Algebraic & Geometric Topology

Spectra associated to symmetric monoidal bicategories

Angélica M Osorno

Full-text: Open access


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].

Article information

Algebr. Geom. Topol., Volume 12, Number 1 (2012), 307-342.

Received: 8 December 2010
Revised: 21 November 2011
Accepted: 28 November 2011
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18D05: Double categories, 2-categories, bicategories and generalizations 55B20 55P42: Stable homotopy theory, spectra
Secondary: 19D23: Symmetric monoidal categories [See also 18D10] 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX}

symmetric monoidal bicategory spectra $K$–theory


Osorno, Angélica M. Spectra associated to symmetric monoidal bicategories. Algebr. Geom. Topol. 12 (2012), no. 1, 307--342. doi:10.2140/agt.2012.12.307.

Export citation


  • N A Baas, B I Dundas, B Richter, J Rognes, Stable bundles over rig categories, J. Topol. 4 (2011) 623–640
  • N A Baas, B I Dundas, J Rognes, Two-vector bundles and forms of elliptic cohomology, from: “Topology, geometry and quantum field theory”, (U Tillmann, editor), London Math. Soc. Lecture Note Ser. 308, Cambridge Univ. Press (2004) 18–45
  • J Bénabou, Introduction to bicategories, from: “Reports of the Midwest Category Seminar”, Springer, Berlin (1967) 1–77
  • P Carrasco, A M Cegarra, A R Garzón, Nerves and classifying spaces for bicategories, Algebr. Geom. Topol. 10 (2010) 219–274
  • B Day, R Street, Monoidal bicategories and Hopf algebroids, Adv. Math. 129 (1997) 99–157
  • A D Elmendorf, M A Mandell, Rings, modules, and algebras in infinite loop space theory, Adv. Math. 205 (2006) 163–228
  • R Gordon, A J Power, R Street, Coherence for tricategories, Mem. Amer. Math. Soc. 117, no. 558, Amer. Math. Soc. (1995)
  • B J Guillou, Strictification of categories weakly enriched in symmetric monoidal categories, Theory Appl. Categ. 24 (2010) No. 20, 564–579
  • 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)”, (W J Haboush, B J Parshall, editors), Proc. Sympos. Pure Math. 56, Amer. Math. Soc. (1994) 177–259
  • G M Kelly, R Street, Review of the elements of $2$–categories, from: “Category Seminar (Proc. Sem., Sydney, 1972/1973)”, (G M Kelly, editor), Lecture Notes in Math. 420, Springer, Berlin (1974) 75–103
  • S Lack, S Paoli, $2$–nerves for bicategories, $K$–Theory 38 (2008) 153–175
  • M L Laplaza, Coherence for distributivity, from: “Coherence in categories”, (S Mac Lane, editor), Lecture Notes in Math. 281, Springer, Berlin (1972) 29–65
  • T Leinster, Higher operads, higher categories, London Math. Soc. Lecture Note Series 298, Cambridge Univ. Press (2004)
  • S Mac Lane, R Paré, Coherence for bicategories and indexed categories, J. Pure Appl. Algebra 37 (1985) 59–80
  • M A Mandell, An inverse $K$–theory functor, Doc. Math. 15 (2010) 765–791
  • J P May, $E\sb{\infty }$ ring spaces and $E\sb{\infty }$ ring spectra, Lecture Notes in Math. 577, Springer, Berlin (1977) With contributions by F Quinn, N Ray, and J Tornehave
  • J P May, The spectra associated to permutative categories, Topology 17 (1978) 225–228
  • J P May, The construction of $E_\infty$ ring spaces from bipermutative categories, from: “New topological contexts for Galois theory and algebraic geometry (BIRS 2008)”, (A Baker, B Richter, editors), Geom. Topol. Monogr. 16, Geom. Topol. Publ., Coventry (2009) 283–330
  • P McCrudden, Balanced coalgebroids, Theory Appl. Categ. 7 (2000) 71–147
  • E G Minian, Loop and suspension functors for small categories and stable homotopy groups, Appl. Categ. Structures 11 (2003) 207–218
  • E G Minian, Spectra of small categories and infinite loop space machines, $K$–Theory 37 (2006) 249–261
  • D Quillen, Higher algebraic $K$–theory. I, from: “Algebraic $K$–theory, I: Higher $K$–theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972)”, (H Bass, editor), Lecture Notes in Math. 341, Springer, Berlin (1973) 85–147
  • C Schlichtkrull, Units of ring spectra and their traces in algebraic $K$–theory, Geom. Topol. 8 (2004) 645–673
  • C Schommer-Pries, The classification of two-dimensional extended topological field theories, preprint (2011) Available at \setbox0\makeatletter\@url {\unhbox0
  • G Segal, Categories and cohomology theories, Topology 13 (1974) 293–312
  • N Shimada, K Shimakawa, Delooping symmetric monoidal categories, Hiroshima Math. J. 9 (1979) 627–645
  • M Shulman, Constructing symmetric monoidal bicategories
  • R Street, Fibrations in bicategories, Cahiers Topologie Géom. Différentielle 21 (1980) 111–160
  • R Street, Categorical structures, from: “Handbook of algebra, Vol. 1”, (M Hazewinkel, editor), North-Holland, Amsterdam (1996) 529–577
  • R W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979) 91–109
  • R W Thomason, Symmetric monoidal categories model all connective spectra, Theory Appl. Categ. 1 (1995) 78–118