| Title:
|
The symmetric monoidal 2-category of permutative categories (English) |
| Author:
|
Johnson, Niles |
| Author:
|
Gurski, Nick |
| Author:
|
Osorno, Angélica M. |
| Language:
|
English |
| Journal:
|
Higher Structures |
| ISSN:
|
2209-0606 |
| Volume:
|
8 |
| Issue:
|
1 |
| Year:
|
2024 |
| Pages:
|
244-320 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We define a tensor product for permutative categories and prove a number of key properties. We show that this product makes the 2-category of permutative categories closed symmetric monoidal as a bicategory. (English) |
| Keyword:
|
permutative category |
| Keyword:
|
symmetric monoidal bicategory |
| MSC:
|
18D15 |
| MSC:
|
18M05 |
| MSC:
|
18N10 |
| MSC:
|
19D23 |
| idZBL:
|
Zbl 1555.18030 |
| idMR:
|
MR4752522 |
| DOI:
|
10.21136/HS.2024.06 |
| . |
| Date available:
|
2026-03-13T14:09:46Z |
| Last updated:
|
2026-03-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153470 |
| . |
| Reference:
|
[1] Blackwell, R., Kelly, G.M., Power, A.J.: Two-dimensional monad theory..J. Pure Appl. Algebra, 59(1):1–41 |
| Reference:
|
[2] Bourke, John: Skew structures in 2-category theory and homotopy theory..J. Homotopy Relat. Struct., 12(1):31–81 MR 3613021 |
| Reference:
|
[3] Carboni, A., Kelly, G. M., Walters, R. F. C., Wood, R. J.: Cartesian bicategories II..Theory Appl. Categ., 19(6):93–124 MR 3656673 |
| Reference:
|
[4] Cheng, Eugenia, Gurski, Nick: Iterated icons..Theory and Applications of Categories, 29(32):929–977 MR 3294506 |
| Reference:
|
[5] Elmendorf, A. D., Mandell, M. A.: Rings, modules, and algebras in infinite loop space theory..Adv. Math., 205(1):163–228 MR 2254311 |
| Reference:
|
[6] Fiore, Thomas M.: Pseudo Limits, Biadjoints, and Pseudo Algebras: Categorical Foundations of Conformal Field Theory..Mem. Amer. Math. Soc., 182(860):x+171 MR 2229946 |
| Reference:
|
[7] Fong, Brendan, Spivak, David I: Supplying bells and whistles in symmetric monoidal categories..Arxiv:1908.02633 http://arxiv.org/pdf/1908.02633 |
| Reference:
|
[8] Gepner, David, Groth, Moritz, Nikolaus, Thomas: Universality of multiplicative infinite loop space machines..Algebr. Geom. Topol., 15(6):3107–3153 MR 3450758 |
| Reference:
|
[9] Gordon, R., Power, A. J., Street, Ross: Coherence for tricategories, volume 117 of Mem. Amer. Math. Soc..Amer. Math. Soc |
| Reference:
|
[10] Gray, John W.: Formal category theory: adjointness for 2-categories, volume 391 of Lecture Notes in Mathematics..Springer-Verlag, Berlin-New York |
| Reference:
|
[11] Gray, John W.: Closed categories, lax limits and homotopy limits..J. Pure Appl. Algebra, 19:127–158 |
| Reference:
|
[12] Gurski, Nick: Loop spaces, and coherence for monoidal and braided monoidal bicategories..Adv. Math., 226(5):4225–4265 MR 2770448 |
| Reference:
|
[13] Gurski, Nick: Biequivalences in tricategories..Theory Appl. Categ., 26(14):349–384 MR 2972968 |
| Reference:
|
[14] Gurski, Nick: Coherence in three-dimensional category theory, volume 201 of Cambridge Tracts in Mathematics..Cambridge University Press, Cambridge MR 3076451 |
| Reference:
|
[15] Gurski, Nick: The monoidal structure of strictification..Theory Appl. Categ., 28(1):1–23 MR 3023916 |
