Let denote the
homotopy fiber of a map
of 2–reduced simplicial sets. Using as input data the strongly homotopy coalgebra structure of the
chain complexes of
and ,
we construct a small, explicit chain algebra, the homology of
which is isomorphic as a graded algebra to the homology of
, the simplicial (Kan)
loop group on .
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
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.
