Algebraic & Geometric Topology

Cubical rigidification, the cobar construction and the based loop space

Manuel Rivera and Mahmoud Zeinalian

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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.

Article information

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

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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

rigidification cobar construction based loop space


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.

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  • J F Adams, On the cobar construction, Proc. Nat. Acad. Sci. U.S.A. 42 (1956) 409–412
  • R Antolini, Geometric realisations of cubical sets with connections, and classifying spaces of categories, Appl. Categ. Structures 10 (2002) 481–494
  • R Brown, P J Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981) 233–260
  • D Dugger, D I Spivak, Mapping spaces in quasi-categories, Algebr. Geom. Topol. 11 (2011) 263–325
  • D Dugger, D I Spivak, Rigidification of quasi-categories, Algebr. Geom. Topol. 11 (2011) 225–261
  • T G Goodwillie, Cyclic homology, derivations, and the free loopspace, Topology 24 (1985) 187–215
  • K Hess, The Hochschild complex of a twisting cochain, J. Algebra 451 (2016) 302–356
  • K Hess, A Tonks, The loop group and the cobar construction, Proc. Amer. Math. Soc. 138 (2010) 1861–1876
  • A Joyal, Quasi-categories and Kan complexes, J. Pure Appl. Algebra 175 (2002) 207–222
  • C Kapulkin, V Voevodsky, Cubical approach to straightening, preprint (2018) Available at \setbox0\makeatletter\@url {\unhbox0
  • M Kontsevich, Symplectic geometry of homological algebra, preprint (2009) Available at \setbox0\makeatletter\@url {\unhbox0
  • B Le Grignou, Composition + homotopy = cubes, preprint (2018)
  • J-L Loday, Cyclic homology, 2nd edition, Grundl. Math. Wissen. 301, Springer (1998)
  • J-L Loday, B Vallette, Algebraic operads, Grundl. Math. Wissen. 346, Springer (2012)
  • J Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton Univ. Press (2009)
  • J Lurie, Higher algebra, book project (2017) Available at \setbox0\makeatletter\@url {\unhbox0
  • E Riehl, Categorical homotopy theory, New Mathematical Monographs 24, Cambridge Univ. Press (2014)