Homotopy coherent category theory or enriched homotopy theory is the attempt to understand those situations that arise in homotopy theory, homotopical algebra, and non-Abelian homological algebra, in which quite naturally occuring diagrams are not commutative, yet are commutative ‘up-to-homotopy’.
If the diagram is just commutative in the homotopy category, that is not much use and one can do little with it. Surprisingly often however, the homotopies involved in such a diagram’s ‘almost commutative’ nature can be specified, but then the question arises as to whether those homotopies form some sort of diagram, so homotopies between composite homotopies become involoved. This begins to look like parts of 2-category theory, or rather ‘higher weak category theory’ and the development of homotopy coherent category theory was initiated in an attempt to merge homotopy theory with categorical tools for handling higher categories.
In particular:
In the case that the category $V$ one is enriching over is itself a model category or at least a category with weak equivalences one wishes to generalize limits and in particular the weighted limits such as as ends and coends in enriched category theory to constructions which satisfy the familiar universal properties only up to coherent homotopy.
These articles deal with the theory of homotopy coherent diagrams:
R. Vogt, Homotopy limits and colimits , Math. Z., 134, (1973), 11 – 52.
J.-M. Cordier and T. Porter, Vogt’s theorems on categories of homotopy coherent diagrams, Math. Proc. Camb. Phil. Soc. 100 (1986), 65–90.
J.-M. Cordier and T. Porter, Maps between homotopy coherent diagrams, Top. and its Applications, 28 (1988) 255-275.
J.-M. Cordier and T. Porter, Fibrant diagrams, rectifications and a construction of Loday, J. Pure Appl. Alg 67 (1990), 111–124.
A discussion of homotopy limits is in
D. Bourn and J.-M. Cordier, A general formulation of homotopy limits , J. Pure Appl. Algebra, 29, (1983), 129–141,
J.-M. Cordier, Sur les limites homotopiques de diagrammes homotopiquement cohérents, Comp. Math. 62 (1987), 367–388.
In
a main point is the definition and discussion of a homotopy coherent end? for the case of enrichment over the model category of simplicial sets.
In
the general issue of enriched homotopy theory is addressed and enriched homotopical categories are introduced, which are a coherent combination of the notion of enriched category with that of homotopical category