Enriched quasicategories and the templicial homotopy coherent nerve

Lowen, Wendy¹ ; Mertens, Arne¹

¹Departement Wiskunde, Universiteit Antwerpen, Antwerpen, Belgium

Algebraic and Geometric Topology, Tome 25 (2025) no. 2, pp. 1029-1074

Résumé

We lay the foundations for a theory of quasicategories in a monoidal category 𝒱 replacing Set ⁡ , aimed at realising weak enrichment in the category S𝒱 of simplicial objects in 𝒱. To accommodate noncartesian monoidal products, we make use of an ambient category S⊗𝒱 of templicial, or “tensor-simplicial”, objects in 𝒱, which are certain colax monoidal functors, following Leinster. Inspired by the description of the categorification functor due to Dugger and Spivak, we construct a templicial analogue of the homotopy coherent nerve functor which goes from S𝒱-enriched categories to S⊗𝒱. We show that an S𝒱-enriched category whose underlying simplicial category is locally Kan is turned into a quasicategory in 𝒱 by this nerve functor.

Détail

DOI : 10.2140/agt.2025.25.1029

Keywords: quasicategory, enriched category, necklace, templicial object, homotopy coherent nerve

Bibliographie

[1] M Aguiar, Internal categories and quantum groups, PhD thesis, Cornell University (1997)

[2] M Aguiar, S Mahajan, Monoidal functors, species and Hopf algebras, 29, Amer. Math. Soc. (2010) | DOI

[3] H V Bacard, Segal enriched categories and applications, Theory Appl. Categ. 35 (2020) 1227

[4] H J Baues, Geometry of loop spaces and the cobar construction, 230, Amer. Math. Soc. (1980) | DOI

[5] C Berger, I Moerdijk, On the homotopy theory of enriched categories, Q. J. Math. 64 (2013) 805 | DOI

[6] J E Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007) 2043 | DOI

[7] J E Bergner, Three models for the homotopy theory of homotopy theories, Topology 46 (2007) 397 | DOI

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

[9] A K Bousfield, Constructions of factorization systems in categories, J. Pure Appl. Algebra 9 (1976/77) 207 | DOI

[10] J M Cordier, Sur la notion de diagramme homotopiquement cohérent, Cahiers Topologie Géom. Différentielle 23 (1982) 93

[11] J M Cordier, T Porter, Vogt’s theorem on categories of homotopy coherent diagrams, Math. Proc. Cambridge Philos. Soc. 100 (1986) 65 | DOI

[12] B Day, On closed categories of functors, from: "Reports of the Midwest Category Seminar, IV" (editor S MacLane), Lecture Notes in Math. 137, Springer (1970) 1 | DOI

[13] E Dimitriadis Bermejo, A new model for dg-categories, PhD thesis, Université de Toulouse (2022)

[14] D Dugger, D I Spivak, Rigidification of quasi-categories, Algebr. Geom. Topol. 11 (2011) 225 | DOI

[15] S Eilenberg, J A Zilber, Semi-simplicial complexes and singular homology, Ann. of Math. 51 (1950) 499 | DOI

[16] D Gepner, R Haugseng, Enriched ∞-categories via non-symmetric ∞-operads, Adv. Math. 279 (2015) 575 | DOI

[17] H Gindi, Coherent nerves for higher quasicategories, Theory Appl. Categ. 37 (2021) 709

[18] D Grady, D Pavlov, Extended field theories are local and have classifying spaces, preprint (2023)

[19] R Haugseng, Rectification of enriched ∞-categories, Algebr. Geom. Topol. 15 (2015) 1931 | DOI

[20] A Hirschowitz, C Simpson, Descente pour les n-champs, preprint (2001)

[21] A Joyal, Disks, duality, and Theta-categories, unpublished manuscript (1997)

[22] A Joyal, Quasi-categories and Kan complexes, J. Pure Appl. Algebra 175 (2002) 207 | DOI

[23] A Joyal, Notes on quasi-categories, unpublished manuscript (2004)

[24] A Joyal, Quasi-categories vs simplicial categories, unpublished manuscript (2007)

[25] A Joyal, M Tierney, Quasi-categories vs Segal spaces, from: "Categories in algebra, geometry and mathematical physics" (editors A Davydov, M Batanin, M Johnson, S Lack, A Neeman), Contemp. Math. 431, Amer. Math. Soc. (2007) 277 | DOI

[26] T Leinster, Homotopy algebras for operads, preprint (2000)

[27] W Lowen, A Mertens, Frobenius templicial modules and the dg-nerve, preprint (2023)

[28] J Lurie, Higher topos theory, 170, Princeton Univ. Press (2009) | DOI

[29] J Lurie, Higher algebra, preprint (2017)

[30] J Lurie, Kerodon, online resource (2018)

[31] S Maclane, Categories for the working mathematician, 5, Springer (1971) | DOI

[32] A Mertens, Templicial objects: simplicial objects in a monoidal category, PhD thesis, University of Antwerp (2022)

[33] A Mertens, Nerves of enriched categories via necklaces, preprint (2024)

[34] L Moser, N Rasekh, M Rovelli, Homotopy coherent nerves of enriched categories, preprint (2022)

[35] F Muro, Dwyer–Kan homotopy theory of enriched categories, J. Topol. 8 (2015) 377 | DOI

[36] D G Quillen, Homotopical algebra, 43, Springer (1967) | DOI

[37] C Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001) 973 | DOI

[38] E Riehl, Categorical homotopy theory, 24, Cambridge Univ. Press (2014) | DOI

[39] C Simpson, Homotopy theory of higher categories, 19, Cambridge Univ. Press (2012) | DOI

[40] A E Stanculescu, Constructing model categories with prescribed fibrant objects, Theory Appl. Categ. 29 (2014) 635 | DOI

[41] R Street, Elementary cosmoi, I, from: "Category seminar" (editor G M Kelly), Lecture Notes in Math. 420, Springer (1974) 134 | DOI