enriched homotopical category


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Basic facts


Enriched category theory



The concept of enriched homotopical category is the generalization of the concept of homotopical category to the context of enriched category theory and hence to homotopy coherent category theory.

The idea is that the homotopy category Ho CHo_C of a category CC which is enriched over a suitable monoidal and homotopical category VV is itself a Ho VHo_V-enriched category.


For VV a closed monoidal homotopical category, a VV-enriched category CC with powers and copowers and with the structure of a homotopical category on its underlying category C 0C_0 is a VV-homotopical category when equipped with a deformation retract for the enrichment.


If VV is a monoidal model category, then any VV-enriched model category is automatically a VV-homotopical category.


Recall that for VV as above, Ho VHo_V is closed monoidal.


With CC a VV-homotopical category, Ho C 0Ho_{C_0} is the underlying category of a Ho VHo_V-enriched category.

Write Ho CHo_C for this Ho VHo_V-enriched category. This is the enriched analogue of the homotopy category of CC.

So schematically we have (with all of the above qualifiers suppressed):

(CVCat)(Ho CHo VCat) (C \in V-Cat) \Rightarrow (Ho_C \in Ho_V-Cat)

Construction of the enriched homotopy category

For CC an enriched homotopical VV-category as above, the Ho VHo_V-category Ho CHo_C is constructed from the homotopy category Ho C 0Ho_{C_0} of the ordinary category underlying CC by constructing a Ho VHo_V-module structure, essentially following section 4.3.2 of Hovey: Model categories.


The definition appears as definition 16.1, p. 46 in Shulman: Homotopy limits and colimits in enriched homotopy theory, the proposition is proposition 16.2, p. 46. The construction of Ho CHo_C follows the proof of proposition 15.4, p. 45.

Revised on August 27, 2017 04:04:44 by Mike Shulman (