Tootfinder

Descend of morphisms of varieties
Supravat Sarkar
https://arxiv.org/abs/2510.11993 https://arxiv.org/pdf/2510.11993…

Descend of morphisms of varieties
Given varieties $X, Y, W$ and dominant morphisms $ϕ:X\to Y$ and $f:X\to W$ such that $f$ is constant on fibres of $ϕ$ , we give sufficient conditions to guarantee that $f$ descends to a rational map or a morphism $Y\to W.$ We pay special attention to the case that the ground field has positive characteristic. This extends previous works of Aichinger and Das, who proved similar results for some classes of affine varieties.

Hall Skew-morphisms and Hall Cayley maps of finite groups
Wendi Di, Zheng Guo, Cai Heng Li
https://arxiv.org/abs/2510.10916 https://arxiv.org/pdf/2510.1091…

Hall Skew-morphisms and Hall Cayley maps of finite groups
A characterization is given of finite groups $H$ that have skew-morphisms of order coprime to the order $|H|$, and their skew-morphisms. A complete classification is then given of the automorphism groups and the underlying graphs of vertex-rotary core-free Hall Cayley maps.

Quantaloid-enriched categories: Factorization, weak classifiers, and symmetry
Javier Guti\'errez Garc\'ia, Ulrich H\"ohle
https://arxiv.org/abs/2510.10162 https://

Quantaloid-enriched categories: Factorization, weak classifiers, and symmetry
This paper provides a comprehensive overview of some of the foundational properties of categories enriched over quantaloids, along with several new results. We demonstrate that the category whose objects are quantaloid-enriched categories and whose morphisms are left adjoint distributors admits an (epi, extremal mono)--factorization system. Furthermore, we prove that the category of cocomplete quantaloid-enriched categories satisfies the weak subobject classifier axiom, under stability conditio…

Stranding $\mathfrak{sl}_n$ webs
Heather M. Russell, Julianna Tymoczko
https://arxiv.org/abs/2510.12035 https://arxiv.org/pdf/2510.12035

Stranding $\mathfrak{sl}_n$ webs
Webs are a kind of planar, directed, edge-labeled graph that encode invariant vectors for quantum representations of $\mathfrak{sl}_n$. The theory of webs developed organically for $\mathfrak{sl}_2$, where they are also known as noncrossing matchings and the Temperley-Lieb algebra, before being formalized by Kuperberg for $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ as the morphisms in a diagrammatic categorification of quantum representations called the spider category. Various models extend webs t…

Linear sections of Grassmannians and resonance of vector bundles
Marian Aprodu, C\u{a}lin Spiridon
https://arxiv.org/abs/2510.09195 https://arxiv.org/pdf/2…

Linear sections of Grassmannians and resonance of vector bundles
This work revolves around the question of whether a given resonance variety is associated with a vector bundle. We show the existence of a family of natural morphisms on a stratification of the resonance variety to a suitable family of a Quot scheme and provide some applications in the curve case. The existence of this family of morphisms represents an obstruction to affirmatively answering the main question. In addition, we study the resonance of restricted universal rank-two quotient bundles …

Denominators of R-matrices, higher Dorey's rules and a generalization of T-systems for quantum affine algebras
Se-jin Oh, Travis Scrimshaw
https://arxiv.org/abs/2510.10874 h…

Denominators of R-matrices, higher Dorey's rules and a generalization of T-systems for quantum affine algebras
We construct a higher level analogue of Dorey's rule, which describe certain surjective morphisms between Kirillov--Reshetikhin (KR) modules over quantum affine algebras. Building on this, we establish a generalized T-system of short exact sequences and prove the denominator formula between KR modules in all nonexceptional types, except with only mild ambiguities persisting in type $C_n^{(1)}$. As a consequence, we can completely classify when a tensor product of KR modules is simple. These res…

Abstract String Domain Defined with Word Equations as a Reduced Product (Extended Version)
Antonina Nepeivoda, Ilya Afanasyev
https://arxiv.org/abs/2510.11007 https://

Abstract String Domain Defined with Word Equations as a Reduced Product (Extended Version)
We introduce a string-interval abstract domain, where string intervals are characterized by systems of word equations (encoding lower bounds on string values) and word disequalities (encoding upper bounds). Building upon the lattice structure of string intervals, we define an abstract string object as a reduced product on a string property semilattice, determined by length-non-increasing morphisms. We consider several reduction strategies for abstract string objects and show that upon these str…

Comparing dg category models for path spaces via $A_\infty$-functors
Manuel Rivera, Yi Wang
https://arxiv.org/abs/2510.01080 https://arxiv.org/pdf/2510.010…

Comparing dg category models for path spaces via $A_\infty$-functors
We construct a many-object dual version of Chen's iterated integral map. For any topological space X, the construction takes the form of an A-infinity functor between two dg categories whose objects are the points of X: the domain has as morphisms the singular (cubical) chains on the space of (Moore) paths in X and the codomain has morphisms arising by totalizing a cosimplicial chain complex determined by the dg coalgebra of singular (simplicial) chains in X. When X is simply connected, we show…

