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Tashi Walde

Publications and source records attributed to Tashi Walde.

11 recordsLinked to original sources
arXiv2026

About the contractibility of the walking coinductive equivalence

We study the marked simplicial set obtained as the Roberts-Street nerve of the walking coinductive equivalence. We show that it is a non-contractible saturated complicial set for which all of its finite truncations are contractible. When regarding saturated complicial sets as a model for right $(\infty,\infty)$-categories, it represents a concrete and explicit example of a right $(\infty,\infty)$-category that is itself non-contractible, but whose reflection to a left $(\infty,\infty)$-category is contractible.

math.AT↗
arXiv2026

Cores and localizations of $(\infty,\infty)$-categories

We consider $(\infty,d)$-categories in the limit $d\to \infty$ via the core or localization functors that forget or invert higher non-invertible arrows, respectively. We compare the two resulting $(\infty,1)$-categories of $(\infty,\infty)$-categories and exhibit the localization-limit as a reflective localization of the core-limit. On the side, we study intermediate localizations that arise from notions of invertibility that only emerge at $d=\infty$ such as the one defined by coinduction.

math.AT↗
arXiv2024

Assembly of Constructible Factorization Algebras

We provide a toolbox of extension, gluing, and assembly techniques for factorization algebras. Using these tools, we fill various gaps in the literature on factorization algebras on stratified manifolds, the main one being that constructible factorization algebras form a sheaf of symmetric monoidal $\infty$-categories. Additionally, we explain how to assemble constructible factorization algebras from the data on the individual strata together with module structures associated to the relative links; thus answering a question by Ayala. Along the way, we give detailed proofs of the following facts which are also of independent interest: constructibility is a local condition; the $\infty$-category of disks is a localization of any sufficiently fine poset of disks; constructibility implies the Weiss condition on disks; constructible factorization algebras are algebras for the $\infty$-operad of embedded disks. For each of these, variants or special cases already existed, but they were either incomplete or not general enough.

math.AT↗
arXiv2024

Lax Additivity

We introduce notions of lax semiadditive and lax additive $(\infty,2)$-categories, categorifying the classical notions of semiadditive and additive 1-categories. To establish a well-behaved axiomatic framework, we develop a calculus of lax matrices and use it to prove that in locally cocomplete $(\infty,2)$-categories lax limits and lax colimits agree and are absolute. In the lax additive setting, we categorify fundamental constructions from homological algebra such as mapping complexes and mapping cones and establish their basic properties.

math.CT↗
arXiv2023

Complexes of stable $\infty$-categories

We study complexes of stable $\infty$-categories, referred to as categorical complexes. As we demonstrate, examples of such complexes arise in a variety of subjects including representation theory, algebraic geometry, symplectic geometry, and differential topology. One of the key techniques we introduce is a totalization construction for categorical cubes which is particularly well-behaved in the presence of Beck-Chevalley conditions. As a direct application we establish a categorical Koszul duality result which generalizes previously known derived Morita equivalences among higher Auslander algebras and puts them into a conceptual context. We explain how spherical categorical complexes can be interpreted as higher-dimensional perverse schobers, and introduce Calabi-Yau structures on categorical complexes to capture noncommutative orientation data. A variant of homological mirror symmetry for categorical complexes is proposed and verified for $\mathbb{C}\mathrm{P}^2$.

math.AG↗
arXiv2021

2-Segal spaces as invertible infinity-operads

We exhibit the simplex category $Δ$ and Segal's category $Γ$ as $\infty$-categorical localizations of the dendroidal categories $Ω_π$ and $Ω$ introduced by Moerdijk and Weiss. As an application we obtain an equivalence of $\infty$-categories between invertible $\infty$-operads and the $2$-Segal spaces of Dyckerhoff and Kapranov. Finally, we describe a cyclic version of the dendroidal category and explain how it $\infty$-localizes to Connes's cyclic category $Λ$.

math.AT↗
arXiv2019

Homotopy coherent theorems of Dold-Kan type

We establish a large class of homotopy coherent Morita-equivalences of Dold-Kan type relating diagrams with values in any weakly idempotent complete additive $\infty$-category; the guiding example is an $\infty$-categorical Dold-Kan correspondence between the $\infty$-categories of simplicial objects and connective coherent chain complexes. Our results generalize many known 1-categorical equivalences such as the classical Dold-Kan correspondence, Pirashvili's Dold-Kan type theorem for abelian $Γ$-groups and, more generally, the combinatorial categorical equivalences of Lack and Street.

math.RT↗
arXiv2019

Higher Segal spaces via higher excision

We show that the various higher Segal conditions of Dyckerhoff and Kapranov can all be characterized in purely categorical terms by higher excision conditions (in the spirit of Goodwillie-Weiss manifold calculus) on the simplex category $Δ$ and the cyclic category $Λ$.

math.AT↗
arXiv2018

Simplicial structures in higher Auslander-Reiten theory

We develop a novel combinatorial perspective on the higher Auslander algebras of type $\mathbb{A}$, a family of algebras arising in the context of Iyama's higher Auslander-Reiten theory. This approach reveals interesting simplicial structures hidden within the representation theory of these algebras and establishes direct connections to Eilenberg-MacLane spaces and higher-dimensional versions of Waldhausen's $\operatorname{S}_\bullet$-construction in algebraic $K$-theory. As an application of our techniques we provide a generalisation of the higher reflection functors of Iyama and Oppermann to representations with values in stable $\infty$-categories. The resulting combinatorial framework of slice mutation can be regarded as a higher-dimensional variant of the abstract representation theory of type $\mathbb{A}$ quivers developed by Groth and Šťov\'ıček. Our simplicial point of view then naturally leads to an interplay between slice mutation, horn filling conditions, and the higher Segal conditions of Dyckerhoff and Kapranov. In this context, we provide a classification of higher Segal objects with values in any abelian category or stable $\infty$-category.

math.RT↗
arXiv2017

Hall monoidal categories and categorical modules

We construct so called Hall monoidal categories (and Hall modules thereover) and exhibit them as a categorification of classical Hall and Hecke algebras (and certain modules thereover). The input of the (functorial!) construction are simplicial groupoids satisfying the $2$-Segal conditions (as introduced by Dyckerhoff and Kapranov), the main examples come from Waldhausen's S-construction. To treat the case of modules, we introduce a relative version of the $2$-Segal conditions. Furthermore, we generalize a classical result about the representation theory of symmetric groups to the case of wreath product groups: We construct a monoidal equivalence between the category of complex $G\wr S_n$-representations (for a fixed finite group $G$ and varying $n\in\mathbb N$) and the category of "$G$-equivariant" polynomial functors; we use this equivalence to prove a version of Schur-Weyl duality for wreath products.

math.CT↗