arXiv subjects

Adrián Doña Mateo

Publications and source records attributed to Adrián Doña Mateo.

3 recordsLinked to original sources

arXiv2026

Magnitude homology equivalence of Euclidean sets

Magnitude homology is an $\mathbf{R}^+$-graded homology theory of metric spaces that captures information on the complexity of geodesics. Here we address the question: when are two metric spaces magnitude homology equivalent, in the sense that there exist back-and-forth maps inducing mutually inverse maps in homology? We give a concrete geometric necessary and sufficient condition in the case of closed Euclidean sets. Along the way, we introduce the convex-geometric concepts of inner boundary and core, and prove a strengthening for closed convex sets of the classical theorem of Carathéodory.

math.MG↗

arXiv2025

Cauchy density

In the paper where he defined the Cauchy completion of a $\mathscr{V}$-category, Lawvere also defined a condition on a $\mathscr{V}$-functor which made it analogous to a map of metric spaces whose image is topologically dense in its codomain. We call this condition Cauchy density. In this note, we focus on the fully faithful Cauchy dense $\mathscr{V}$-functors, and show that the Cauchy completion of $\mathscr{A}$ is the largest $\mathscr{V}$-category that admits a fully faithful Cauchy dense $\mathscr{V}$-functor from $\mathscr{A}$. Moreover, we show that $F \colon \mathscr{A} \to \mathscr{B}$ is fully faithful and Cauchy dense iff $[F,\mathscr{C}] \colon [\mathscr{B},\mathscr{C}] \to [\mathscr{A},\mathscr{C}]$ is an equivalence for any Cauchy complete $\mathscr{C}$. Finally, we provide examples and characterisations of Cauchy dense functors in various contexts.

math.CT↗

arXiv2025

Pushforward monads

Given a monad $T$ on $\mathscr{A}$ and a functor $G \colon \mathscr{A} \to \mathscr{B}$, one can construct a monad $G_\#T$ on $\mathscr{B}$ subject to the existence of a certain Kan extension; this is the pushforward of $T$ along $G$. We develop the general theory of this construction in a $2$-category, giving two universal properties it satisfies. In the case of monads in $\mathsf{CAT}$, this gives, among other things, two adjunctions between categories of monads on $\mathscr{A}$ and $\mathscr{B}$. We conclude by computing the pushforward of several familiar monads on the category of finite sets along the inclusion $\mathsf{FinSet} \hookrightarrow \mathsf{FinSet}$, which produces the monad for continuous lattices, among others. We also show that, with two trivial exceptions, these pushforwards never have rank.

math.CT↗