Maps into homotopy coalgebras

Homotopy Classification of Maps into Homogeneous Spaces

We give an alternative to Postnikov’s homotopy classification of maps from 3-dimensional CW-complexes to homogeneous spaces G/H of Lie groups. It describes homotopy classes in terms of lifts to the group G and is suitable for extending the notion of homotopy to Sobolev maps. This is required in applications to variational problems of mathematical physics.

Lie Coalgebras and Rational Homotopy Theory, I: Graph Coalgebras

Homotopy Theory of Coalgebras over Operads

This paper constructs model structures on the categories of coalgebras and pointed irreducible coalgebras over an operad whose components are projective, finitely generated in each dimension, and satisfy a condition that allows one to take tensor products with a unit interval. The underlying chain-complex is assumed to be unbounded and the results for bounded coalgebras over an operad are deriv...

The Homotopy Theory of Coalgebras over a Comonad

Let K be a comonad on a model category M. We provide conditions under which the associated category MK of K-coalgebras admits a model category structure such that the forgetful functor MK →M creates both cofibrations and weak equivalences. We provide concrete examples that satisfy our conditions and are relevant in descent theory and in the theory of Hopf-Galois extensions. These examples are s...

Crossed Modules as Homotopy Normal Maps

In this note we consider crossed modules of groups (N → G, G→ Aut(N)), as a homotopy version of the inclusion N ⊂ G of a normal subgroup. Our main observation is a characterization of the underlying map N → G of a crossed module, in terms of a simplicial group structure on the associated bar construction. This approach allows for “natural” generalizations to other monoidal categories, in partic...