Instances of models of double-categorical theories
Kevin Carlson, Evan Patterson
https://arxiv.org/abs/2510.08861 https://arxiv.org/pdf/2510.08861

Instances of models of double-categorical theories
We contribute a chapter in common to categorical database theory and to the study of higher morphisms between double categories. The common thread here is the notion of instance, or right module, which we generalize from functors from a plain category into Set to the models of a (cartesian) double theory. This provides a concept of instance for such objects as a category equipped with a monad, or a (symmetric) multicategory, recovering the multifunctors into Set in the latter case. We also show…

Hofer-continuous quasi-morphisms on Liouville manifolds
Frol Zapolsky
https://arxiv.org/abs/2509.23277 https://arxiv.org/pdf/2509.23277

Hofer-continuous quasi-morphisms on Liouville manifolds
We show that, given a complete Liouville manifold, any homogeneous quasi-morphism on its Hamiltonian group, which satisfies a strengthened version of Hofer continuity called stability, must vanish. This partially addresses a conjecture due to L. Polterovich.

Braden-MacPherson sheaves on alcoves
Noriyuki Abe
https://arxiv.org/abs/2510.11093 https://arxiv.org/pdf/2510.11093…

Braden-MacPherson sheaves on alcoves
We study Braden-MacPherson sheaves on the moment graph associated to the set of of alcoves. We define an action of Soergel bimodules on the category of Braden-MacPherson sheaves. We also prove a certain stability of morphisms between Braden-MacPherson sheaves.

Saturated base change of toroidal morphisms
Santai Qu
https://arxiv.org/abs/2509.15590 https://arxiv.org/pdf/2509.15590

Saturated base change of toroidal morphisms
We show that saturated base change of a dominant toroidal morphism is also toroidal. For completeness, we give full details on equivalence between definitions regarding toroidal embeddings and toroidal morphisms in literature. Moreover, we show in detail the equivalence between toroidal embeddings and logarithmically regular log-varieties, respectively, toroidal morphisms and logarithmically smooth morphisms.

Adhesive category theory for graph rewriting in Rocq
Samuel Arsac (PLUME, LIP, ENS de Lyon), Russ Harmer (PLUME, LIP), Damien Pous (PLUME, LIP)
https://arxiv.org/abs/2509.17392 …

Adhesive category theory for graph rewriting in Rocq
We design a Rocq library about adhesive categories, using Hierarchy Builder (HB). It is built around two hierarchies. The first is for categories, with usual categories at the bottom and adhesive categories at the top, with weaker variants of adhesive categories in between. The second is for morphisms (notably isomorphisms, monomorphisms and regular monomorphisms). Each level of these hierarchies is equipped with several interfaces to define instances. We cover basic categorical concepts such a…

Concentration structures on categories and horizontal categorification
Yangxiao Luo, Shunyu Wan
https://arxiv.org/abs/2510.07553 https://arxiv.org/pdf/2510…

Concentration structures on categories and horizontal categorification
We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration structure we can functorially construct a $\textit{concentration monoid}$, which can be used to give a precise definition of horizontal categorification and decategorification. Moreover, by studying concentration structures on fundamental groupoids, we show that ev…

Coassociative structures on self-injective algebras
Alexandru Chirvasitu
https://arxiv.org/abs/2509.21435 https://arxiv.org/pdf/2509.21435

Coassociative structures on self-injective algebras
For general finite-dimensional self-injective algebra $A$ we construct a family of injective coassociative coproducts $A\to A\otimes A$, all $A$-bimodule morphisms. In particular such structures always exist, confirming a conjecture of Hernandez, Walton and Yadav. The coproducts are indexed by subsets of $\{1,\cdots,m(i)\}\times \{1,\cdots,m(ν^{-1}i)\}$, where $A\cong \mathrm{End}_Λ(M)$ is the general form of a self-injective algebra in terms of a basic Frobenius $Λ$, the $m(i)$, $1\le i\le …

The Theory of Topo-Symmetric Extensions of Topological Groups
Es-said En-naoui
https://arxiv.org/abs/2510.00018 https://arxiv.org/pdf/2510.00018

The Theory of Topo-Symmetric Extensions of Topological Groups
We introduce the notion of \emph{topo-symmetric extensions} of topological groups, a new generalization of classical group extensions that incorporates both topological and symmetry constraints. We define morphisms between such extensions, construct the associated groupoid, and develop classification results in terms of adapted cohomology. Several new invariants are introduced, including dimension, stabilizer, and density invariants, which characterize the fine structure of these extensions. Ap…

Zigzags and free adjunctions
Lorenzo Riva, Martina Rovelli
https://arxiv.org/abs/2510.05371 https://arxiv.org/pdf/2510.05371…

Zigzags and free adjunctions
We construct an explicit combinatorial model of the functor which adds right adjoints to the morphisms of an $\infty$-category, and we speculate on possible extensions to higher dimensions.

