## Algebraic & Geometric Topology

### Cubical rigidification, the cobar construction and the based loop space

#### 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 $ℭ □ c$ from simplicial sets to categories enriched over cubical sets with connections, which, after triangulation of their mapping spaces, coincides with Lurie’s rigidification functor $ℭ$ from simplicial sets to simplicial categories. Taking normalized chains of the mapping spaces of $ℭ □ c$ 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

Source
Algebr. Geom. Topol., Volume 18, Number 7 (2018), 3789-3820.

Dates
Revised: 26 July 2018
Accepted: 3 August 2018
First available in Project Euclid: 18 December 2018

https://projecteuclid.org/euclid.agt/1545102054

Digital Object Identifier
doi:10.2140/agt.2018.18.3789

Mathematical Reviews number (MathSciNet)
MR3892231

Zentralblatt MATH identifier
07006377

#### Citation

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

#### References

• 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 https://tinyurl.com/Kapulkin-Voevodsky-2018 {\unhbox0
• M Kontsevich, Symplectic geometry of homological algebra, preprint (2009) Available at \setbox0\makeatletter\@url https://www.ihes.fr/~maxim/TEXTS/Symplectic_AT2009.pdf {\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 https://tinyurl.com/Lurie-Higher-alg {\unhbox0
• E Riehl, Categorical homotopy theory, New Mathematical Monographs 24, Cambridge Univ. Press (2014)