Transfinite mutations in the completed infinity-gon

We introduce mutation along infinite admissible sequences for infinitely marked surfaces, that is surfaces with infinitely many marked points on the boundary. We show that mutation along such admissible sequences produces a preorder on the set of triangulations of a fixed infinitely marked surface. We provide a complete classification of the strong mutation equivalence classes of triangulations of the infinity-gon and the completed infinity-gon respectively, where strong mutation equivalence is the equivalence relation induced by this preorder. Finally, we introduce the notion of transfinite mutations in the completed infinity-gon and show that all its triangulations are transfinitely mutation equivalent, that is we can reach any triangulation of the completed infinity-gon from any other triangulation via a transfinite mutation

The Grothendieck Groups of Discrete Cluster Categories of Dynkin Type A∞A_{\infty}

In this work we compute the triangulated Grothendieck groups for each of the family of discrete cluster categories of Dynkin type $A_{\infty}$ as introduced by Holm-Jorgensen. Subsequently, we also compute the Grothendieck group of a completion of these discrete cluster categories in the sense of Paquette-Yildirim.Comment: V2: 18 pages, updated some figures, restated Theorem 4.4, comments welcome

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.Comment: 30 pages, 18 figures. For the special JAA issue on Cluster Algebras and Related Topic

Continuous quivers of type A (I) Foundations

We generalize type $A$ quivers to continuous type $A$ quivers and prove initial results about pointwise finite-dimensional (pwf) representations. We classify the indecomosable pwf representations and provide a decomposition theorem, recovering results of Botnan and Crawley-Boevey. We also classify the indecomposable pwf projective representations. Finally, we prove that many of the properties of finite-dimensional type $A_n$ representations are present in finitely generated pwf representations. This is the self-contained foundational part of a series of works to study a generalization of continuous clusters categories and their relationship to other type $A$ cluster structures.Comment: 29 pages. Revised to be a self-contained treatment with a focus on representation theoretic method

Higher Segal spaces I

This is the first paper in a series on new higher categorical structures called higher Segal spaces. For every d > 0, we introduce the notion of a d-Segal space which is a simplicial space satisfying locality conditions related to triangulations of cyclic polytopes of dimension d. In the case d=1, we recover Rezk's theory of Segal spaces. The present paper focuses on 2-Segal spaces. The starting point of the theory is the observation that Hall algebras, as previously studied, are only the shadow of a much richer structure governed by a system of higher coherences captured in the datum of a 2-Segal space. This 2-Segal space is given by Waldhausen's S-construction, a simplicial space familiar in algebraic K-theory. Other examples of 2-Segal spaces arise naturally in classical topics such as Hecke algebras, cyclic bar constructions, configuration spaces of flags, solutions of the pentagon equation, and mapping class groups.Comment: 221 page

From Grothendieck groups to generators: the discrete cluster categories of type A∞

In this thesis we look at two closely related families of categories: the discrete cluster categories of Dynkin type A∞, and their completions in the sense of Paquette and Yıldırım. We compute the triangulated Grothendieck group of the discrete cluster categories of Dynkin type A∞, as well as their Paquette-Yıldırım completions. Further, we provide a counterexample to a theorem by Palu and provide a corrected statement of the result. We also introduce the concept of homologically connected objects, and show that any object in the Paquette-Yıldırım completion of a discrete cluster category of Dynkin type A∞ can be decomposed into homologically connected direct summands, and that the smallest thick subcategory containing an object is determined by its decomposition into homologically connected direct summands. This allows us to classify the classical generators of the Paquette-Yıldırım completions of the discrete cluster categories of Dynkin type A∞, and associate an integer to each classical generator that is an upper bound on their generation time. This allows us to compute an upper bound for the Orlov spectrum, and to compute the Rouquier dimension of the Paquette-Yıldırım completions. Further, we compute the graded endomorphism ring of a chosen classical generator as a Z graded, upper triangular matrix ring with polynomial rings and Laurent polynomial rings as entries