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David Mosquera-Lois

Publications and source records attributed to David Mosquera-Lois.

10 recordsLinked to original sources
arXiv2026

Whitehead's theorem for minimal finite models

We investigate the extent to which Whitehead's theorem remains valid for minimal finite models. We show that it fails in this setting, answering negatively a question posed by Barmak. More precisely, for every $n\geq 2$, we construct a weak homotopy equivalence between two $(2n+4)$-point minimal finite models of $S^n\vee S^{n-1}\vee S^{n-1}$ which are not homotopy equivalent. The minimality of these examples follows from a near-extremal classification theorem: if a connected finite space has at most $2n+3$ points and nonzero $n$th homology over a field, then its order complex is homotopy equivalent either to $S^n$ or to $S^n\vee S^k$ for some $1\leq k\leq n$. Finally, we prove a positive Whitehead-type result: under a natural cohomological rigidity hypothesis, every weak homotopy equivalence between minimal finite models is a homotopy equivalence.

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arXiv2026

Morse theory for loop-free categories

We extend discrete Morse-Bott theory to the setting of loop-free (or acyclic) categories. First of all, we state a homological version of Quillen's Theorem A in this context and introduce the notion of cellular categories. Second, we present a notion of vector field for loop-free categories. Third, we prove a homological collapsing theorem in the absence of critical objects in order to obtain the Morse inequalities. Examples are provided through the exposition. This answers partially a question by T. John: whether there is a Morse theory for loop-free (or acyclic) categories? [14].

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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.

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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.

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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 $f$.

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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.

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arXiv2025

A Combinatorial Study of the Fixed Point Index

We introduce a theory of integration with respect to the fixed point index, offering a substantial improvement over previous approaches based on the Lefschetz number. This framework eliminates several restrictive assumptions -- such as the need for definability, openness, or f-invariance of subspaces -- thereby allowing broader applicability. We also present a natural combinatorial adaptation of the fixed point index that extends the combinatorial Lefschetz number. This extension yields new topological and homotopical invariance results and facilitates the integration of real-valued functions with respect to fixed points.

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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.

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arXiv2023

Integration with respect to the Lefschetz number

We develop a theory of integration with respect to the Lefschetz number in the context of o-minimal structures containing the semilinear sets. We prove several results and we apply the theory to the field of object detection using sensors.

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