Related papers: Higher-dimensional normalisation strategies for ac…
We show that diagrammatic sets, a topologically sound alternative to polygraphs and strict $\omega$-categories, admit an internal notion of equivalence in the sense of coinductive weak invertibility. We prove that equivalences have the…
We define standardized constructions of finite fields, and standardized generators of (multiplicative) cyclic subgroups in these fields. The motivation is to provide a substitute for Conway polynomials which can be used by various software…
We provide a foundation for working with homological and homotopical methods in categorical algebra. This involves two mutually complementary components, namely (a) the strategic selection of suitable axiomatic frameworks, some well known…
In this paper, we introduce a new method to compute magnitude homology of general graphs. To each direct sum component of magnitude chain complexes, we assign a pair of simplicial complexes whose simplicial chain complex is isomorphic to…
We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining an even dimensional nodal hypersurface. This implies the validity of formulas due to M. Saito, L. Wotzlaw and…
In classical homotopy theory, two spaces are homotopy equivalent if one space can be continuously deformed into the other. This theory, however, does not respect the discrete nature of graphs. For this reason, a discrete homotopy theory…
This document is centered around a main idea: simplicial categories, by which we mean simplicial objects in the category of categories, can be treated as a two-fold categorical structure and their double category theory is homotopically…
We summarize our axioms for higher categories, and describe the blob complex. Fixing an n-category C, the blob complex associates a chain complex B_*(W;C)$ to any n-manifold W. The 0-th homology of this chain complex recovers the usual…
The computational cost of micromechanics for heterogeneous materials can be reduced in certain cases where symmetric boundary conditions are applicable. We derived an eighth symmetric formulation of the Generalized Method of Cells for…
We construct a finite convergent semi-quadratic presentation for the Chinese monoid by adding column generators and using combinatorial properties of insertion algorithms on Chinese staircases. We extend this presentation into a coherent…
In this article we study higher homological properties of $n$-levelled algebras and connect them to properties of the underlying graphs. Notably, to each $2$-representation-finite quadratic monomial algebra $\Lambda$ we associate a…
We show that the category of graphs has the structure of a 2-category with homotopy as the 2-cells. We then develop an explicit description of homotopies for finite graphs, in terms of what we call `spider moves'. We then create a category…
We consider the multilinear polytope defined as the convex hull of the set of binary points satisfying a collection of multilinear equations. The complexity of the facial structure of the multilinear polytope is closely related to the…
The biologically inspired framework of port-graphs has been successfully used to specify complex systems. It is the basis of the PORGY modelling tool. To facilitate the specification of proof normalisation procedures via graph rewriting, in…
We present a new notion of non-positively curved groups: the collection of discrete countable groups acting (AU-)acylindrically on finite products of $\delta$-hyperbolic spaces with general type factors. Inspired by the classical theory of…
Relationship between agents can be conveniently represented by graphs. When these relationships have different modalities, they are better modelled by multilayer graphs where each layer is associated with one modality. Such graphs arise…
Sigma models effectively describe ordered phases of systems with spontaneously broken symmetries. At low energies, field configurations fall into solitonic sectors, which are homotopically distinct classes of maps. Depending on context,…
We introduce cyclic diagram monoids, a generalisation of classical diagram monoids that adds elements of arbitrary period by including internal components, with a view towards cryptography. We classify their simple representations and…
Graph generation techniques are increasingly being adopted for drug discovery. Previous graph generation approaches have utilized relatively small molecular building blocks such as atoms or simple cycles, limiting their effectiveness to…
We introduce the notion of cyclic cohomology of an A-infinity algebra and show that the deformations of an A-infinity algebra which preserve an invariant inner product are classified by this cohomology. We use this result to construct some…