Segal spaces, spans, and semicategories

Haugseng, Rune

doi:10.1090/proc/15197

Segal spaces, spans, and semicategories
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Proc. Amer. Math. Soc. 149 (2021), 961-975 Request permission

Abstract:

We show that Segal spaces, and more generally category objects in an $\infty$-category $\mathcal {C}$, can be identified with associative algebras in the double $\infty$-category of spans in $\mathcal {C}$. We use this observation to prove that “having identities” is a property of a non-unital $(\infty ,n)$-category.

References

Clark Barwick, $(\infty , n)$-Cat as a closed model category, ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)–University of Pennsylvania. MR 2706984
Clark Barwick, Spectral Mackey functors and equivariant algebraic $K$-theory (I), Adv. Math. 304 (2017), 646–727. MR 3558219, DOI 10.1016/j.aim.2016.08.043
Albert Burroni, $T$-catégories (catégories dans un triple), Cahiers Topologie Géom. Différentielle 12 (1971), 215–321 (French). MR 308236
Jean Bénabou, Introduction to bicategories, Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1–77. MR 0220789
G. S. H. Cruttwell and Michael A. Shulman, A unified framework for generalized multicategories, Theory Appl. Categ. 24 (2010), No. 21, 580–655. MR 2770076
Tobias Dyckerhoff and Mikhail Kapranov, Higher Segal spaces, Lecture Notes in Mathematics, vol. 2244, Springer, Cham, 2019. MR 3970975, DOI 10.1007/978-3-030-27124-4
Thomas M. Fiore, Nicola Gambino, and Joachim Kock, Monads in double categories, J. Pure Appl. Algebra 215 (2011), no. 6, 1174–1197. MR 2769225, DOI 10.1016/j.jpaa.2010.08.003
Dennis Gaitsgory and Nick Rozenblyum, A study in derived algebraic geometry. Vol. I. Correspondences and duality, Mathematical Surveys and Monographs, vol. 221, American Mathematical Society, Providence, RI, 2017. MR 3701352, DOI 10.1090/surv/221.1
David Gepner and Rune Haugseng, Enriched $\infty$-categories via non-symmetric $\infty$-operads, Adv. Math. 279 (2015), 575–716. MR 3345192, DOI 10.1016/j.aim.2015.02.007
David Gepner, Rune Haugseng, and Thomas Nikolaus, Lax colimits and free fibrations in $\infty$-categories, Doc. Math. 22 (2017), 1225–1266. MR 3690268
Marco Grandis and Robert Pare, Limits in double categories, Cahiers Topologie Géom. Différentielle Catég. 40 (1999), no. 3, 162–220 (English, with French summary). MR 1716779
Yonatan Harpaz, Quasi-unital $\infty$-categories, Algebr. Geom. Topol. 15 (2015), no. 4, 2303–2381. MR 3402342, DOI 10.2140/agt.2015.15.2303
Rune Haugseng, Iterated spans and classical topological field theories, Math. Z. 289 (2018), no. 3-4, 1427–1488. MR 3830256, DOI 10.1007/s00209-017-2005-x
Rune Haugseng, The higher Morita category of $\Bbb {E}_n$-algebras, Geom. Topol. 21 (2017), no. 3, 1631–1730. MR 3650080, DOI 10.2140/gt.2017.21.1631
André Joyal and Joachim Kock, Coherence for weak units, Doc. Math. 18 (2013), 71–110. MR 3035770
Tom Leinster, Generalized enrichment of categories, J. Pure Appl. Algebra 168 (2002), no. 2-3, 391–406. Category theory 1999 (Coimbra). MR 1887166, DOI 10.1016/S0022-4049(01)00105-0
Tom Leinster, Higher operads, higher categories, London Mathematical Society Lecture Note Series, vol. 298, Cambridge University Press, Cambridge, 2004. MR 2094071, DOI 10.1017/CBO9780511525896
Jacob Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. MR 2522659, DOI 10.1515/9781400830558
Jacob Lurie, Higher Algebra, 2017. Available at http://math.ias.edu/~lurie/.
Volodymyr Lyubashenko and Oleksandr Manzyuk, $A_\infty$-algebras, $A_\infty$-categories and $A_\infty$-functors, Handbook of algebra. Vol. 5, Handb. Algebr., vol. 5, Elsevier/North-Holland, Amsterdam, 2008, pp. 143–188. MR 2523451, DOI 10.1016/S1570-7954(07)05003-6
Charles Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001), no. 3, 973–1007. MR 1804411, DOI 10.1090/S0002-9947-00-02653-2
Michael Shulman, Framed bicategories and monoidal fibrations, Theory Appl. Categ. 20 (2008), No. 18, 650–738. MR 2534210
Wolfgang Steimle, Degeneracies in quasi-categories, J. Homotopy Relat. Struct. 13 (2018), no. 4, 703–714. MR 3870770, DOI 10.1007/s40062-018-0199-1
Hiro Lee Tanaka, Functors (between $\infty$-categories) that aren’t strictly unital, J. Homotopy Relat. Struct. 13 (2018), no. 2, 273–286. MR 3802796, DOI 10.1007/s40062-017-0182-2