Rewriting modulo isotopies in pivotal linear $(2,2)$-categories

Abstract

In this paper, we study rewriting modulo a set of algebraic axioms in categories enriched in linear categories, called linear (2, 2)-categories. We introduce the structure of linear (3, 2)polygraph modulo as a presentation of a linear (2, 2)-category by a rewriting system modulo algebraic axioms. We introduce a symbolic computation method in order to compute linear bases for the vector spaces of 2-cells of these categories. In particular, we study the case of pivotal 2-categories using the isotopy relations given by biadjunctions on 1-cells and cyclicity conditions on 2-cells as axioms for which we rewrite modulo. By this constructive method, we recover the bases of normally ordered dotted oriented Brauer diagrams in te affine oriented Brauer linear (2, 2)-category.

Categorification of infinite-dimensional $\mathfrak{sl}_2$-modules and braid group 2-actions I: tensor products

“…However, by the results of Toen [44], if k is a field and pA, d A q and pA 1 , d A 1 q are dg-algebras, then it is possible to compute the space of 'coproduct preserving' quasifunctors RHom cop Hqe p D dg pA, d A q, D dg pA 1 , d A 1 qq. Indeed, in the same way as the category of coproducts preserving functors between categories of modules is equivalent to the category of bimodules, there is a triangulated quasi-equivalence (10) RHom cop Hqe p D dg pA, d A q, D dg pA 1 , d A 1 qq -D dg ppA 1 , d A 1 q, pA, d A qq, where D dg ppA 1 , d A 1 q, pA, d A qq is the dg-derived category of dg-bimodules. Composition of functors is equivalent to derived tensor product, and understanding the triangulated structure of RHom cop Hqe p D dg pA, d A q, D dg pA 1 , d A 1 qq becomes as easy as to understand the structure of DppA, d A q, pA 1 , d A 1 qq.…”

“…The linear 2-polygraph modulo pR, E, Sq is said to be confluent modulo E if any of its branching modulo is confluent modulo E. We refer the reader to [12,10] for rewriting properties of polygraphs and linear polygraphs modulo. The local confluence criteria in terms of critical branchings for terminating linear rewriting systems has been extended in [10] in the context of linear rewriting modulo.…”

“…The linear 2-polygraph modulo pR, E, Sq is said to be confluent modulo E if any of its branching modulo is confluent modulo E. We refer the reader to [12,10] for rewriting properties of polygraphs and linear polygraphs modulo. The local confluence criteria in terms of critical branchings for terminating linear rewriting systems has been extended in [10] in the context of linear rewriting modulo. When E R E is terminating, in order to prove that the linear 2-polygraph E R E is confluent modulo, it suffices to prove that the critical branchings modulo pα, βq where α is a rewriting step of R and β is a rewriting step of E R E are confluent.…”

“…Namely, these critical branchings modulo are given by application of a rewriting step α of R and a rewriting step γ of R on two 1-cells that are E-equivalent, with pα, e, γq being minimal for the order pα, e, βq Ď phαh 1 , heh 1 , hγh 1 q. Moreover, following [10], when the linear 2-polygraph E is convergent, the basis theorem of [15] extends to that context of rewriting modulo. Explicitely, given an algebra A presented by a linear 2-polygraph P that we split into two parts E (non-oriented) and R (oriented), if E is convergent, E R E is terminating and E R E is confluent modulo E, then the set of E-normal forms of monomials in normal form with respect to E R E yields a basis of A.…”

“…It is used mainly to split confluence proofs into many incremental steps, by first proving that the set E forms a convergent rewriting system, and then study the remaining relations on Eequivalence classes. Following [10] the usual basis result given by the irreducible monomials of a convergent presentation is extended in that setting, by considering E-normal forms of irreducible monomials with respect to S.…”