoint takes the “underlying topologically enriched category” of a category enriched in -spaces (its morphisms are the fixed-point spaces of the original -space-enriched category; we think of these as the “-equivariant maps” while the original spaces consisted of not-necessarily-equivariant maps acted on by conjugation). Similarly, any category enriched in spectra has an underlying category enriched in spaces (whose hom-spaces are the 0-spaces of the original hom-spectra), any dg-category has an underlying Ab-category (whose morphisms are the degree-0 ones in the original category), and so on.
The geometric realization and total singular complex? adjunction between simplicial sets and topological spaces induces another adjunction between simplicially enriched categories and topologically enriched categories.
If SSet or Top, then the set of connected components is lax monoidal. The resulting change of enrichment functor takes a -category to its “naive homotopy category” obtained by “identifying homotopic morphisms”.
Similarly, the fundamental groupoid functor is lax monoidal, so any -category has an underlying (2,1)-category.
An enhancement of the last example is that if is any monoidal model category, then its homotopy category comes with a lax monoidal functor . Thus any -category has an underlying -enriched “homotopy category” . Usually the underlying ordinary category of is the naive homotopy category from the previous example.
The nerve functor preserves products and has a left adjoint that also preserves products. Thus, we have a change of enrichment adjunction in which any 2-category can be regarded as a simplicially enriched category, and similarly any simplicially enriched category has a “homotopy 2-category”. The latter plays an important role in the theory of quasi-categories.
For the power set-functor, the change of base functor sends plain categories to plain categories. For any category, the morphisms of are called the poly-morphisms of in Mochizuki 12, section 0.
Samuel Eilenberg, Max Kelly, Closed categories, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), Springer (1966) [doi:10.1007/978-3-642-99902-4_22]
Geoff Cruttwell, chapter 4 of: Normed spaces and the Change of Base for Enriched Categories, 2014 (pdf)
Emily Riehl, lemma 3.4.3 inL Categorical Homotopy Theory, Cambridge University Press (2014)
Last revised on January 14, 2024 at 16:31:14. See the history of this page for a list of all contributions to it.