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.
"An algebraic model for the loop space homology of a homotopy fiber." Algebr. Geom. Topol. 7 (4) 1699 - 1765, 2007. https://doi.org/10.2140/agt.2007.7.1699