Related papers: On the Categorified Wrapping Number Conjecture
We generalize the classical probability frame by adopting a wider family of random variables that includes nondeterministic ones. The frame that emerges is known to host a ''classical'' extension of quantum mechanics. We discuss the notion…
We exhibit infinite families of annular links for which the maximum non-zero annular Khovanov grading grows infinitely large but the maximum non-zero annular Floer-theoretic gradings are bounded. We also show this phenomenon exists at the…
The celebrated union-closed conjecture is concerned with the cardinalities of various subsets of the Boolean $d$-cube. The cardinality of such a set is equivalent, up to a constant, to its measure under the uniform distribution, so we can…
A bridgeless cubic graph $G$ is said to have a 2-bisection if there exists a 2-vertex-colouring of $G$ (not necessarily proper) such that: (i) the colour classes have the same cardinality, and (ii) the monochromatic components are either an…
In this paper we present some applications of Ann-category theory to classification of crossed bimodules over rings, classification of ring extensions of the type of a crossed bimodule.
We use categorical annular evaluation to give a uniform construction of both $\mathfrak{sl}_n$ and HOMFLYPT Khovanov-Rozansky link homology, as well as annular versions of these theories. Variations on our construction yield…
We construct a family of rings. To a plane diagram of a tangle we associate a complex of bimodules over these rings. Chain homotopy equivalence class of this complex is an invariant of the tangle. On the level of Grothendieck groups this…
We generalize the notion of a crossed module of groups to that of a crossed module of racks. We investigate the relation to categorified racks, namely strict 2-racks, and trunk-like objects in the category of racks, generalizing the…
We offer a solution to the long-standing problem of group completing within the context of rig categories (also known as bimonoidal categories). Given a rig category R we construct a natural additive group completion R' that retains the…
In this paper, we introduce a category of graded commutative rings with certain algebraic morphisms, to investigate the cobordism category of plumbed 3-manifolds. In particular, we define a non-associative distributive algebra that gives…
We generalize the constructions of [17,19] to layered semirings, in order to enrich the structure and provide finite examples for applications in arithmetic (including finite examples). The layered category theory of [19] is extended…
We provide a framework for abstract reconstruction problems using the $K$-theory of categories with covering families, which we then apply to reformulate the edge reconstruction conjecture in graph theory. Along the way, we state some…
We build extensions of the arc rings, relate their centers to the cohomology rings of the Springer varieties, and categorify all level two representations of quantum sl(N).
We review the notions of a multiplier category and the $W^{*}$-envelope of a $C^{*}$-category. We then consider the notion of an orthogonal sum of a (possibly infinite) family of objects in a $C^{*}$-category. Furthermore, we construct…
We prove that for any prime $p$ the finite $p$-groups of fixed coclass have only finitely many different mod-$p$ cohomology rings between them. This was conjectured by Carlson; we prove it by first proving a stronger version for groups of…
We prove that any fusion category over $\mathbb{C}$ with exactly one non-invertible simple object is spherical. Furthermore, we classify all such categories that come equipped with a braiding.
Sharing of notations and theories across an inheritance hierarchy of mathematical structures, e.g., groups and rings, is important for productivity when formalizing mathematics in proof assistants. The packed classes methodology is a…
We propose a new classification scheme for quantum entanglement based on topological links. This is done by identifying a non-rigid ring to a particle, attributing the act of cutting and removing a ring to the operation of tracing out the…
The theory of rack and quandle modules is developed - in particular a tensor product is defined, and shown to satisfy an appropriate adjointness condition. Notions of free rack and quandle modules are introduced, and used to define an…
The purpose of this paper is to make the theory of vertex algebras trivial. We do this by setting up some categorical machinery so that vertex algebras are just ``singular commutative rings'' in a certain category. This makes it easy to…