## Algebraic & Geometric Topology

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

#### Abstract

Let $F$ denote the homotopy fiber of a map $f:K→L$ 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 7, Number 4 (2007), 1699-1765.

Dates
Received: 26 April 2007
Accepted: 2 October 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796771

Digital Object Identifier
doi:10.2140/agt.2007.7.1699

Mathematical Reviews number (MathSciNet)
MR2366176

Zentralblatt MATH identifier
1182.55008

#### Citation

Hess, Kathryn; Levi, Ran. An algebraic model for the loop space homology of a homotopy fiber. Algebr. Geom. Topol. 7 (2007), no. 4, 1699--1765. doi:10.2140/agt.2007.7.1699. https://projecteuclid.org/euclid.agt/1513796771

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