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On operads, bimodules and analytic functors
About this Title
Nicola Gambino, School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom and André Joyal, Department de Mathèmatiques, Universitè du Quèbec à Montréal, Case Postale 8888, Succursale Centre-Ville, Montŕeal (Quèbec) H3C 3P8, Canada
Publication: Memoirs of the American Mathematical Society
Publication Year:
2017; Volume 249, Number 1184
ISBNs: 978-1-4704-2576-0 (print); 978-1-4704-4135-7 (online)
DOI: https://doi.org/10.1090/memo/1184
Published electronically: August 9, 2017
Keywords: Operad,
bimodule,
bicategory,
symmetric sequence,
analytic functor
MSC: Primary 18D50; Secondary 55P48, 18D05, 18C15
Table of Contents
Chapters
- Introduction
- 1. Background
- 2. Monoidal distributors
- 3. Symmetric sequences
- 4. The bicategory of operad bimodules
- 5. Cartesian closure of operad bimodules
- A. A compendium of bicategorical definitions
- B. A technical proof
Abstract
We develop further the theory of operads and analytic functors. In particular, we introduce the bicategory $\operatorname {OpdBim}_{\mathcal {V}}$ of operad bimodules, that has operads as $0$-cells, operad bimodules as $1$-cells and operad bimodule maps as 2-cells, and prove that it is cartesian closed. In order to obtain this result, we extend the theory of distributors and the formal theory of monads.- J. Adámek, J. Rosický, and E. M. Vitale, Algebraic theories, Cambridge Tracts in Mathematics, vol. 184, Cambridge University Press, Cambridge, 2011. A categorical introduction to general algebra; With a foreword by F. W. Lawvere. MR 2757312
- J. Adámek and J. Velebil, Analytic functors and weak pullbacks, Theory Appl. Categ. 21 (2008), No. 11, 191–209. MR 2476852
- Marcelo Aguiar and Swapneel Mahajan, Monoidal functors, species and Hopf algebras, CRM Monograph Series, vol. 29, American Mathematical Society, Providence, RI, 2010. With forewords by Kenneth Brown and Stephen Chase and André Joyal. MR 2724388
- John C. Baez and James Dolan, Higher-dimensional algebra. III. $n$-categories and the algebra of opetopes, Adv. Math. 135 (1998), no. 2, 145–206. MR 1620826, DOI 10.1006/aima.1997.1695
- Michael Barr, The Chu construction, Theory Appl. Categ. 2 (1996), No. 2, 17–35. MR 1393936
- Michael Barr and Charles Wells, Toposes, triples and theories, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 278, Springer-Verlag, New York, 1985. MR 771116
- M. A. Batanin, Monoidal globular categories as a natural environment for the theory of weak $n$-categories, Adv. Math. 136 (1998), no. 1, 39–103. MR 1623672, DOI 10.1006/aima.1998.1724
- Jean Bénabou, Introduction to bicategories, Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1–77. MR 0220789
- J. Bénabou. Les distributeurs. Technical Report 33, Séminaire de Mathématiques Pures, Université Catholique de Louvain, 1973.
- J. Bénabou. Distributors at work. Available from T. Streicher’s web page, 2000. Notes by T. Streicher of a course given at TU Darmstadt.
- Clemens Berger and Ieke Moerdijk, Axiomatic homotopy theory for operads, Comment. Math. Helv. 78 (2003), no. 4, 805–831. MR 2016697, DOI 10.1007/s00014-003-0772-y
- Clemens Berger and Ieke Moerdijk, Resolution of coloured operads and rectification of homotopy algebras, Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, Amer. Math. Soc., Providence, RI, 2007, pp. 31–58. MR 2342815, DOI 10.1090/conm/431/08265
- F. Bergeron, G. Labelle, and P. Leroux, Combinatorial species and tree-like structures, Encyclopedia of Mathematics and its Applications, vol. 67, Cambridge University Press, Cambridge, 1998. Translated from the 1994 French original by Margaret Readdy; With a foreword by Gian-Carlo Rota. MR 1629341
- Renato Betti, Aurelio Carboni, Ross Street, and Robert Walters, Variation through enrichment, J. Pure Appl. Algebra 29 (1983), no. 2, 109–127. MR 707614, DOI 10.1016/0022-4049(83)90100-7
- R. Blackwell, G. M. Kelly, and A. J. Power, Two-dimensional monad theory, J. Pure Appl. Algebra 59 (1989), no. 1, 1–41. MR 1007911, DOI 10.1016/0022-4049(89)90160-6
- J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin-New York, 1973. MR 0420609
- F. Borceux. Handbook of categorical algebra. Cambridge University Press, 1994.
- Aurelio Carboni, Stefano Kasangian, and Robert Walters, An axiomatics for bicategories of modules, J. Pure Appl. Algebra 45 (1987), no. 2, 127–141. MR 889588, DOI 10.1016/0022-4049(87)90065-X
- Gian Luca Cattani and Glynn Winskel, Profunctors, open maps and bisimulation, Math. Structures Comput. Sci. 15 (2005), no. 3, 553–614. MR 2142929, DOI 10.1017/S0960129505004718
- Eugenia Cheng and Nick Gurski, The periodic table of $n$-categories II: Degenerate tricategories, Cah. Topol. Géom. Différ. Catég. 52 (2011), no. 2, 82–125 (English, with English and French summaries). MR 2839900
- E. Cheng, M. Hyland, and J. Power. Pseudo-distributive laws. Electronic Notes in Theoretical Computer Science, 83(1–20), 2004.
