Classification of braids which give rise to interchange

Forcey, Stefan¹ ; Humes, Felita²

¹Department of Physics and Mathematics, Tennessee State University, Nashville TN 37209, USA
²Department of Mathematics, College of the Bahamas, P O Box N4912 Nassau, Bahamas

Algebraic and Geometric Topology, Tome 7 (2007) no. 3, pp. 1233-1274

Résumé

It is well known that the existence of a braiding in a monoidal category V allows many higher structures to be built upon that foundation. These include a monoidal 2–category V–Cat of enriched categories and functors over V, a monoidal bicategory V–Mod of enriched categories and modules, a category of operads in V and a 2–fold monoidal category structure on V. These all rely on the braiding to provide the existence of an interchange morphism η necessary for either their structure or its properties. We ask, given a braiding on V, what non-equal structures of a given kind from this list exist which are based upon the braiding. For example, what non-equal monoidal structures are available on V–Cat, or what non-equal operad structures are available which base their associative structure on the braiding in V. The basic question is the same as asking what non-equal 2–fold monoidal structures exist on a given braided category. The main results are that the possible 2–fold monoidal structures are classified by a particular set of four strand braids which we completely characterize, and that these 2–fold monoidal categories are divided into two equivalence classes by the relation of 2–fold monoidal equivalence.

Détail

DOI : 10.2140/agt.2007.7.1233

Keywords: iterated monoidal categories, enriched categories, braided categories

Bibliographie

[1] J C Baez, Higher-dimensional algebra II: 2–Hilbert spaces, Adv. Math. 127 (1997) 125

[2] J C Baez, J Dolan, Categorification, from: "Higher category theory (Evanston, IL, 1997)", Contemp. Math. 230, Amer. Math. Soc. (1998) 1

[3] C Balteanu, Z Fiedorowicz, R Schwänzl, R Vogt, Iterated monoidal categories, Adv. Math. 176 (2003) 277

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

[5] S Eilenberg, G M Kelly, Closed categories, from: "Proc. Conf. Categorical Algebra (La Jolla, CA, 1965)", Springer (1966) 421

[6] S Forcey, Enrichment over iterated monoidal categories, Algebr. Geom. Topol. 4 (2004) 95

[7] S Forcey, J Siehler, E S Sowers, Operads in iterated monoidal categories, J. Homotopy Relat. Struct. 2 (2007) 1

[8] F A Garside, The braid group and other groups, Quart. J. Math. Oxford Ser. $(2)$ 20 (1969) 235

[9] A Joyal, R Street, Braided tensor categories, Adv. Math. 102 (1993) 20

[10] G M Kelly, Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series 64, Cambridge University Press (1982) 245

[11] S Macnbsp;Lane, Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer (1998)

[12] J P May, The geometry of iterated loop spaces, Lecture Notes in Mathematics 271, Springer (1972)

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

[14] M C Shum, Tortile tensor categories, J. Pure Appl. Algebra 93 (1994) 57