## An algebraic model for the loop space homology of a homotopy fiber.(English)Zbl 1182.55008

Summary: Let $$F$$ denote the homotopy fiber of a map $$f:K\rightarrow L$$ 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 $$\mathbf{DCSH}$$ of chain coalgebras and of strongly homotopy coalgebra maps given by K. Hess, P.-E. Parent and J. 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.

### MSC:

 55P35 Loop spaces 55U35 Abstract and axiomatic homotopy theory in algebraic topology 57T30 Bar and cobar constructions 16T05 Hopf algebras and their applications 18D50 Operads (MSC2010) 55U10 Simplicial sets and complexes in algebraic topology 57T35 Applications of Eilenberg-Moore spectral sequences
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