Homotopy Coherent Representations

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...

Representations and Homotopy Theory

Let X = ΣY be a suspension. A question in homotopy theory is how to decompose the n-fold self smash product X into a wedge of spaces. Consider the set of homotopy classes [X, X]. Let the symmetric group Sn act on X by permuting positions. So for each σ ∈ Sn there is a map σ : X → X. This gives a function θ : Sn → [X, X]. By assuming that X is a suspension, [X, X] is an abelian group and so ther...

Homotopy sphere representations for matroids

For any rank r oriented matroid M , a construction is given of a ”topological representation” of M by an arrangement of homotopy spheres in a simplicial complex which is homotopy equivalent to Sr−1. The construction is completely explicit and depends only on a choice of maximal flag in M . If M is orientable, then all Folkman-Lawrence representations of all orientations of M embed in this repre...

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...