This is the first part of a series of two papers aiming to construct a categorification of the braiding on tensor products of Verma modules, and in particular of the Lawrence--Krammer--Bigelow representations. In this part, we categorify all tensor
products of Verma modules and integrable modules for quantum $mathfrak{sl_2}$. The categorification is given by derived categories of
We develop the rewriting theory for monoidal supercategories and 2-supercategories. This extends the theory of higher-dimensional rewriting established for (linear) 2-categories to the super setting, providing a suite of tools for constructing bases
and normal forms for 2-supercategories given by generators and relations. We then employ this newly developed theory to prove the non-degeneracy conjecture for the odd categorification of quantum sl(2) from arXiv:1307.7816 and arXiv:1701.04133. As a corollary, this gives a classification of dg-structures on the odd 2-category conjectured in arXiv:1808.04924.
Convergent rewriting systems on algebraic structures give methods to solve decision problems, to prove coherence results, and to compute homological invariants. These methods are based on higher-dimensional extensions of the critical branching lemma
that characterizes local confluence from confluence of the critical branchings. The analysis of local confluence of rewriting systems on algebraic structures, such as groups or linear algebras, is complicated because of the underlying algebraic axioms, and in some situations, local confluence properties require additional termination conditions. This article introduces the structure of algebraic polygraph modulo that formalizes the interaction between the rules of an algebraic rewriting system and the inherent algebraic axioms, and we show a critical branching lemma for algebraic polygraphs. We deduce from this result a critical branching lemma for rewriting systems on algebraic objects whose axioms are specified by convergent modulo rewriting systems. We illustrate our constructions for string, linear, and group rewriting systems.
We study a presentation of Khovanov - Lauda - Rouquiers candidate $2$-categorification of a quantum group using algebraic rewriting methods. We use a computational approach based on rewriting modulo the isotopy axioms of its pivotal structure to comp
ute a family of linear bases for all the vector spaces of $2$-cells in this $2$-category. We show that these bases correspond to Khovanov and Laudas conjectured generating sets, proving the non-degeneracy of their diagrammatic calculus. This implies that this $2$-category is a categorification of Lusztigs idempotent and integral quantum group $bf{U}_{q}(mathfrak{g})$ associated to a symmetrizable simply-laced Kac-Moody algebra $mathfrak{g}$.
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~$(2,2)$-cate
gory 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.
A coherent presentation of an n-category is a presentation by generators, relations and relations among relations. Completions of presentations by rewriting systems give coherent presentations, whose relations among relations are generated by conflue
nce diagrams induced by critical branchings. This article extends this construction to presentations by polygraphs defined modulo a set of relations. Our coherence results are formulated using the structure of n-category enriched in double groupoids, whose horizontal cells represent rewriting sequences, vertical cells represent the congruence generated by relations modulo and square cells represent coherence cells induced by confluence modulo. We illustrate these constructions for rewriting modulo commutation relations in monoids and isotopy relations in pivotal monoidal categories.
