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2018 Cubical rigidification, the cobar construction and the based loop space
Manuel Rivera, Mahmoud Zeinalian
Algebr. Geom. Topol. 18(7): 3789-3820 (2018). DOI: 10.2140/agt.2018.18.3789


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 S 0 = { x } , Λ ( S ) ( x , x ) is a dga isomorphic to Ω Q Δ ( S ) , the cobar construction on the dg coalgebra Q Δ ( S ) of normalized chains on S . We use these facts to show that Q Δ 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.


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Manuel Rivera. Mahmoud Zeinalian. "Cubical rigidification, the cobar construction and the based loop space." Algebr. Geom. Topol. 18 (7) 3789 - 3820, 2018.


Received: 25 February 2017; Revised: 26 July 2018; Accepted: 3 August 2018; Published: 2018
First available in Project Euclid: 18 December 2018

zbMATH: 07006377
MathSciNet: MR3892231
Digital Object Identifier: 10.2140/agt.2018.18.3789

Primary: 18G30, 55P35, 55U10, 57T30
Secondary: 18D20, 55U35, 55U40

Rights: Copyright © 2018 Mathematical Sciences Publishers


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Vol.18 • No. 7 • 2018
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