Spectra associated to symmetric monoidal bicategories

Osorno, Angélica M¹

¹Department of Mathematics, University of Chicago, 5734 S University Ave, Chicago IL 60637, USA

Algebraic and Geometric Topology, Tome 12 (2012) no. 1, pp. 307-342

Résumé

We show how to construct a Γ–bicategory from a symmetric monoidal bicategory and use that to show that the classifying space is an infinite loop space upon group completion. We also show a way to relate this construction to the classic Γ–category construction for a permutative category. As an example, we use this machinery to construct a delooping of the K–theory of a rig category as defined by Baas, Dundas and Rognes [London Math. Soc. Lecture Note Ser. 308, Cambridge Univ. Press (2004) 18–45].

Détail

DOI : 10.2140/agt.2012.12.307

Classification : 18D05, 55B20, 55P42, 19D23, 55N15
Keywords: symmetric monoidal bicategory, spectra, $K$–theory

Bibliographie

[1] N A Baas, B I Dundas, B Richter, J Rognes, Stable bundles over rig categories, J. Topol. 4 (2011) 623

[2] N A Baas, B I Dundas, J Rognes, Two-vector bundles and forms of elliptic cohomology, from: "Topology, geometry and quantum field theory" (editor U Tillmann), London Math. Soc. Lecture Note Ser. 308, Cambridge Univ. Press (2004) 18

[3] J Bénabou, Introduction to bicategories, from: "Reports of the Midwest Category Seminar", Springer (1967) 1

[4] P Carrasco, A M Cegarra, A R Garzón, Nerves and classifying spaces for bicategories, Algebr. Geom. Topol. 10 (2010) 219

[5] B Day, R Street, Monoidal bicategories and Hopf algebroids, Adv. Math. 129 (1997) 99

[6] A D Elmendorf, M A Mandell, Rings, modules, and algebras in infinite loop space theory, Adv. Math. 205 (2006) 163

[7] R Gordon, A J Power, R Street, Coherence for tricategories, 117, no. 558, Amer. Math. Soc. (1995)

[8] B J Guillou, Strictification of categories weakly enriched in symmetric monoidal categories, Theory Appl. Categ. 24 (2010) 564

[9] M M Kapranov, V A Voevodsky, 2–categories and Zamolodchikov tetrahedra equations, from: "Algebraic groups and their generalizations : quantum and infinite-dimensional methods (University Park, PA, 1991)" (editors W J Haboush, B J Parshall), Proc. Sympos. Pure Math. 56, Amer. Math. Soc. (1994) 177

[10] G M Kelly, R Street, Review of the elements of 2–categories, from: "Category Seminar (Proc. Sem., Sydney, 1972/1973)" (editor G M Kelly), Lecture Notes in Math. 420, Springer (1974) 75

[11] S Lack, S Paoli, 2–nerves for bicategories, K–Theory 38 (2008) 153

[12] M L Laplaza, Coherence for distributivity, from: "Coherence in categories" (editor S Mac Lane), Lecture Notes in Math. 281, Springer (1972) 29

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

[14] S Mac Lane, R Paré, Coherence for bicategories and indexed categories, J. Pure Appl. Algebra 37 (1985) 59

[15] M A Mandell, An inverse K–theory functor, Doc. Math. 15 (2010) 765

[16] J P May, E∞ ring spaces and E∞ ring spectra, 577, Springer (1977) 268

[17] J P May, The spectra associated to permutative categories, Topology 17 (1978) 225

[18] J P May, The construction of E∞ ring spaces from bipermutative categories, from: "New topological contexts for Galois theory and algebraic geometry (BIRS 2008)" (editors A Baker, B Richter), Geom. Topol. Monogr., Geom. Topol. Publ. (2009) 283

[19] P Mccrudden, Balanced coalgebroids, Theory Appl. Categ. 7 (2000) 71

[20] E G Minian, Loop and suspension functors for small categories and stable homotopy groups, Appl. Categ. Structures 11 (2003) 207

[21] E G Minian, Spectra of small categories and infinite loop space machines, K–Theory 37 (2006) 249

[22] D Quillen, Higher algebraic K–theory. I, from: "Algebraic K–theory, I : Higher K–theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972)" (editor H Bass), Lecture Notes in Math. 341, Springer (1973) 85

[23] C Schlichtkrull, Units of ring spectra and their traces in algebraic K–theory, Geom. Topol. 8 (2004) 645

[24] C Schommer-Pries, The classification of two-dimensional extended topological field theories, preprint (2011)

[25] G Segal, Categories and cohomology theories, Topology 13 (1974) 293

[26] N Shimada, K Shimakawa, Delooping symmetric monoidal categories, Hiroshima Math. J. 9 (1979) 627

[27] M Shulman, Constructing symmetric monoidal bicategories

[28] R Street, Fibrations in bicategories, Cahiers Topologie Géom. Différentielle 21 (1980) 111

[29] R Street, Categorical structures, from: "Handbook of algebra, Vol. 1" (editor M Hazewinkel), North-Holland (1996) 529

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

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