arXiv math.CT bot

A Critical Pair Enumeration Algorithm for String Diagram Rewriting

Anna Matsui (Johns Hopkins University, USA), Innocent Obi (University of Washington, USA), Guillaume Sabbagh (University of Technology of Compi\`egne, France), Leo Torres (Universidad Nacional de C\`ordoba, Argentina), Diana Kessler (Tallinn University of Technology, Estonia), Juan F. Meleiro (University of S\~ao Paulo, Brazil), Koko Muroya (National Institute of Informatics, Japan,Ochanomizu University, Japan)
https://arxiv.org/abs/2603.09433 https://arxiv.org/pdf/2603.09433 https://arxiv.org/html/2603.09433

arXiv:2603.09433v1 Announce Type: new
Abstract: Critical pair analysis provides a convenient and computable criterion of confluence, which is a fundamental property in rewriting theory, for a wide variety of rewriting systems. Bonchi et al. showed validity of critical pair analysis for rewriting on string diagrams in symmetric monoidal categories. This work aims at automation of critical pair analysis for string diagram rewriting, and develops an algorithm that implements the core part of critical pair analysis. The algorithm enumerates all critical pairs of a given left-connected string diagram rewriting system, and it can be realised by concrete manipulation of hypergraphs. We prove correctness and exhaustiveness of the algorithm, for string diagrams in symmetric monoidal categories without a Frobenius structure.

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Clock systems for stochastic and non-deterministic categorical systems theories

Owen Lynch, David Jaz Myers, Eigil Fjeldgren Rischel, Sam Staton
https://arxiv.org/abs/2603.29573 https://arxiv.org/pdf/2603.29573 https://arxiv.org/html/2603.29573

arXiv:2603.29573v1 Announce Type: new
Abstract: One of the characteristic features of categorical systems theory is that the behavior of systems can be characterized by certain morphisms into them. In other words, behaviors form a representable covariant functor to Set. And more generally, in the compositional setting, behaviors form a representable double functor to Span. Clock systems are convenient because behavior functors represented by clock systems are automatically well-behaved.
It was previously not known whether stochastic and non-deterministic systems theories have clock systems. In this paper, we show that indeed they do have clock systems. Moreover, the clock systems for non-deterministic systems point to generalized notions of behavior for non-linear time.

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Day convolution for algebraic patterns

Thomas Blom, F\'elix Loubaton, Jaco Ruit
https://arxiv.org/abs/2603.29815 https://arxiv.org/pdf/2603.29815 https://arxiv.org/html/2603.29815

arXiv:2603.29815v1 Announce Type: new
Abstract: We characterize the exponentiable objects for a wide range of structures prevalent in $\infty$-categorical algebra, extending the construction of Day convolution to more general structures than $\infty$-operads. More precisely, we give a criterion that is both necessary and sufficient for many of these structures encountered in practice, such as (equivariant) $\infty$-operads and virtual double $\infty$-categories. We work within the framework of algebraic patterns of Chu-Haugseng that describe these structures in terms of weak Segal fibrations. As part of the proof, we give a new description of weak Segal fibrations in terms of generalized Segal spaces on certain "tree" categories. We also define the "underlying graph" of a weak Segal fibration, extending the notion of the underlying $\infty$-category for $\infty$-operads, and explicitly describe the underlying graph of exponential objects in weak Segal fibrations.

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Unitary, Inner product, and Dagger categories

Robin Cockett, Durgesh Kumar, Priyaa Varshinee Srinivasan
https://arxiv.org/abs/2603.27614 https://arxiv.org/pdf/2603.27614 https://arxiv.org/html/2603.27614

arXiv:2603.27614v1 Announce Type: new
Abstract: This article provides an alternate characterization of dagger categories, which are central to the study of categorical quantum mechanics, in terms of inner product categories. An inner product category is an "achiral involutive" category with an inner product combinator. Inner product categories are, in turn, precisely the same as unitary categories, which are a weaker form of dagger categories. In unitary categories, there is an isomorphism between an object and its dagger, instead of the identity function as in the case of dagger categories. Every unitary category is equipped with a global inner product structure, which allows one to strictify the involutive structure on the unitary category to obtain a dagger category, making unitary categories 2-categorically equivalent to dagger categories.
By regarding the inner product as an abstract metric on an (achiral) involutive category, one can define metric-preserving maps (isometries) in inner product categories, and also develop other notions of special maps -- unitary, Hermitian, positive, and normal maps -- in this setting.

