arXiv subjects

Job Daisie Rock

Publications and source records attributed to Job Daisie Rock.

6 recordsLinked to original sources

arXiv2026

Representations of infinite species

We consider species, consisting of a possibly infinite set of rings, and bimodules between them. Simson realised the category of representations as a functor category, which we prove is hereditary when each of the rings is semisimple. We use purity to provide sufficient conditions, in order for a representation to decompose into indecomposables with local endomorphism rings. For any bifunctor valued in bimodules, we functorially construct species equipped with commutativity conditions. This generates examples coming from a range of topics, such as subobject lattices in abelian length categories, the field choice problem in persistent homology, and topological field theories with defects.

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arXiv2025

Preprojective categories of type A

We introduce a continuous version of preprojective algebras of type $A$. In particular, we are interested in the preprojective category over an open, bounded subinterval $\mathbb{I}$ of $\mathbb{R}$, denoted $Λ_{\mathbb{I}}$. We study the representable projective modules and define a useful type of sub- and quotient module called decorous modules. These are completely described by a function from the closure $\overline{\mathbb{I}}$ of $\mathbb{I}$ to $\mathbb{R}$ whose 'slopes' are not too steep anywhere. We later use these to describe permuton ideals, a generalization of the support $τ$-tilting ideals of preprojective algebras of type $A_n$, which we call permutation ideals. Once we have our generalization, we show that permutation ideals can be recovered from permuton ideals. Moreover, permutation ideals are $τ$-rigid and we show an analogous property for our permuton ideals. Along the way, we classify all the brick $Λ_{\mathbb{I}}$-modules.

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arXiv2025

Categories of generalized thread quivers

We study the representation category of thread quivers and their quotients. A thread quiver is a quiver in which some arrows have been replaced by totally ordered sets. Pointwise finite-dimensional (pwf) representations of such a thread quiver admit a Krull-Remak-Schmidt-Azumaya decomposition. We show that an indecomposable representation is induced from an indecomposable representation of a quiver obtained from the original quiver by replacing some of its arrows by a finite linear $\mathbb{A}_n$ quiver. We study injective and projective pwf indecomposable representations and we fully classify them when the quiver satisfies a mild condition. We give a characterization of the indecomposable pwf representations of certain categories whose representation theory has similar properties to finite type or tame type. We further construct new hereditary abelian categories, including a Serre subcategory of pwf representations that includes every indecomposable representation.

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arXiv2023

Continuous Stability Conditions of Type A and Measured Laminations of the Hyperbolic Plane

We introduce stability conditions (in the sense of King) for representable modules of continuous quivers of type A along with a special criteria called the four point condition. The stability conditions are defined using a generalization of delta functions, called half-delta functions. We show that for a continuous quiver of type A with finitely many sinks and sources, the stability conditions satisfying the four point condition are in bijection with measured laminations of the hyperbolic plane. Along the way, we extend an earlier result by the first author and Todorov regarding continuous cluster categories for linear continuous quivers of type A and laminations of the hyperbolic plane to all continuous quivers of type A with finitely many sinks and sources. We also give a formula for the continuous cluster character.

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arXiv2022

Progress on Infinite Cluster Categories Related to Triangulations of the (Punctured) Disk

In this mostly expository paper, we present recent progress on infinite (weak) cluster categories that are related to triangulations of the disk, with and without a puncture. First we recall the notion of a cluster category. Then we move to the infinite setting and survey recent work on infinite cluster categories of types $\mathbb{A}$ and $\mathbb{D}$. We conclude with our contributions, two infinite families of infinite (weak) cluster categories of type $\mathbb{D}$. We first present a discrete, infinite version of Schiffler's combinatorial model of the punctured disk with marked points. We then produce each (weak) cluster category starting with representations of thread quivers, taking the derived category, and then taking the appropriate orbit category. We show that the combinatorics in the (weak) cluster categories match with the corresponding combinatorics of the punctured disk with countably-many marked points. We also state two conjectures concerning weak cluster structures inside our (weak) cluster categories.

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arXiv2021

Cluster Theories and Cluster Structures of Type A

In the present paper we examine the relationship between several type $A$ cluster theories and structures. We define a 2D geometric model of a cluster theory, which generalizes cluster algebras from surfaces, and encode several existing type $A$ cluster theories into a 2D geometric model. We review two other cluster theories of type $A$. Then we introduce an abstraction of cluster structures. We prove two results: the first relates several existing type $A$ cluster theories and the second relates some of these cluster structures using the new abstraction.

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