A refinement of the notion of homotopy category of a model category (more generally, relative category) that keeps track of homotopies between morphisms.

Dwyer–Kan implement the notion of a simplicial homotopy category as a simplicial category that is Dwyer–Kan equivalent to the hammock localization.

See §5.1 in

- William G. Dwyer, Daniel M. Kan,
*Function complexes in homotopical algebra*, Topology 19 (1980), 427-440 [doi:10.1016/0040-9383(80)90025-7, pdf]

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