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Tortile Yang-Baxter operators in tensor categories. (English) Zbl 0726.18004
The relationship between the theories of knots and categories began with D. Yetter’s definition of the category of tangles [“Markov algebras”, Contemp. Math. 78, 705-730 (1988; Zbl 0665.57004)] and the authors’ notion of braided tensor category [“Braided monoidal categories”, Macquarie Mathematics Reports 850067 (December 1985)]. Then P. Freyd and D. Yetter [“Braided compact closed categories with applications to low dimensional topology”, Adv. Math. 77, 156-182 (1989; Zbl 0679.57003)] noticed the relevance of dual objects, V. G. Turaev [“The Yang-Baxter equations and invariants of links”, Invent. Math. 92, 527-553 (1988; Zbl 0648.57003)] made the connexion with the Yang-Baxter equation, the authors introduced balanced and tortile tensor categories to relate to framed braids and links, and M. C. Shum [“Tortile tensor categories”, PhD Thesis (Macquarie University, 1989)] proved the freeness of the tortile tensor category of framed tangles (or, tangles on ribbons).
The present paper gives a purely algebraic proof that the free tortile tensor category is also the free tensor category containing an object equipped with a “tortile” Yang-Baxter operator. For related ideas see the authors’ expository article [“An introduction to Tannaka duality and quantum groups”, Proc. Conf. Category Theory at Lake Como (Italy, 1990), to appear].
Reviewer: R.Street

##### MSC:
 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010) 57M25 Knots and links in the $$3$$-sphere (MSC2010)
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##### References:
 [1] A. Joyal and R. Street, Braided tensor categories, Adv. in Math., to appear. · Zbl 0726.18004 [2] A. Joyal and R. Street, The geometry of tensor calculus I, Adv. in Math., to appear. · Zbl 0738.18005 [3] S. Majid, Representations, duals and quantum doubles of monoidal categories, Rend. Circ. Mat. Palermo (2) Suppl., to appear. · Zbl 0762.18005 [4] Shum, M.C., Tortile tensor categories, (), Macquarie mathematics reports 900047, (April 1990) [5] Turacv, V.G., The Yang-Baxter equation and invariants of links, LOMI preprints E-3-87, (January 1987), Leningrad [6] Freyd, P.; Yetter, D., Braided compact closed categories with applications to low dimensional topology, Adv. in math., 77, 156-182, (1989) · Zbl 0679.57003
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