- Alex Chirvasitu and Theo Johnson-Freyd, The fundamental pro-groupoid of an affine 2-scheme, Appl. Categ. Structures 21 (2013), no. 5, 469–522. MR 3097055, DOI 10.1007/s10485-011-9275-y
- Pierre-Louis Curien, Operads, clones, and distributive laws, Operads and universal algebra, Nankai Ser. Pure Appl. Math. Theoret. Phys., vol. 9, World Sci. Publ., Hackensack, NJ, 2012, pp. 25–49. MR 3013081, DOI 10.1142/9789814365123_{0}002
- B. J. Day. Construction of Biclosed Categories. PhD thesis, University of New South Wales, 1970.
- Brian Day, On closed categories of functors, Reports of the Midwest Category Seminar, IV, Lecture Notes in Mathematics, Vol. 137, Springer, Berlin, 1970, pp. 1–38. MR 0272852
- W. Dwyer and K. Hess. The Boardman-Vogt tensor product of operadic bimodules. arXiv:1302.3711, 2013.
- Thomas Ehrhard and Laurent Regnier, The differential lambda-calculus, Theoret. Comput. Sci. 309 (2003), no. 1-3, 1–41. MR 2016523, DOI 10.1016/S0304-3975(03)00392-X
- A. D. Elmendorf and M. A. Mandell, Rings, modules, and algebras in infinite loop space theory, Adv. Math. 205 (2006), no. 1, 163–228. MR 2254311, DOI 10.1016/j.aim.2005.07.007
- A. D. Elmendorf and M. A. Mandell, Permutative categories, multicategories and algebraic $K$-theory, Algebr. Geom. Topol. 9 (2009), no. 4, 2391–2441. MR 2558315, DOI 10.2140/agt.2009.9.2391
- M. Fiore, N. Gambino, M. Hyland, and G. Winskel, The Cartesian closed bicategory of generalised species of structures, J. Lond. Math. Soc. (2) 77 (2008), no. 1, 203–220. MR 2389925, DOI 10.1112/jlms/jdm096
- Marcelo P. Fiore, Mathematical models of computational and combinatorial structures (invited address), Foundations of software science and computation structures, Lecture Notes in Comput. Sci., vol. 3441, Springer, Berlin, 2005, pp. 25–46. MR 2179106, DOI 10.1007/978-3-540-31982-5_{2}
- Benoit Fresse, Modules over operads and functors, Lecture Notes in Mathematics, vol. 1967, Springer-Verlag, Berlin, 2009. MR 2494775
- R. Garner and M. Shulman. Enriched categories as a free cocompletion. arXiv:1301.3191, 2013.
- Ezra Getzler, Operads revisited, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Progr. Math., vol. 269, Birkhäuser Boston, Boston, MA, 2009, pp. 675–698. MR 2641184, DOI 10.1007/978-0-8176-4745-2_{1}6
- Victor Ginzburg and Mikhail Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), no. 1, 203–272. MR 1301191, DOI 10.1215/S0012-7094-94-07608-4
- R. Gordon, A. J. Power, and Ross Street, Coherence for tricategories, Mem. Amer. Math. Soc. 117 (1995), no. 558, vi+81. MR 1261589, DOI 10.1090/memo/0558
- Nick Gurski, Coherence in three-dimensional category theory, Cambridge Tracts in Mathematics, vol. 201, Cambridge University Press, Cambridge, 2013. MR 3076451
- Nick Gurski and Angélica M. Osorno, Infinite loop spaces, and coherence for symmetric monoidal bicategories, Adv. Math. 246 (2013), 1–32. MR 3091798, DOI 10.1016/j.aim.2013.06.028
- Martin Hyland, Some reasons for generalising domain theory, Math. Structures Comput. Sci. 20 (2010), no. 2, 239–265. MR 2607096, DOI 10.1017/S0960129509990375
- Geun Bin Im and G. M. Kelly, A universal property of the convolution monoidal structure, J. Pure Appl. Algebra 43 (1986), no. 1, 75–88. MR 862873, DOI 10.1016/0022-4049(86)90005-8
- André Joyal, Une théorie combinatoire des séries formelles, Adv. in Math. 42 (1981), no. 1, 1–82 (French, with English summary). MR 633783, DOI 10.1016/0001-8708(81)90052-9
- André Joyal, Foncteurs analytiques et espèces de structures, Combinatoire énumérative (Montreal, Que., 1985/Quebec, Que., 1985) Lecture Notes in Math., vol. 1234, Springer, Berlin, 1986, pp. 126–159 (French). MR 927763, DOI 10.1007/BFb0072514
- M. Kapranov and Yu. Manin, Modules and Morita theorem for operads, Amer. J. Math. 123 (2001), no. 5, 811–838. MR 1854112
- Gregory Maxwell Kelly, Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series, vol. 64, Cambridge University Press, Cambridge-New York, 1982. MR 651714
