Homotopy coherent theorems of Dold–Kan type

Homotopy Coherent Category Theory

Homotopy Coherent Structures

Naturally occurring diagrams in algebraic topology are commutative up to homotopy, but not on the nose. It was quickly realized that very little can be done with this information. Homotopy coherent category theory arose out of a desire to catalog the higher homotopical information required to restore constructibility (or more precisely, functoriality) in such “up to homotopy” settings. The firs...

Homotopy Coherent Adjunctions of Quasi-categories

We show that an adjoint functor between quasi-categories may be extended to a simplicially enriched functor whose domain is an explicitly presented “homotopy coherent adjunction”. This enriched functor encapsulates both the coherent monad and the coherent comonad generated by the adjunction. Furthermore, because its domain is cofibrant, this data can be used to construct explicit quasi-categori...

Colimit Theorems for Relative Homotopy Groups

The first paper [10] (whose results were announced in [8]) developed the necessary ‘algebra of cubes’. Categories G of ω-groupoids and C of crossed complexes were defined, and the principal result of [10] was an equivalence of categories γ : G → C. Also established were a version of the homotopy addition lemma, and properties of ‘thin’ elements, in an ω-groupoid. In particular it was proved tha...

Coherent Functors in Stable Homotopy Theory

such that C and D are compact objects in S (an object X in S is compact if the representable functor Hom(X,−) preserves arbitrary coproducts). The concept of a coherent functor has been introduced explicitly for abelian categories by Auslander [1], but it is also implicit in the work of Freyd on stable homotopy [9]. In this paper we characterize coherent functors in a number of ways and use the...