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Enrique Macías-Virgós

Publications and source records attributed to Enrique Macías-Virgós.

12 recordsLinked to original sources
arXiv2026

Weak and Strong Fibrations of Functors

We develop a homotopical framework for small categories that extends classical invarints of algebraic topology to the categorical setting. Our approach is based on the construction of genuine path category, obtained trough a localization procedure, which allows us to define strong and weak fibrations for functor. We establish their basic properties, introduce a fibrant replacement for functors, and extend homotopical invariants such as the Svarc genus and sectional category to small categories. Finally, we apply this framework to motion planning in small categories, providing categorical analogues of Farber's topological complexity while removing finiteness constraints typical of existing approaches.

math.CT↗
arXiv2026

Baues-Wirsching Cohomology and Svarc Genus in Small Categories

We prove that for a bifibration P between small categories, the lenght of the cup product in the kernel of the induced morphism in the Baues-Wirsching cohomology with coefficients in any natural system is a lower bound for the homotopic sectional category (also called Svarc genus). Our results extend classical Svarc type inequalties to the categorical setting and introduce a computationally efficient method via a reduced cochain complex for Baues-Wirching cohomology.

math.CT↗
arXiv2026

Morse-Bott inequalities for endomorphisms

Let $K$ be a finite simplicial complex, let $g\colon K\to K$ be a simplicial map and let $f$ be a discrete Morse-Bott function on $K$ satisfying $f(g(σ))\leq f(σ)$ for all simplices $σ$ in $K$. We establish a set of inequalities (generalizing the Morse-Bott inequalities which we recover as a particular case when $g$ is the identity) relating the dynamics of $g$ and $f$.

math.AT↗
arXiv2025

A Computational (Co)homological Approach to Contiguity Distance

We introduce two new algebraic invariants, the (co)homological distances between continuous maps, which provide computable lower bounds for the homotopic distance and strictly refine the classical cup-length estimates. We then define the simplicial cohomological distance between simplicial maps and prove a convergence theorem showing that, after sufficiently many barycentric subdivisions, it recovers the cohomological distance between the corresponding continuous maps. Several explicit computations are presented to illustrate the effectiveness of the proposed approach.

math.AT↗
arXiv2025

Subspace {L}usternik-{S}chnirelmann category of quasi-projective quaternionic spaces

Let $Q_n$ be the quasi-projective subspace of the symplectic group $\mathrm{Sp}(n)$. In this short note, we prove that the subspace Lusternik-Schnirelmann category of $Q_n$ in $\mathrm{Sp}(n)$ is 2. For that, we use a quaternionic logarithm, as Singhof did in the complex case for the determination of the Lusternik-Schnirelmann category of the unitary group. Our result generalizes the known case $n=2$ (by L. Fernández-Suárez, A. Gómez-Tato and D. Tanré) and has to be compared to the equality $\mathrm{cat}\,Q_{3}=3$, established by N. Iwase and T. Miyauchi.

math.AT↗
arXiv2024

Relative De Rham cohomology in diffeological spaces

We define two different versions of the relative De Rham cohomology groups of a diffeological space. Additionally, we study a variant of the Mayer-Vietoris sequence and discuss the existence of a relative cup product. Our approach is categorical, replacing open coverings with generating families and unions and intersections with pull-back and join constructions.

math.AT↗
arXiv2024

Lefschetz fixed-object and fixed-morphism theorems for acyclic categories

We introduce two novel complementary notions of the Lefschetz number for a functor from a finite acyclic category to itself and we prove a Lefschetz fixed-object theorem and a Lefschetz fixed-morphism theorem. In order to do so, we use the connection between these type of categories and simplicial structures, such as trisps or delta complexes. Through the use of a pair of functors that, when composed, form the barycentric subdivision, we are not only able to identify fixed objects but also fixed chains of morphisms.

math.AT↗
arXiv2020

Non-linear Morse-Bott functions on quaternionic Stiefel manifolds

In the Stiefel manifold $X_{n,k}$, we replace Frankel linear height function by a quadratic one. We prove this is still a Morse-Bott function, whose structure of critical levels presents a dichotomy according to the sign of $n-2k$. The critical submanifolds are no longer Grassmannians but total spaces of fibrations of basis a product of two Grassmannians. We explicitly integrate the gradient flow.

math.DG↗
arXiv2016

Strong discrete Morse theory and simplicial L-S category: A discrete version of the Lusternik-Schnirelmann Theorem

We prove a discrete version of the Lusternik-Schnirelmann theorem for discrete Morse functions and the recently introduced simplicial Lusternik-Schnirelmann category of a simplicial complex. To accomplish this, a new notion of critical object of a discrete Morse function is presented, which generalizes the usual concept of critical simplex (in the sense of R. Forman). We show that the non-existence of such critical objects guarantees the strong homotopy equivalence (in the Barmak and Minian's sense) between the corresponding sublevel complexes. Finally, we establish that the number of critical objects of a discrete Morse function defined on $K$ is an upper bound for the non-normalized simplicial Lusternik-Schnirelmann category of $K$.

math.AT↗
arXiv2016

Simplicial Lusternik-Schnirelmann category

The simplicial LS-category of a finite abstract simplicial complex is a new invariant of the strong homotopy type, defined in purely combinatorial terms, that generalizes to arbitrary simplicial complexes the well known notion of arboricity of a graph, and that allows to develop all the machinery of algebraic topology which is costumary in the classical theory of Lusternik-Schnirelmann category.

math.AT↗