Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 18, Number 7 (2018), 3789-3820.
Cubical rigidification, the cobar construction and the based loop space
We prove the following generalization of a classical result of Adams: for any pointed path-connected topological space , that is not necessarily simply connected, the cobar construction of the differential graded (dg) coalgebra of normalized singular chains in with vertices at is weakly equivalent as a differential graded associative algebra (dga) to the singular chains on the Moore based loop space of at . We deduce this statement from several more general categorical results of independent interest. We construct a functor 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 yields a functor from simplicial sets to dg categories which is the left adjoint to the dg nerve functor. For any simplicial set with , is a dga isomorphic to , the cobar construction on the dg coalgebra of normalized chains on . We use these facts to show that 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.
Algebr. Geom. Topol., Volume 18, Number 7 (2018), 3789-3820.
Received: 25 February 2017
Revised: 26 July 2018
Accepted: 3 August 2018
First available in Project Euclid: 18 December 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 18G30: Simplicial sets, simplicial objects (in a category) [See also 55U10] 55P35: Loop spaces 55U10: Simplicial sets and complexes 57T30: Bar and cobar constructions [See also 18G55, 55Uxx]
Secondary: 18D20: Enriched categories (over closed or monoidal categories) 55U35: Abstract and axiomatic homotopy theory 55U40: Topological categories, foundations of homotopy theory
Rivera, Manuel; Zeinalian, Mahmoud. Cubical rigidification, the cobar construction and the based loop space. Algebr. Geom. Topol. 18 (2018), no. 7, 3789--3820. doi:10.2140/agt.2018.18.3789. https://projecteuclid.org/euclid.agt/1545102054