| Reference:
|
[16] Gurski, Nick, Osorno, Angélica M: Infinite loop spaces, and coherence for symmetric monoidal bicategories..Adv. Math., 246:1–32 MR 3091798 |
| Reference:
|
[17] Hyland, Martin, Power, John: Pseudo-commutative monads and pseudo-closed 2-categories..J. Pure Appl. Algebra, 175(1-3):141–185 MR 1935977 |
| Reference:
|
[18] Johnson, Niles, Yau, Donald: Bimonoidal Categories, E_(n)-Monoidal Categories, and Algebraic K-Theory. Volume III: From Categories to Structured Ring Spectra..available at https://nilesjohnson.net MR 4806166 |
| Reference:
|
[19] Johnson, Niles, Yau, Donald: 2-Dimensional Categories..Oxford University Press, New York MR 4261588 |
| Reference:
|
[20] Joyal, André, Street, Ross: Braided tensor categories..Adv. Math., 102(1):20–78 |
| Reference:
|
[21] Kelly, G. M.: Doctrinal adjunction..In Category Seminar (Proc. Sem., Sydney, 1972/1973), volume 420 of Lecture Notes in Mathematics, pages 257–280. Springer, Berlin |
| Reference:
|
[22] Kelly, G. M.: Elementary observations on 2-categorical limits..Bull. Austral. Math. Soc., 39(2):301–317 |
| Reference:
|
[23] Lack, Stephen, Shulman, Michael: Enhanced 2-categories and limits for lax morphisms..Adv. Math., 229(1):294–356 MR 2854177 |
| Reference:
|
[24] Lane, Saunders Mac: Categories for the working mathematician, volume 5 of Grad. Texts Math..New York, NY: Springer, 2nd edition |
| Reference:
|
[25] May, J. Peter: E_(∞) spaces, group completions, and permutative categories..In New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), volume 11 of London Math. Soc. Lecture Note Ser, pages 61–93. Cambridge Univ. Press, London |
| Reference:
|
[26] May, J Peter: Pairings of categories and spectra..Journal of Pure and Applied Algebra, 19:299–346 |
| Reference:
|
[27] McCrudden, Paddy: Balanced coalgebroids..Theory and Applications of Categories, 7(6):71–147 |
| Reference:
|
[28] Pstrągowski, Piotr: On dualizable objects in monoidal bicategories..Theory Appl. Categ., 38(9):257–310 MR 4373026 |
| Reference:
|
[29] Schäppi, Daniel: Ind-abelian categories and quasi-coherent sheaves..In Mathematical Proceedings of the Cambridge Philosophical Society, volume 157, pages 391–423. Cambridge University Press MR 3286515 |
| Reference:
|
[30] Schmitt, Vincent: Tensor product for symmetric monoidal categories..Arxiv:0711.0324 http://arxiv.org/pdf/0711.0324 |
| Reference:
|
[31] Schommer-Pries, Christopher: The classification of two-dimensional extended topological field theories..Arxiv:1112.1000 http://arxiv.org/pdf/1112.1000 MR 2713992 |
| Reference:
|
[32] Street, R.: Fibrations in bicategories..Cahiers de Topologie et Geometrie Differentielle, 21(2):111–160 |
| Reference:
|
[33] Street, Ross: Elementary cosmoi. I..In Category Seminar (Proc. Sem., Sydney, 1972/1973), Lecture Notes in Math., Vol. 420, pages 134–180. Springer, Berlin 10.1007/BFb0063103 |
| Reference:
|
[34] Street, Ross: Categorical structures..In Handbook of algebra, Vol. 1, volume 1 of Handb. Algebr., pages 529–577. Elsevier/North-Holland, Amsterdam |
| Reference:
|
[35] Szlachányi, Kornel: Skew-monoidal categories and bialgebroids..Advances in Mathematics, 231(3-4):1694–1730 MR 2964621 |
| . |
