Tootfinder

Sites and Grothendieck toposes: an introduction
Olivia Caramello, Laurent Lafforgue
https://arxiv.org/abs/2508.21609 https://arxiv.org/pdf/2508.21609

Sites and Grothendieck toposes: an introduction
Topos theory occupies a singular place in contemporary mathematics: born from Grothendieck's algebraic geometry, it has emerged as a unifying language for geometry, topology, algebra, and logic. This book offers a progressive introduction that moves from familiar ground - groups and their actions, topological spaces, categories, and sheaves - to Grothendieck topologies and sites, then to the axiomatic and categorical foundations of toposes (via Giraud's theorem), to their geometry (morphisms, p…

A Diagrammatic Calculus for a Functional Model of Natural Language Semantics
Matthieu Pierre Boyer
https://arxiv.org/abs/2507.00782 https://

A Diagrammatic Calculus for a Functional Model of Natural Language Semantics
In this paper, we study a functional programming approach to natural language semantics, allowing us to increase the expressivity of a more traditional denotation style. We will formalize a category based type and effect system, and construct a diagrammatic calculus to model parsing and handling of effects, and use it to efficiently compute the denotations for sentences.

A Survey of LLM-based Automated Program Repair: Taxonomies, Design Paradigms, and Applications
Boyang Yang, Zijian Cai, Fengling Liu, Bach Le, Lingming Zhang, Tegawend\'e F. Bissyand\'e, Yang Liu, Haoye Tian
https://arxiv.org/abs/2506.23749

A Survey of LLM-based Automated Program Repair: Taxonomies, Design Paradigms, and Applications
Large language models (LLMs) are reshaping automated program repair (APR). We categorize the recent 63 LLM-based APR systems published from January 2022 to June 2025 into four paradigms, and show how retrieval- or analysis-augmented contexts strengthen any of them. This taxonomy clarifies key trade-offs: fine-tuning delivers strong task alignment at high training cost; prompting enables rapid deployment but is limited by prompt design and context windows; procedural pipelines offer reproducible…

A Chimera domain decomposition method with weak Dirichlet-Robin coupling for finite element simulation of particulate flows
Raphael M\"unster, Otto Mierka, Dmitri Kuzmin, Stefan Turek
https://arxiv.org/abs/2506.22831

A Chimera domain decomposition method with weak Dirichlet-Robin coupling for finite element simulation of particulate flows
We introduce a new multimesh finite element method for direct numerical simulation of incompressible particulate flows. The proposed approach falls into the category of overlapping domain decomposition / Chimera / overset grid meshes. In addition to calculating the velocity and pressure of the fictitious fluid on a fixed background mesh, we solve the incompressible Navier-Stokes equations on body-fitted submeshes that are attached to moving particles. The submesh velocity and pressure are used …

This https://arxiv.org/abs/2408.02664 has been replaced.
initial toot: https://mastoxiv.page/@arX…

Noninvertible Gauge Symmetry in (2+1)d Topological Orders: A String-Net Model Realization
We develop a systematic framework for understanding symmetries in topological phases in 2+1 dimensions using the string-net model, encompassing both gauge symmetries that preserve anyon species and global symmetries permuting anyon species, including both invertible symmetries describable by groups and noninvertible symmetries described by categories. As an archetypal example, we reveal the first noninvertible categorical gauge symmetry of topological orders in 2+1 dimensions: the Fibonacci gau…

Why does Kindle have different categories for paperback and ebooks? Like Programming Languages is a category for paperback but not for ebook?

Program Logics via Distributive Monoidal Categories
Filippo Bonchi, Elena Di Lavore, Mario Rom\'an, Sam Staton
https://arxiv.org/abs/2507.18238 https://

Program Logics via Distributive Monoidal Categories
We derive multiple program logics, including correctness, incorrectness, and relational Hoare logic, from the axioms of imperative categories: uniformly traced distributive copy-discard categories. We introduce an internal language for imperative multicategories, on top of which we derive combinators for an adaptation of Dijkstra's guarded command language. Rules of program logics are derived from this internal language.

Functional programming isn't just for Haskell developers. It's for #PHP developers, too. "Thinking Functionally in PHP" is available from LeanPub.
https://leanpub.com/thinking-functiona

Equivalence via surjections
Tom Leinster
https://arxiv.org/abs/2508.20555 https://arxiv.org/pdf/2508.20555

Equivalence via surjections
Many types of categorical structure obey the following principle: the natural notion of equivalence is generated, as an equivalence relation, by identifying $A$ with $B$ when there exists a strictly structure-preserving map $A \to B$ that is genuinely (not just essentially) surjective in each dimension and faithful in the top dimension. We prove this principle for four types of structure: categories, monoidal categories, bicategories and double categories. The last of these theorems suggests th…

Super funny that when you try to create an entry for a book on Kindle, Smalltalk is its own separate category at the same level as, say, “Algorithms”, “Pattern Recognition”, and “Microsoft Programming”.