Related papers: Spaces of directed paths on pre-cubical sets II

We construct an algebraic weak factorization system $(L, R)$ on the cartesian cubical sets, in which the canonical path object factorization $A \to A^I \to A\times A$ induced by the 1-cube $I$ is an $L$-$R$ factorization for any $R$-object…

For a finite group $G$, we define an equivariant cobordism category $\mathcal{C}_d^G$. Objects of the category are $(d-1)$-dimensional closed smooth $G$-manifolds and morphisms are smooth $d$-dimensional equivariant cobordisms. We identify…

We construct a many-object dual version of Chen's iterated integral map. For any topological space X, the construction takes the form of an A-infinity functor between two dg categories whose objects are the points of X: the domain has as…

We prove affirmatively the conjecture raised by J. Mostovoy; the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in $\mathbb{R}^3$. We make use of techniques…

In this paper we study the topology of three different kinds of spaces associated to polynomial knots of degree at most $d$, for $d\geq2$. We denote these spaces by $\mathcal{O}_d$, $\mathcal{P}_d$ and $\mathcal{Q}_d$. For $d\geq3$, we show…

One of the primary methods of studying the topology of configurations of points in a graph and configurations of disks in a planar region has been to examine discrete combinatorial models arising from the underlying spaces. Despite the…

Let $I$ be a small category with finite dimensional nerve, and $X\colon I\to Cat$ a diagram of small categories. We show that, under a "Reedy quasi-fibrancy condition", the homotopy limit of the geometric realization of $X$ is itself the…

A semantics of concurrent programs can be given using precubical sets, in order to study (higher) commutations between the actions, thus encoding the "geometry" of the space of possible executions of the program. Here, we study the…

This paper gives a uniform-theoretic refinement of classical homotopy theory. Both cubical sets (with connections) and uniform spaces admit classes of weak equivalences, special cases of classical weak equivalences, appropriate for the…

We construct a topological space $\mathcal{B}$ consisting of translation invariant injective matrix product states (MPS) of all physical and bond dimensions and show that it has the weak homotopy type $K(\mathbb{Z}, 2) \times K(\mathbb{Z},…

Let $V$ be a finite-dimensional symplectic vector space over a field of characteristic 0, and let $G \subset Sp(V)$ be a finite subgroup. We prove that for any crepant resolution $X \to V/G$, the bounded derived category $D^b(Coh(X))$ of…

Gerbes are locally connected presheaves of groupoids. They are classified up to local weak equivalence by path components in a 2-cocycle category taking values in all sheaves of groups, their isomorphisms and homotopies. If F is a full…

We introduce and show preliminary results of a fast randomized method that finds a set of K paths lying in distinct homotopy classes. We frame the path planning task as a graph search problem, where the navigation graph is based on a…

We show that discrete and classical homotopy theories are equivalent after localizing at n-equivalences for any non-negative integer n. By constructing an explicit homotopy inverse to the graph nerve functor associating an n-fibrant cubical…

We prove that a weak equivalence between two cofibrant (colored) props in chain complexes induces a Dwyer-Kan equivalence between the simplicial localizations of the associated categories of algebras. This homotopy invariance under base…

For a closed topological $n$--manifold $K$ and a map $p:K\to B$ inducing an isomorphism $\pi_1(K)\to\pi_1(B)$, there is a canonicaly defined morphism $b:H_{n+1}(B,K,\mathbb{L})\to \mathbb{S} (K)$, where $\mathbb{L}$ is the periodic…

Bousfield and Kan's $\mathbb{Q}$-completion and fiberwise $\mathbb{Q}$-completion of spaces lead to two different approaches to the rational homotopy theory of non-simply connected spaces. In the first approach, a map is a weak equivalence…

We prove that the loop space of the directed suspension of a directed space is homotopy equivalent to the James construction. In particular, it does not depend on the directed structure of a given directed space.

In this article, we construct a cofibrantly generated model structure on the category of spaces stratified over a fixed poset, and show that it is Quillen-equivalent to a category of diagrams of simplicial sets. Then, considering all those…

Pseudotopological spaces are the Cartesian closed hull of the category of \v{C}ech closure spaces. In this paper, we give a direct proof that the model category of the pseudotopological spaces constructed by Rieser is Quillen equivalent to…