Geodesic
Quasi-unital ∞–categories
Harpaz, Yonatan1
1Département de mathématiques et applications, École Normale Supérieure, 45 rue d’Ulm, 75005 Paris, France
Algebraic and Geometric Topology, Tome 15 (2015) no. 4, pp. 2303-2381
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Inspired by Lurie’s theory of quasi-unital algebras we prove an analogous result for ∞–categories. By constructing a suitable model category of non-unital complete Segal spaces, we show that the unital structure of an ∞–category can be uniquely recovered from the underlying non-unital structure once suitable candidates for units have been identified. The main result of this paper can be used to produce a proof of the 1–dimensional cobordism hypothesis, as described in a forthcoming paper of the author.

DOI : 10.2140/agt.2015.15.2303
Classification : 55U35, 55U40
Keywords: higher category theory, complete Segal spaces, units

Harpaz, Yonatan  1

1 Département de mathématiques et applications, École Normale Supérieure, 45 rue d’Ulm, 75005 Paris, France 
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Harpaz, Yonatan. Quasi-unital ∞–categories. Algebraic and Geometric Topology, Tome 15 (2015) no. 4, pp. 2303-2381. doi: 10.2140/agt.2015.15.2303

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

[2] W G Dwyer, D M Kan, Function complexes in homotopical algebra, Topology 19 (1980) 427

[3] W G Dwyer, D M Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17 (1980) 267

[4] Y Harpaz, The cobordism hypothesis in dimension $1$,

[5] V Hinich, Dwyer–Kan localization revisited,

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

[7] M Hovey, Model categories, Math. Surv. Monogr. 63, Amer. Math. Soc. (1999)

[8] A Joyal, J Kock, Weak units and homotopy $3$–types, 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) 257

[9] A Joyal, J Kock, Coherence for weak units, Doc. Math. 18 (2013) 71

[10] J Kock, Weak identity arrows in higher categories, Int. Math. Res. Pap. (2006) 1

[11] J Lurie, Higher topos theory, Ann. Math. Studies 170, Princeton Univ. Press (2009)

[12] J Lurie, On the classification of topological field theories, from: "Current developments in mathematics, 2008" (editors D Jerison, B Mazur, T Mrowka, W Schmid, R Stanley, S T Yau), Int. Press (2009) 129

[13] J Lurie, Higher algebra, book in preparation (2014)

[14] V Puppe, A remark on “homotopy fibrations”, Manuscripta Math. 12 (1974) 113

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

[16] C P Rourke, B J Sanderson, $\Delta$–sets, I: Homotopy theory, Quart. J. Math. Oxford Ser. 22 (1971) 321

[17] S Schwede, B Shipley, Equivalences of monoidal model categories, Algebr. Geom. Topol. 3 (2003) 287

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