nLab
enriched (infinity,1)-category

Context

(,1)(\infty,1)-Category theory

Enriched category theory

Contents

Idea

The notion of enriched (,1)(\infty,1)-category is the generalization of the notion of enriched category from category theory to (∞,1)-category theory. For 𝒱\mathcal{V} a monoidal (∞,1)-category, a 𝒱\mathcal{V}-enriched (,1)(\infty,1)-category CC is

  • a set/class of objects Obj(C)Obj(C);

  • for every tuple (aC,bC)(a \in C,b \in C) an object C(a,b)𝒱C(a,b) \in \mathcal{V} “of morphisms” from aa to bb in CC;

  • for each sequence (a iC) i=0 n(a_i \in C)_{i = 0}^n of objects in CC an object C(a 0,,a n)𝒱C(a_0, \cdots, a_n) \in \mathcal{V} “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 CC to be a ∞-algebra over an (∞,1)-operad in 𝒱 \mathcal{V}^{\otimes} over Assoc Obj(C){}_{Obj(C)}, the Obj(C)Obj(C)-colored version of the associative operad;

CAlg Assoc Obj(C)(𝒱 ). C \in Alg_{Assoc_{Obj(C)}}(\mathcal{V}^\otimes) \,.

For 𝒱\mathcal{V} \in ∞Grpd, this should be equivalent to ordinary (∞,1)-categories. This is for instance in (Lurie, def. 4.2.1.12).

A construction that should be a model for this notion in terms of a model category presentation for 𝒱\mathcal{V} is discussed in (Simpson). For the case that 𝒱=\mathcal{V} = ∞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 VV for 𝒱\mathcal{V} has been chosen, one can consider semi-strict \infty-enrichments given by ordinary VV-enriched categories equipped with a notion of weak equivalence that remembers that these are presentations for enriched (,1)(\infty,1)-categories. See also enriched homotopical category.

Examples

A stable (∞,1)-category is naturally enriched in the (∞,1)-category of spectra.

More generally, for RR an E-∞ ring then an RR-linear (∞,1)-category? is naturally enriched in RR-∞-modules. (This includes the previous case for RR the sphere spectrum.)

A closed monoidal (∞,1)-category is naturally enriched over itself.

(Gepner-Haugseng 13)

References

Further discussion of (infinity,n)-categories as \infty-categories enriched in (,n1)(\infty,n-1)-categories is (via Theta-spaces) in

Revised on June 18, 2014 01:09:42 by Urs Schreiber (89.204.154.27)