Related papers: The branching nerve of HDA and the Kan condition
We introduce a new simplicial nerve of higher dimensional automata whose homology groups yield a new definition of the globular homology. With this new definition, the drawbacks noticed with the construction of math.CT/9902151 disappear.…
The most natural notion of a simplicial nerve for a (weak) bicategory was given by Duskin, who showed that a simplicial set is isomorphic to the nerve of a $(2,1)$-category (i.e. a bicategory with invertible $2$-morphisms) if and only if it…
We explore the combinatorial properties of the branching areas of execution paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner (or branching) homology of a globular…
We show that the Street nerve of a strict $\omega$-category $C$ is a Kan complex (respectively a quasi-category) if and only if the $n$-cells of $C$ for $n\geq 1$ (respectively $n> 1$) are weakly invertible. Moreover, we equip…
This paper continues the development of a simplicial theory of weak omega-categories, by studying categories which are enriched in weak complicial sets. These complicial Gray-categories generalise both the Kan complex enriched categories of…
In this dissertation, we compare the "classical" homology of an $\omega$-category (defined as the homology of its Street nerve) with its polygraphic homology. More precisely, we prove that both homologies generally do not coincide and call…
Given a simplicial complex and a collection of subcomplexes covering it, the nerve theorem, a fundamental tool in topological combinatorics, guarantees a certain connectivity of the simplicial complex when connectivity conditions on the…
The notion of geometric nerve of a 2-category (Street, \cite{refstreet}) provides a full and faithful functor if regarded as defined on the category of 2-categories and lax 2-functors. Furthermore, lax 2-natural transformations between lax…
We show that the nerve of a strict omega-category can be described algebraically as a simplicial set with additional operations subject to certain identities. The resulting structures are called sets with complicial identities. We also…
There are infinitely many variants of the notion of Kan fibration that, together with suitable choices of cofibrations and the usual notion of weak equivalence of simplicial sets, satisfy Quillen's axioms for a homotopy model category. The…
It is shown that the cubical nerve of a strict omega-category is a sequence of sets with cubical face operations and distinguished subclasses of thin elements satisfying certain thin filler conditions. It is also shown that a sequence of…
In this note we show that a particular homological nerve theorem, which was originally proved for a finite cover of a simplicial complex by subcomplexes, also holds for an open cover of an arbitrary topological space. The motivation for…
We show that a regular cover of a general topological space provides structure similar to a triangulation. In this general setting we define analogues of simplicial maps and prove their existence and uniqueness up to homotopy. As an…
The general goal of this paper is to gather and review several methods from homotopy and combinatorial topology and formal concepts analysis (FCA) and analyze their connections. FCA appears naturally in the problem of combinatorial…
The subject of this paper is a nerve construction for bicategories introduced by Leinster, which defines a fully faithful functor from the category of bicategories and normal pseudofunctors to the category of presheaves over Joyal's…
The Barratt nerve, denoted $B$, is the endofunctor that takes a simplicial set to the nerve of the poset of its non-degenerate simplices. The ordered simplicial complex $BSd\, X$, namely the Barratt nerve of the Kan subdivision $Sd\, X$, is…
We discuss various concepts of $\infty$-homotopies, as well as the relations between them (focussing on the Leibniz type). In particular $\infty$-$n$-homotopies appear as the $n$-simplices of the nerve of a complete Lie ${\infty}$-algebra.…
We describe a Cat-valued nerve of bicategories, which associates to every bicategory a simplicial object in Cat, called the 2-nerve. We define a 2-category NHom whose objects are bicategories and whose 1-cells are normal homomorphisms of…
We construct a nerve from double categories into double $(\infty,1)$-categories and show that it gives a right Quillen and homotopically fully faithful functor between the model structure for weakly horizontally invariant double categories…
Many supervised machine learning methods are naturally cast as optimization problems. For prediction models which are linear in their parameters, this often leads to convex problems for which many mathematical guarantees exist. Models which…