- G. M. Kelly, Structures defined by finite limits in the enriched context. I, Cahiers Topologie Géom. Différentielle 23 (1982), no. 1, 3–42. Third Colloquium on Categories, Part VI (Amiens, 1980). MR 648793
- G. M. Kelly, On the operads of J. P. May, Repr. Theory Appl. Categ. 13 (2005), 1–13. MR 2177746
- G. M. Kelly and A. J. Power, Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads, J. Pure Appl. Algebra 89 (1993), no. 1-2, 163–179. MR 1239558, DOI 10.1016/0022-4049(93)90092-8
- Jürgen Koslowski, Monads and interpolads in bicategories, Theory Appl. Categ. 3 (1997), No. 8, 182–212. MR 1472221
- Stephen Lack, A 2-categories companion, Towards higher categories, IMA Vol. Math. Appl., vol. 152, Springer, New York, 2010, pp. 105–191. MR 2664622, DOI 10.1007/978-1-4419-1524-5_{4}
- Stephen Lack and Ross Street, The formal theory of monads. II, J. Pure Appl. Algebra 175 (2002), no. 1-3, 243–265. Special volume celebrating the 70th birthday of Professor Max Kelly. MR 1935981, DOI 10.1016/S0022-4049(02)00137-8
- Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
- Saunders Mac Lane and Robert Paré, Coherence for bicategories and indexed categories, J. Pure Appl. Algebra 37 (1985), no. 1, 59–80. MR 794793, DOI 10.1016/0022-4049(85)90087-8
- F. William Lawvere, Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. Fis. Milano 43 (1973), 135–166 (1974) (English, with Italian summary). MR 352214, DOI 10.1007/BF02924844
- Tom Leinster, Higher operads, higher categories, London Mathematical Society Lecture Note Series, vol. 298, Cambridge University Press, Cambridge, 2004. MR 2094071
- Tom Leinster, Are operads algebraic theories?, Bull. London Math. Soc. 38 (2006), no. 2, 233–238. MR 2214475, DOI 10.1112/S002460930601825X
- Muriel Livernet, From left modules to algebras over an operad: application to combinatorial Hopf algebras, Ann. Math. Blaise Pascal 17 (2010), no. 1, 47–96 (English, with English and French summaries). MR 2674654
- Jean-Louis Loday and Bruno Vallette, Algebraic operads, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346, Springer, Heidelberg, 2012. MR 2954392
- M. A. Mandell, J. P. May, S. Schwede, and B. Shipley, Model categories of diagram spectra, Proc. London Math. Soc. (3) 82 (2001), no. 2, 441–512. MR 1806878, DOI 10.1112/S0024611501012692
- Martin Markl, Models for operads, Comm. Algebra 24 (1996), no. 4, 1471–1500. MR 1380606, DOI 10.1080/00927879608825647
- M. Markl, S. Schnider, and J. Stasheff. Operads in Algebra, Topology and Physics. American Mathematical Society, 2007.
- F. Marmolejo, Distributive laws for pseudomonads, Theory Appl. Categ. 5 (1999), No. 5, 91–147. MR 1673316
- F. Marmolejo and R. J. Wood, Coherence for pseudodistributive laws revisited, Theory Appl. Categ. 20 (2008), No. 5, 74–84. MR 2395242
- J. P. May, The geometry of iterated loop spaces, Springer-Verlag, Berlin-New York, 1972. Lectures Notes in Mathematics, Vol. 271. MR 0420610
- Joan Millès, André-Quillen cohomology of algebras over an operad, Adv. Math. 226 (2011), no. 6, 5120–5164. MR 2775896, DOI 10.1016/j.aim.2011.01.002
- Charles W. Rezk, Spaces of algebra structures and cohomology of operads, ProQuest LLC, Ann Arbor, MI, 1996. Thesis (Ph.D.)–Massachusetts Institute of Technology. MR 2716655
- V. A. Smirnov, On the cochain complex of topological spaces, Mat. Sb. (N.S.) 115(157) (1981), no. 1, 146–158, 160 (Russian). MR 618592
- V. A. Smirnov, Simplicial and operad methods in algebraic topology, Translations of Mathematical Monographs, vol. 198, American Mathematical Society, Providence, RI, 2001. Translated from the Russian manuscript by G. L. Rybnikov. MR 1811110
- Michael Stay, Compact closed bicategories, Theory Appl. Categ. 31 (2016), Paper No. 26, 755–798. MR 3542382
- Ross Street, The formal theory of monads, J. Pure Appl. Algebra 2 (1972), no. 2, 149–168. MR 299653, DOI 10.1016/0022-4049(72)90019-9
- Ross Street, Fibrations in bicategories, Cahiers Topologie Géom. Différentielle 21 (1980), no. 2, 111–160. MR 574662
- Ross Street, Enriched categories and cohomology, Proceedings of the Symposium on Categorical Algebra and Topology (Cape Town, 1981), 1983, pp. 265–283. MR 700252