An algebraic model for the loop space homology of a homotopy fiber

Abstract

Let $F$ denote the homotopy fiber of a map $f:K\to L$ of 2–reduced simplicial sets. Using as input data the strongly homotopy coalgebra structure of the chain complexes of $K$ and $L$, we construct a small, explicit chain algebra, the homology of which is isomorphic as a graded algebra to the homology of $GF$, the simplicial (Kan) loop group on $F$. To construct this model, we develop machinery for modeling the homotopy fiber of a morphism of chain Hopf algebras.

Essential to our construction is a generalization of the operadic description of the category $DCSH$ of chain coalgebras and of strongly homotopy coalgebra maps given by Hess, Parent and Scott [Co-rings over operads characterize morphisms arxiv:math.AT/0505559] to strongly homotopy morphisms of comodules over Hopf algebras. This operadic description is expressed in terms of a general theory of monoidal structures in categories with morphism sets parametrized by co-rings, which we elaborate here.