Enriched category theory
The notion of enriched -category is the generalization of the notion of enriched category from category theory to (∞,1)-category theory. For a monoidal (∞,1)-category, a -enriched -category is
a set/class of objects ;
for every tuple an object “of morphisms” from to in ;
for each sequence of objects in an object “of sequences of composable morphisms and their composites”;
such that these composites exist essentially uniquely and satisfy associativity in a coherent fashion.
One way to make this precise in a general abstract way should be to define to be a ∞-algebra over an (∞,1)-operad in over Assoc, the -colored version of the associative operad;
For ∞Grpd, this should be equivalent to ordinary (∞,1)-categories. This is for instance in (Lurie, def. 220.127.116.11).
A construction that should be a model for this notion in terms of a model category presentation for is discussed in (Simpson). For the case that ∞Grpd presented by the standard model structure on simplicial sets this reproduces the notion of Segal categories. (See there for further details and references.) The iteration of this construction yields Segal n-categories, a model for (∞,n)-categories.
Once a model category for has been chosen, one can consider semi-strict -enrichments given by ordinary -enriched categories equipped with a notion of weak equivalence that remembers that these are presentations for enriched -categories. See also enriched homotopical category.
A stable (∞,1)-category is naturally enriched in the (∞,1)-category of spectra.
More generally, for an E-∞ ring then an -linear (∞,1)-category? is naturally enriched in -∞-modules. (This includes the previous case for the sphere spectrum.)
A closed monoidal (∞,1)-category is naturally enriched over itself.
Revised on March 10, 2014 10:27:08
by Urs Schreiber