The condensed homotopy type of a scheme
Peter J. Haine, Tim Holzschuh, Marcin Lara, Catrin Mair, Louis Martini, Sebastian Wolf with an appendix by Bogdan Zavyalov
https://arxiv.org/abs/2510.07443

The condensed homotopy type of a scheme
We study a condensed version of the étale homotopy type of a scheme, which refines both the usual étale homotopy type of Friedlander-Artin-Mazur and the proétale fundamental group of Bhatt-Scholze. In the first part of this paper, we prove that this condensed homotopy type satisfies descent along integral morphisms and that the expected fiber sequences hold. We also provide explicit computations, for example, for rings of continuous functions. A key ingredient in many of our arguments is a d…

Double categorical equivalences
Lyne Moser, Maru Sarazola, Paula Verdugo
https://arxiv.org/abs/2509.23181 https://arxiv.org/pdf/2509.23181

Double categorical equivalences
We present an efficient and user-friendly method for constructing any cofibrantly generated model structure on the category of double categories whose trivial fibrations are the "canonical" ones: the double functors which are surjective on objects, full on both horizontal and vertical morphisms, and fully faithful on squares. We show that all of these model structures are left proper and that they are localizations of the gregarious model structure introduced by Campbell. As a notable consequen…

Power Towers: Purely Inseparable Galois Theory and Foliations in Positive Characteristic
Przemys{\l}aw Grabowski
https://arxiv.org/abs/2510.06058 https://a…

Power Towers: Purely Inseparable Galois Theory and Foliations in Positive Characteristic
We build a purely inseparable Galois theory using non-derived commutative algebra. Our theory works on fields and on normal varieties. It says that a purely inseparable morphism corresponds to a finite (saturated) subalgebra of differential operators. Our approach unifies most of the literature about purely inseparable morphisms and shows new research directions. Indeed, our theory extends to a theory that covers all (saturated) subalgebras of differential operators. The extended theory gives…

Expansion of Integer Matrices over Various Rings
Jakub Szyma\'nski
https://arxiv.org/abs/2509.17823 https://arxiv.org/pdf/2509.17823

Expansion of Integer Matrices over Various Rings
In this article, we explore the problem of constructing high-dimensional expanders through the study of relations between expansion constants over different rings. We investigate expansion constants of integer matrices regarded as morphisms between free modules over $\mathbb{R}$, $\mathbb{Z}$, and $\mathbb{Z}/p\mathbb{Z}$. We introduce a new condition which we call integral spanning regarding kernels of integer matrices, and prove that it ensures equality of real and integral expansions. In add…

Colored Petri Nets are Lax Double Functors
Jade Master, Joe Moeller
https://arxiv.org/abs/2510.01946 https://arxiv.org/pdf/2510.01946

Colored Petri Nets are Lax Double Functors
We give a characterization of colored Petri nets as lax double functors. Framing colored Petri nets in terms of category theory allows for canonical definitions of various well-known constructions on colored Petri nets. In particular, we show how morphisms of colored Petri nets may be understood as natural transformations. A result from folklore, which we sketch in the appendix, shows that lax double functors are equivalent to functors with codomain their former domain. We use this result to ch…

Replaced article(s) found for math.AT. https://arxiv.org/list/math.AT/new
[1/1]:
- A minimal model for prestacks and morphisms of operadic algebras via Koszul duality for box operads
Lander Hermans

Symmetric monoidal extensions and graph cobordisms between finite sets
Andrea Bianchi
https://arxiv.org/abs/2509.22575 https://arxiv.org/pdf/2509.22575

Symmetric monoidal extensions and graph cobordisms between finite sets
Given a symmetric monoidal $(\infty,n)$-category $\mathcal{C}$ and a space $X$, we address the problem of explicitly describing the symmetric monoidal $(\infty,n)$-category freely obtained from $\mathcal{C}$ by adjoining $X$ new $n$-morphisms with prescribed sources and targets. We develop an apparatus of tools that allow one to detect in concrete situations such a free symmetric monoidal extension. As motivating application, we introduce a symmetric monoidal $(\infty,2)$-category ${\mathbb{G}\…

Learners are Almost Free Compact Closed
Mitchell Riley
https://arxiv.org/abs/2509.20930 https://arxiv.org/pdf/2509.20930

Learners are Almost Free Compact Closed
The category of learners has a pleasant symmetric formulation when the morphisms are considered up to a coarser equivalence than the one originally described in the paper "Backprop as Functor". A quotient of this modified category gives a new construction of the free compact closed category on a symmetric monoidal category.

Approximate Fiber Products of Schemes and Their \'Etale Homotopical Invariants
Dongfang Zhao
https://arxiv.org/abs/2509.15616 https://arxiv.org/pdf/250…

Approximate Fiber Products of Schemes and Their Étale Homotopical Invariants
The classical fiber product in algebraic geometry provides a powerful tool for studying loci where two morphisms to a base scheme, $ϕ: X \to S$ and $ψ: Y \to S$, coincide exactly. This condition of strict equality, however, is insufficient for describing the geometric structure of semantic spaces in modern large language models, whose foundational architecture is the Transformer. The token spaces of these models are fundamentally arithmetic, and recent work has revealed complex geometric sing…