nLab enriched homotopy theory

Idea

Enriched homotopy theory is about combining the notion of categories that model aspects of homotopy theory (such as homotopical categories and model categories) with the ideas and tools of enriched category theory.

On the one hand, enriched category theory is already used in many places in ordinary homotopy theory, usually for the particular case of enrichment over topological spaces or simplicial sets. Just as an ordinary category has a set of morphisms between any two objects, an ordinary homotopy theory has a space of morphisms between any two objects (including morphisms, homotopies between them, higher homotopies, and so on). From a higher categorical viewpoint, a homotopy theory is a presentation of an $(\infty,1)$-category, and $(\infty,1)$-categories can be viewed as categories enriched (perhaps weakly) over $\infty$-groupoids (that is, spaces). Many of the model categories used in homotopy theory come with natural topological or simplicial enrichments that make them better-behaved and easier to work with. The enrichment can also be used in a detailed study of homotopy coherence, without necessarily needing model-categorical tools. See homotopy coherent category theory.

On the other hand, instead of using enriched category theory as a tool to understand ordinary homotopy theory, one can look for a common generalization of them both. Now we take the point of view that spaces (or $\infty$-groupoids, or simplicial sets) are the higher-categorical counterpart of sets, and look for an analogue of the passage from Set-enriched categories to $V$-enriched categories that starts from ordinary (space-enriched) homotopy theories and ends up at $V$-enriched homotopy theories. Note that now $V$ need not be any space-like category, so that $V$-enriched homotopy theories need not be very similar to $(\infty,1)$-categories. One use of this approach is to get at a sort of higher categories with noninvertible $k$-cells for $k\gt 1$, by taking $V$ to be categories, 2-categories, and so on (perhaps iteratively). But $V$ could also be chain complexes, or $W$-categories for some other $W$, or spectra, or some other category with even less resemblance to “spaces.”

Fortunately, both of these approaches can be handled by many of the same technical tools, including but not limited to:

• Monoidal model categories and enriched model categories.

• Closed monoidal homotopical categories and enriched homotopical categories.

• Operads and $A_\infty$-categories.