Labelled cospan categories and properads. (English) Zbl 1527.18020
Properads are generalizations of operads that allow operations with multiple inputs and multiple outputs, with composition along connected graphs. A labelled cospan category in the sense of Steinebrunner is a symmetric monoidal functor \(\pi \colon C \to \mathbf{Csp}\) satisfying some axioms. Here \(\mathbf{Csp}\) is the category with finite sets as objects, and a morphism from \(A\) to \(B\) is an equivalence class of cospans \(A \to C \leftarrow B\). The authors prove the following conjecture of Steinebrunner.
{Theorem A}. The 2-category of properads is biequivalent to the 2-category of labelled cospan categories.
The proof of Theorem A uses the following variant.
{Theorem B}. There is a strict 2-equivalence between the 2-category of properads and the 2-category of strict labelled cospan categories.
{Theorem A}. The 2-category of properads is biequivalent to the 2-category of labelled cospan categories.
The proof of Theorem A uses the following variant.
{Theorem B}. There is a strict 2-equivalence between the 2-category of properads and the 2-category of strict labelled cospan categories.
Reviewer: Donald Yau (Newark)
MSC:
| 18M85 | Polycategories/dioperads, properads, PROPs, cyclic operads, modular operads |
| 18B10 | Categories of spans/cospans, relations, or partial maps |
| 18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |
| 18M60 | Operads (general) |
| 55P48 | Loop space machines and operads in algebraic topology |
| 55U10 | Simplicial sets and complexes in algebraic topology |
References:
| [1] | Barkan, Shaul; Haugseng, Rune; Steinebrunner, Jan, Envelopes for algebraic patterns, preprint |
| [2] | Barkan, Shaul; Steinebrunner, Jan, The equifibered approach to ∞-properads, preprint |
| [3] | Batanin, M. A.; Berger, C., Homotopy theory for algebras over polynomial monads, Theory Appl. Categ., 32, 6, 148-253 (2017), MR 3607212 · Zbl 1368.18006 |
| [4] | Carboni, Aurelio, Matrices, relations, and group representations, J. Algebra, 136, 2, 497-529 (1991), MR 1089310 · Zbl 0723.18007 |
| [5] | Chu, Hongyi; Hackney, Philip, On rectification and enrichment of infinity properads, J. Lond. Math. Soc. (2), 105, 1, 1418-1517 (2022), MR 4389306 · Zbl 1525.18007 |
| [6] | Chu, Hongyi; Haugseng, Rune, Homotopy-coherent algebra via Segal conditions, Adv. Math., 385, Article 107733 pp. (2021), 95. MR 4256131 · Zbl 1473.18017 |
| [7] | Coya, Brandon; Fong, Brendan, Corelations are the prop for extraspecial commutative Frobenius monoids, Theory Appl. Categ., 32, 11, 380-395 (2017), MR 3633708 · Zbl 1372.18002 |
| [8] | Duncan, Ross, Types for quantum computing (2006), Oxford University, Ph.D. thesis |
| [9] | Gray, John W., Fibred and cofibred categories, (Proc. Conf. Categorical Algebra. Proc. Conf. Categorical Algebra, La Jolla, Calif., 1965 (1966), Springer: Springer New York), 21-83, MR 0213413 · Zbl 0192.10701 |
| [10] | Philip Hackney, The 2-category of properads, (working title), in preparation. · Zbl 1525.18007 |
| [11] | Hackney, Philip; Robertson, Marcy, On the category of props, Appl. Categ. Struct., 23, 4, 543-573 (2015), MR 3367131 · Zbl 1320.18010 |
| [12] | Hackney, Philip; Robertson, Marcy, The homotopy theory of simplicial props, Isr. J. Math., 219, 2, 835-902 (2017), MR 3649609 · Zbl 1386.18055 |
| [13] | Hackney, Philip; Robertson, Marcy; Yau, Donald, Infinity Properads and Infinity Wheeled Properads, Lecture Notes in Mathematics, vol. 2147 (2015), Springer: Springer Cham, MR 3408444 · Zbl 1338.18002 |
| [14] | Hackney, Philip; Robertson, Marcy; Yau, Donald, A simplicial model for infinity properads, High. Struct., 1, 1, 1-21 (2017), MR 3912049 · Zbl 1426.18005 |
| [15] | Haugseng, Rune; Kock, Joachim, ∞-operads as symmetric monoidal ∞-categories, Forthcoming in Publ. Mat. · Zbl 1494.18019 |
| [16] | Johnson, Niles; Yau, Donald, 2-Dimensional Categories (2021), Oxford University Press: Oxford University Press Oxford, MR 4261588 · Zbl 1471.18002 |
| [17] | Kaufmann, Ralph M.; Monaco, Michael, Plus constructions, plethysm, and unique factorization categories with applications to graphs and operad-like theories, preprint |
| [18] | Lack, Stephen, Composing PROPS, Theory Appl. Categ., 13, 9, 147-163 (2004), MR 2116328 · Zbl 1062.18007 |
| [19] | Loday, Jean-Louis; Vallette, Bruno, Algebraic Operads, Grundlehren der mathematischen Wissenschaften, vol. 346 (2012), Springer: Springer Heidelberg, MR 2954392 · Zbl 1260.18001 |
| [20] | Loregian, Fosco; Riehl, Emily, Categorical notions of fibration, Expo. Math., 38, 4, 496-514 (2020), MR 4177953 · Zbl 1464.18010 |
| [21] | Lurie, Jacob, Higher Topos Theory, Annals of Mathematics Studies, vol. 170 (2009), Princeton University Press: Princeton University Press Princeton, NJ, MR 2522659 · Zbl 1175.18001 |
| [22] | Mac Lane, Saunders, Categorical algebra, Bull. Am. Math. Soc., 71, 40-106 (1965), MR 171826 · Zbl 0324.55001 |
| [23] | May, J. P., \( E_\infty\) spaces, group completions, and permutative categories, (New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972). New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), London Math. Soc. Lecture Note Ser., vol. 11 (1974), Cambridge University Press), 61-93 · Zbl 0281.55003 |
| [24] | nLab authors, Kan extension along (op)fibration |
| [25] | Riehl, Emily; Verity, Dominic, On the construction of limits and colimits in ∞-categories, Theory Appl. Categ., 35, 30, 1101-1158 (2020), MR 4127725 · Zbl 1451.18043 |
| [26] | Riehl, Emily; Verity, Dominic, Elements of ∞-Category Theory, Cambridge Studies in Advanced Mathematics, vol. 194 (2022), Cambridge University Press: Cambridge University Press Cambridge, MR 4354541 · Zbl 1492.18001 |
| [27] | Rosebrugh, R.; Sabadini, N.; Walters, R. F.C., Generic commutative separable algebras and cospans of graphs, Theory Appl. Categ., 15, 6, 164-177 (2005/06), MR 2210579 · Zbl 1087.18003 |
| [28] | Steinebrunner, Jan, The surface category and tropical curves, preprint · Zbl 1471.57037 |
| [29] | Vallette, Bruno, A Koszul duality for PROPs, Trans. Am. Math. Soc., 359, 10, 4865-4943 (2007), MR 2320654 (2008e:18020) · Zbl 1140.18006 |
| [30] | Yau, Donald; Johnson, Mark W., A Foundation for PROPs, Algebras, and Modules, Mathematical Surveys and Monographs, vol. 203 (2015), American Mathematical Society: American Mathematical Society Providence, RI, MR 3329226 · Zbl 1328.18014 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
