Rectification of weak product algebras over an operad in 𝒞at and 𝒯op and applications

Fiedorowicz, Zbigniew¹ ; Stelzer, Manfred² ; Vogt, Rainer³

¹Department of Mathematics, The Ohio State University, 100 Mathematics Building, 231 West 18th Avenue, Columbus, OH 43210-1174, USA
²Fachbereich Mathematik/Informatik, Universität Osnabrück, Albrechtstrasse 28a, D-49076 Osnabrück, Germany
³Fachbereich Mathematik/Informatik, Universität Osnabrück, Albrechtstrasse 28a, D-49069 Osnabrück, Germany

Algebraic and Geometric Topology, Tome 16 (2016) no. 2, pp. 711-755

Résumé

We develop an alternative to the May–Thomason construction used to compare operad-based infinite loop machines to those of Segal, which rely on weak products. Our construction has the advantage that it can be carried out in Cat, whereas their construction gives rise to simplicial categories. As an application we show that a simplicial algebra over a Σ–free Cat operad O is functorially weakly equivalent to a Cat algebra over O. When combined with the results of a previous paper, this allows us to conclude that, up to weak equivalences, the category of O–categories is equivalent to the category of BO–spaces, where B: Cat →Top is the classifying space functor. In particular, n–fold loop spaces (and more generally En spaces) are functorially weakly equivalent to classifying spaces of n–fold monoidal categories. Another application is a change of operads construction within Cat.

Détail

DOI : 10.2140/agt.2016.16.711

Classification : 18D50, 55P48
Keywords: operads, categories, loop space machines

Bibliographie

[1] M Artin, A Grothendieck, J L Verdier, Théorie des topos et cohomologie étale des schémas, Tome 1 : Théorie des topos, Exposés I–IV (SGA 41), 269, Springer (1972)

[2] C Balteanu, Z Fiedorowicz, R Schwänzl, R Vogt, Iterated monoidal categories, Adv. Math. 176 (2003) 277

[3] C Berger, I Moerdijk, Axiomatic homotopy theory for operads, Comment. Math. Helv. 78 (2003) 805

[4] C Berger, I Moerdijk, The Boardman–Vogt resolution of operads in monoidal model categories, Topology 45 (2006) 807

[5] J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, 347, Springer (1973)

[6] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, 304, Springer (1972)

[7] Z Fiedorowicz, S Gubkin, R M Vogt, Associahedra and weak monoidal structures on categories, Algebr. Geom. Topol. 12 (2012) 469

[8] Z Fiedorowicz, M Stelzer, R M Vogt, Homotopy colimits of algebras over Cat–operads and iterated loop spaces, Adv. Math. 248 (2013) 1089

[9] Z Fiedorowicz, R Vogt, Simplicial n–fold monoidal categories model all loop spaces, Cah. Topol. Géom. Différ. Catég. 44 (2003) 105

[10] P S Hirschhorn, Model categories and their localizations, 99, Amer. Math. Soc. (2003)

[11] J Hollender, R M Vogt, Modules of topological spaces, applications to homotopy limits and E∞ structures, Arch. Math. (Basel) 59 (1992) 115

[12] T Leinster, Higher operads, higher categories, 298, Cambridge Univ. Press (2004)

[13] J P May, The geometry of iterated loop spaces, 271, Springer (1972)

[14] J P May, R Thomason, The uniqueness of infinite loop space machines, Topology 17 (1978) 205

[15] C Schlichtkrull, M Solberg, Braided injections and double loop spaces, preprint (2015)

[16] R Schwänzl, R M Vogt, E∞–spaces and injective Γ–spaces, Manuscripta Math. 61 (1988) 203

[17] R Schwänzl, R M Vogt, The categories of A∞– and E∞–monoids and ring spaces as closed simplicial and topological model categories, Arch. Math. 56 (1991) 405

[18] R W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979) 91

[19] R W Thomason, First quadrant spectral sequences in algebraic K–theory via homotopy colimits, Comm. Algebra 10 (1982) 1589

[20] R W Thomason, Symmetric monoidal categories model all connective spectra, Theory Appl. Categ. 1 (1995) 78