We give a new criterion for when two Bourn-normal monomorphisms with the same co domain in a unital category Huq-commute. We use this to prove that in a unital category, satisfying the (additional) condition that a morphism is a monomorphism if and only if its kernel is a zero morphism, two Bourn-normal monomorphisms with the same co domain Huq-commute as soon as they have trivial pullback. As corollaries, we show that several facts known only in the protomodular context are in fact true in more general contexts.
Following the theory of principal 8-bundles of Niklaus-Schreiber-Steveson, we develop a homotopy categorification of Hopf algebras, which model quantum groups. We study their higher-representation theory in the setting of 2Vect(hBC), which is a homotopy refinement of the notion of 2-vector spaces due to Baez-Crans that allows for higher coherence data. We construct in particular the 2-quantum double as a homotopy double crossed product, and prove its duality and factorization properties. We also define and characterize "2-R-matrices", which can be seen as an extension of the usual notion of R-matrix in an ordinary Hopf algebra. We found that the 2-Yang-Baxter equations describe the braiding of extended defects in 4d, distinct from but not unlike the the Zamolodchikov tetrahedron equations. The main results we prove in this paper is that the 2-representation 2-category of a weak 2-bialgebra is braided monoidal if it is equipped with a universal 2-R-matrix, and that our homotopy quantization admits the theory of Lie 2-bialgebras as a semiclassical limit.
In this article, we introduce an extrinsic approach to the notion of semi-direct product, an intrinsic one having been already done in [Bourn-Janelidze, 2003] and [Borceux-Janelidze-Kelly, 2005-1]. This led us to focus our attention on two algebraic structures, hyper-Slom & imath;nski setting and hypersubtraction which produce pointed protomodular categories, and will allow us to characterize this extrinsic approach.
The theory of (infinity, 1)-categories can be developed synthetically in an augmentation of homotopy type theory introduced by Riehl-Shulman. Central to their development is an additional type forming operation called extensions. The original article sketches the semantics of this formal system, explaining how the simplicial homotopy theory can be used to reason about (infinity, 1)-categories presented using the Segal space model. However, they leave it open to demonstrate the strict stability of extension types. We prove this using the splitting method of Voevodsky, later generalized by Lumsdaine-Warren to local universes. The practical upshot is that this system has semantics in simplicial objects of an oo-topos, and thus can be used to prove theorems about internal oo-categories in the sense of Martini-Wolf.
In a monoidal category, there are generally two types of ideals: thick and tensor ideals. The current paper's main focus is on ideals of the former type, which are defined on the objects of the ambient category. In the setting of a semisimple pivotal category, we introduce a particular class of pseudo-negligible morphisms and establish some related results. Moreover, given a one-sided thick ideal I of a semisimple pure pivotal category, we call I admissible when it is equipped with a one-sided trace function whose associated modified dimension is invertible on the simple objects therein. We then introduce and study a determinant function on such ideals and establish analogous properties to those of the usual determinants. As an application, we construct new topological invariants of planar graphs from these determinant functions on traced ideals.
