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Cubical rigidification, the cobar construction and the based loop space

Manuel Rivera and Mahmoud Zeinalian

Algebraic & Geometric Topology 18 (2018) 3789–3820
Abstract

We prove the following generalization of a classical result of Adams: for any pointed path-connected topological space (X,b), that is not necessarily simply connected, the cobar construction of the differential graded (dg) coalgebra of normalized singular chains in X with vertices at b is weakly equivalent as a differential graded associative algebra (dga) to the singular chains on the Moore based loop space of X at b. We deduce this statement from several more general categorical results of independent interest. We construct a functor c from simplicial sets to categories enriched over cubical sets with connections, which, after triangulation of their mapping spaces, coincides with Lurie’s rigidification functor from simplicial sets to simplicial categories. Taking normalized chains of the mapping spaces of c yields a functor Λ from simplicial sets to dg categories which is the left adjoint to the dg nerve functor. For any simplicial set S with S0 = {x}, Λ(S)(x,x) is a dga isomorphic to ΩQΔ(S), the cobar construction on the dg coalgebra QΔ(S) of normalized chains on S. We use these facts to show that QΔ sends categorical equivalences between simplicial sets to maps of connected dg coalgebras which induce quasi-isomorphisms of dgas under the cobar functor, which is strictly stronger than saying the resulting dg coalgebras are quasi-isomorphic.

Keywords
rigidification, cobar construction, based loop space
Mathematical Subject Classification 2010
Primary: 18G30, 55P35, 55U10, 57T30
Secondary: 18D20, 55U35, 55U40
References
Publication
Received: 25 February 2017
Revised: 26 July 2018
Accepted: 3 August 2018
Published: 11 December 2018
Authors
Manuel Rivera
Department of Mathematics
University of Miami
Coral Gables, FL
United States
Mahmoud Zeinalian
Department of Mathematics
City University of New York, Lehman College
Bronx, NY
United States