nLab enriched (infinity,1)-category

Idea

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

• a set/class of objects $Obj(C)$;

• for every tuple $(a \in C,b \in C)$ an object $C(a,b) \in \mathcal{V}$ “of morphisms” from $a$ to $b$ in $C$;

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

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

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

Examples

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

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

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

(Gepner-Haugseng 13)

Related concepts

• enriched (∞,1)-functor

• Segal category, Segal n-category

• enriched homotopical category, weak enrichment

• internal category in an (∞,1)-category

• table - models for (∞,1)-operads

• enriched type

References

• Carlos Simpson, Homotopy Theory of Higher Categories (arXiv:1001.4071)

• Jacob Lurie, Higher Algebra

• David Gepner, Rune Haugseng, Enriched ∞-categories via non-symmetric ∞-operads (arXiv:1312.3178)

• Hadrian Heine, An equivalence between enriched ∞-categories and ∞-categories with weak action, arxiv, 2020

• John D. Berman, Enriched infinity categories I: enriched presheaves (arXiv:2008.11323)

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

• Julie Bergner, Charles Rezk, Comparison of models for $(\infty,n)$-categories, I (arXiv:1204.2013)

• Julie Bergner, Charles Rezk, Comparison of models for $(\infty,n)$-categories, II (arXiv:1406.4182)

See also

• Rune Haugseng, Bimodules and natural transformations for enriched ∞-categories (arXiv:1506.07341)

On $\infty$-colimits and Day convolution in the context of enriched $\infty$-categories:

• Vladimir Hinich, Colimits in enriched ∞-categories and Day convolution, Theory and Applications of Categories 39 12 (2023) 365-422 [tac:39-12, arXiv:2101.09538]