Affinization of monoidal categories (Q2660434)

The article defines and studies a general construction called affinization which produces a category for any given strict monoidal category \(\mathcal{C}\). The affinization of \(\mathcal{C}\) can be thought of as the category of \(\mathcal{C}\)-(string )diagrams on the cylinder. If \(\mathcal{C}\) is braided, the affinization is strict monoidal and can be obtained from \(\mathcal{C}\) by adjoining so-called ``dot''-morphisms for all objects, which represent identity morphisms ``wrapping around the cylinder''. Properties of the affinization are examined. For instance, it is shown that any action of \(\mathcal{C}\) on a balanced strict monoidal category extends to an action of the affinization of \(\mathcal{C}\), where the action of the dot-morphisms involve twists. A comparison of the affinization with the so-called horizontal trace construction yields that the two result in isomorphic categories if \(\mathcal{C}\) is rigid, but not in general, in which case the affinization still naturally corresponds to the category of \(\mathcal{C}\)-diagrams on the cylinder. It is explained that the so-called vertical trace of the affinization can be regarded as the category of \(\mathcal{C}\)-diagrams on the torus. On the other hand, for a left-rigid or right-rigid braided \(\mathcal{C}\), the endomorphism algebra of the identity object in the affinization is identified with the vertical trace of \(\mathcal{C}\). Affinizations are computed for several categories \(\mathcal{C}\). In particular, it is shown that the affinization of various categories of braids or (framed/oriented) tangles over the disc are exactly the corresponding categories over the annulus. Similarly, the affinization of the framed HOMFLYPT skein category or the Kauffman skein category over the disc is the corresponding category over the annulus. The affinization of the tower of Iwahori-Hecke algebras of type A, viewed as a monoidal category, is the tower of affine Hecke algebras of type A, and the affinization of the Temperley-Lieb category is the affine Temperley-Lieb category. An affinization construction for 2-categories generalizing the construction for strict monoidal categories is defined and discussed briefly. It is stated that the construction reproduces the horizontal trace construction in the rigid situation, while it produces a different category in general, which still corresponds to the category of string diagrams for the 2-category on the cylinder.