Simplicial Structures and Function Spaces in General Model Categories

MapC(A,X) = MorC(A⊗∆, X), where we take the morphisms on each component of the cosimplicial objectA⊗∆• to get a simplicial set of morphisms associated to the pair A,X ∈ C. We have a dual approach involving simplicial function objects X 7→ X∆ instead of cosimplicial objects A 7→ A ⊗ ∆•, but in a simplicial model category, we have an adjunction relation between these functors, so that both constructions give the same function space. The main purpose of this chapter is to explain that this definition of function spaces extends to general model categories, without assuming more than the axioms of §1.1.4. The idea is to consider cosimplicial objects A⊗∆•, referred to as cosimplicial frames, which we characterize in terms of a model structure on the category of cosimplicial objects. We lack relations which enable us to provide the function spaces associated to our framing with the composition operation of an enriched category structure in general, nonetheless we still have a composition structure at the homotopy level, and we can use the function spaces associated with generalized cosimplicial frames as models for a generalized simplicial category structure. The first applications of these ideas go back to the work of W. Dwyer and D. Kan [60]. We may also consider simplicial frames instead of cosimplicial frames in our function space construction, but we still have a homotopy version of the adjunction relation of a simplicial model category, so that the cosimplicial and simplicial frame constructions return homotopy equivalent function spaces in general. We explain the preliminary definition of a particular model structure, the Reedy model structure, on the category of cosimplicial (respectively, simplicial) objects in a model category in the first section of the chapter (§3.1). We use this model structure in order to formalize the homotopy properties attached to our cosimplicial (respectively, simplicial) frames. We address the definition of cosimplicial (respectively, simplicial) frames and function spaces in general model categories in the second section of the chapter (§3.2). We can also use cosimplicial frames to define an analogue of the geometric realization functor | − | in any model category, and we can use simplicial frames to define generalized totalization functors on cosimplicial objects. We provide a survey of these constructions in a third section (§3.3). We devote an appendix section §3.4