arXiv subjects

Luigi Caputi

Publications and source records attributed to Luigi Caputi.

24 records · Page 2Linked to original sources

arXiv2022

Multipath cohomology of directed graphs

This work is part of a series of papers focusing on multipath cohomology of directed graphs. Multipath cohomology is defined as the (poset) homology of the path poset -- i.e., the poset of disjoint simple paths in a graph -- with respect to a certain functor. This construction is essentially equivalent, albeit more computable, to taking the higher limits of said functor on (a certain modification of) the path poset. We investigate the functorial properties of multipath cohomology. We provide a number of sample computations, show that the multipath cohomology does not vanish on trees, and that, when evaluated at the coherently oriented polygon, it recovers Hochschild homology. Finally, we use the same techniques employed to study the functoriality to investigate the connection with the chromatic homology of (undirected) graphs introduced by L. Helme-Guizon and Y. Rong.

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arXiv2022

Hochschild homology, and a persistent approach via connectivity digraphs

We introduce a persistent Hochschild homology framework for directed graphs. Hochschild homology groups of (path algebras of) directed graphs vanish in degree $i\geq 2$. To extend them to higher degrees, we introduce the notion of connectivity digraphs and analyse two main examples; the first, arising from Atkin's $q$-connectivity, and the second, here called $n$-path digraphs, generalising the classical notion of line graphs. Based on a categorical setting for persistent homology, we propose a stable pipeline for computing persistent Hochschild homology groups. This pipeline is also amenable to other homology theories; for this reason, we complement our work with a survey on homology theories of digraphs.

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arXiv2022

Monotone cohomologies and oriented matchings

In this paper, we extend the definition of cohomology associated to monotone graph properties, to encompass twisted functor coefficients. We introduce oriented matchings on graphs, and focus on their (twisted) cohomology groups. We characterise oriented matchings in terms of induced free-flow pseudoforests, and explicitly determine the homotopy type of the associated simplicial complexes. Furthermore, we provide a connection between the cohomology of oriented matchings with certain functor coefficients, and the recently defined multipath cohomology. Finally, we define a further oriented homology for graphs and interpret it as a count of free-flow orientations.

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arXiv2022

Categorifying connected domination via graph überhomology

Überhomology is a recently defined homology theory for simplicial complexes, which yields subtle information on graphs. We prove that bold homology, a certain specialisation of überhomology, is related to dominating sets in graphs. To this end, we interpret überhomology as a poset homology, and investigate its functoriality properties. We then show that the Euler characteristic of the bold homology of a graph coincides with an evaluation of its connected domination polynomial. Even more, the bold chain complex retracts onto a complex generated by connected dominating sets. We conclude with several computations of this homology on families of graphs; these include a vanishing result for trees, and a characterisation result for complete graphs.

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arXiv2020

Cyclic homology for bornological coarse spaces

We define Hochschild and cyclic homologies for bornological coarse spaces: for a fixed field $k$ and group $G$, these are lax symmetric monoidal functors $\mathcal{X}HH_{k}^G$ and $\mathcal{X}HC_{k}^G$ from the category of equivariant bornological coarse spaces $G\mathbf{BornCoarse}$ to the cocomplete stable $\infty$-category of chain complexes $\mathbf{Ch}_\infty$. We relate these equivariant coarse homology theories to coarse algebraic $K$-theory $\mathcal{X} K^G_{k}$ and to coarse ordinary homology $\mathcal{X} H^G$ by constructing a trace-like natural transformation $\mathcal{X} K_{k}^G\to \mathcal{X} H^G$ that factors through coarse Hochschild (or cyclic) homology. We further compare the forget-control map for coarse Hochschild homology with the associated generalized assembly map.

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arXiv2019

Controlled objects as a symmetric monoidal functor

The goal of this paper is to associate functorially to every symmetric monoidal additive category $\mathbf{A}$ with a strict $G$-action a lax symmetric monoidal functor $\mathbf{V}_{\mathbf{A}}^{G}:G\mathbf{BornCoarse}\to \mathbf{Add}_{\infty}$ from the symmetric monoidal category of $G$-bornological coarse spaces $G\mathbf{BornCoarse}$ to the symmetric monoidal $\infty$-category of additive categories $ \mathbf{Add}_{\infty}$. This allows to refine equivariant coarse algebraic $K$-homology to a lax symmetric monoidal functor.

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