50 papers
This paper investigates the split Grothendieck group of a -rigid subcategory within an extriangulated category, establishing isomorphisms between the Grothendieck group of the ambient category and specific subcategory groups for silting and -cluster tilting cases, while also determining the explicit structure of the Grothendieck group for -cluster categories of type .
This paper introduces a local treatment of finite alignment for arbitrary higher-rank graphs by identifying a finitely aligned sub-constellation, which enables the construction of novel locally compact path and boundary-path groupoids that extend existing models and are proven to be amenable.
This paper introduces an operad of dependent directed wiring diagrams to enable the composition of instantaneous systems like Mealy machines and parameterized stock-and-flow diagrams, providing a formal algebraic framework and a semantic morphism that maps these diagrams into Mealy machines.
This paper establishes that the -category of normed algebras in genuine -spectra is modeled by strictly commutative algebras in -symmetric spectra, providing a higher-categorical description of ultra-commutative global ring spectra as a partially lax limit of genuine -spectra categories.
This paper extends support variety theory from the compact to the non-compact part of a monoidal triangulated category in the noncommutative setting, establishing conditions under which the extended theory detects the zero object and thereby confirming a portion of a conjecture by the second author, Nakano, and Yakimov regarding central cohomological support in stable categories of finite tensor categories.
This paper investigates how key properties of probability monads, including their Kleisli laws, monoidal structures, and affinity, arise from their codensity presentations, establishing new universal properties for these monads and characterizing their tensor products via Day convolution while highlighting the limitations of the Giry monad on standard Borel spaces.
This paper extends fundamental homotopical concepts to -categories and presentable enriched categories by introducing homotopy posets indexed by categorical disk boundaries, which assemble into a Postnikov tower converging for -categories and characterize Postnikov complete -categories as the limit of -categories under truncation.
This paper constructs a fractured structure on the -topos of condensed anima to identify a jointly conservative collection of points and resolves a question by Clausen by demonstrating that the category of extremally disconnected spaces lacks all fibers.
This paper presents and proves the correctness of an algorithm that automates critical pair analysis for string diagram rewriting in symmetric monoidal categories (without Frobenius structure) by enumerating all critical pairs through concrete hypergraph manipulation.
This paper introduces a change-of-base construction using symmetric monoidal monads on Markov categories to extend symmetric monoidal categories of open systems, such as design problems, into 2-categories that compositionaly model various types of uncertainty while preserving their underlying structural properties.
This paper proposes a categorical framework for convex optimization by defining a category where optimization problems are objects, utilizing this structure to rederive key results such as a minimax-type theorem and the involution property of the Legendre dual.
This paper demonstrates that a lightweight, automated AI pipeline integrating next-generation large language models with citation-based verification can successfully generate and solve sophisticated, research-grade mathematical problems, including previously unpublished questions, with verified results and open-sourced tools.
This paper investigates homotopy decompositions of classifying spaces of compact connected Lie groups via a relative fiber-cofiber construction, establishing rational sharpness and Cohen-Macaulay properties for the resulting spaces under specific cohomological conditions while providing explicit cohomological and K-theoretic computations for examples involving maximal tori and commuting elements.
This paper establishes that the filter quotient construction preserves model structures under suitable conditions, demonstrating that while the resulting model inherits properties like being simplicial or proper, it may not remain cofibrantly generated, and further proves its compatibility with filter quotient -categories.
This paper constructs a finite-dimensional Kan simplicial manifold that integrates the universal Lie -algebroid of a singular foliation admitting a geometric resolution, utilizing a recursive application of bi-submersions to generalize the Androulidakis-Skandalis holonomy groupoid to a higher Lie groupoid setting.
This paper demonstrates that even in a non-additive setting, the existence of a codereliction within a monoidal coalgebra modality inherently induces an additive enrichment via bialgebra convolution, thereby uniquely characterizing differential linear categories and establishing the uniqueness of coderelictions.
This paper establishes a correspondence between model structures and a single hereditary complete cotorsion pair on weakly idempotent complete extriangulated categories, thereby generalizing previous results by Beligiannis-Reiten and Cui et al., and provides methods to construct such model structures from silting objects and co--structures.
This paper establishes the theorem of the heart for Weibel's homotopy -theory (), proving that the realization functor induces an equivalence for small stable -categories with bounded -structures, a result derived from a strengthened version of Barwick's theorem that provides precise isomorphism ranges for classical -theory and demonstrates the sharpness of these bounds.
This paper establishes a natural -multicategory structure on moduli spaces of pseudo-holomorphic polygons to provide a uniform operadic framework for various -type structures, including Fukaya categories, as algebras over differential graded -multicategories.
This paper establishes the foundational theory of covers and envelopes within proto-exact categories and applies these results to prove that categories of Banach modules over arbitrary Banach rings possess enough injectives.