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Signs in objective linear algebra, exemplified with exterior powers and determinants

Joachim Kock, Jesper Michael M{\o}ller
https://arxiv.org/abs/2603.19437 https://arxiv.org/pdf/2603.19437 https://arxiv.org/html/2603.19437

arXiv:2603.19437v1 Announce Type: new
Abstract: We develop objective linear algebra in a new setting with a cardinality functor that can take negative values. The signs arise as little homotopies, as ratios between orientations. To illustrate the workings of the theory we give an objective treatment of exterior powers and determinants.

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Presenting Neural Networks via Coherent Functors

Matthew Pugh, Jo Grundy, Corina Cirstea, Nick Harris
https://arxiv.org/abs/2604.15100 https://arxiv.org/pdf/2604.15100 https://arxiv.org/html/2604.15100

arXiv:2604.15100v1 Announce Type: new
Abstract: This paper develops a methodology for representing machine learning models as models of formal theories, grounded in the perspective that machine learning models are a form of database and that databases are models of theories in coherent logic. Two intermediate results support this approach: any functorial database schema has an associated $\kappa$-coherent theory whose models coincide with its instances, and data may be hard-coded into a coherent category such that any model of the resulting theory necessarily contains it. These tools are used to show that any dense feed-forward neural network architecture over the floating point numbers may be presented as a coherent category $G$ whose $Set$-models are the networks of that architecture, with inference arising as the precomposition functor $Coh(\iota, Set)$ along a coherent functor $\iota : RSpan(a_0, a_n) \rightarrow G$. This representation is extended to networks with weight and bias fixing and tying, encompassing sparse and convolutional architectures, via a 2-coequaliser construction in $Coh_\sim$. Taken together, these results recast neural network inference as an extension problem in the 2-category $Coh_\sim$ of coherent categories, supporting the interpretation of a network architecture as a formal hypothesis about the structure of data and of model training as a lifting of a dataset into a more constrained theory.

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Replaced article(s) found for math.CT. https://arxiv.org/list/math.CT/new
[1/1]:
- Thomason cohomology and Quillen's Theorem A
Mehmet Kirtisoglu, Ergun Yalcin
https://arxiv.org/abs/2503.14659 @arXiv_mathAT_bot@mastoxiv.page

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Dioperads, Frobenius monoidal functors and duality

Valerio Melani, Hugo Pourcelot
https://arxiv.org/abs/2604.01080 https://arxiv.org/pdf/2604.01080 https://arxiv.org/html/2604.01080

arXiv:2604.01080v1 Announce Type: new
Abstract: Motivated by duality phenomena for derived global sections on derived local systems on compact oriented manifolds, we introduce the notion of a $d$-duality context between symmetric monoidal enriched categories. In this setting, the right adjoint of a symmetric monoidal functor carries compatible lax and colax structures twisted by an invertible object $d$.
For any enriched dioperad $\mathcal{P}$, we define a $d$-twist $\mathcal{P}\{d\}$ and prove that, in a $d$-duality context, the right adjoint sends $\mathcal{P}$-algebras to $\mathcal{P}\{-d\}$-algebras. To achieve this, the key conceptual result is that Frobenius monoidal functors between symmetric monoidal categories are precisely those functors inducing morphisms between the underlying dioperads. We also develop a dioperadic Day convolution, yielding an alternative proof of the main theorem and suggesting an $\infty$-categorical extension of the theory.

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Replaced article(s) found for math.CT. https://arxiv.org/list/math.CT/new
[1/1]:
- Relativized universal algebra via partial Horn logic
Yuto Kawase
https://arxiv.org/abs/2403.19661 @arXiv_mathCT_bot@mastoxiv.page

- Nonabelian $H^2$ with coefficients in a group and with coefficients in a crossed module
Mikhail Borovoi
https://arxiv.org/abs/1608.07366

- $(\infty,n)$-Limits I: Definition and first consistency results
Lyne Moser, Nima Rasekh, Martina Rovelli
https://arxiv.org/abs/2312.11101 @arXiv_mathAT_bot@mastoxiv.page

- 2-Functoriality of Initial Semantics, and Applications
Benedikt Ahrens, Ambroise Lafont, Thomas Lamiaux
https://arxiv.org/abs/2503.10863 @arXiv_csPL_bot@mastoxiv.page

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Higher algebra in $t$-structured tensor triangulated $\infty$-categories

Jiacheng Liang
https://arxiv.org/abs/2603.27786 https://arxiv.org/pdf/2603.27786 https://arxiv.org/html/2603.27786

arXiv:2603.27786v1 Announce Type: new
Abstract: We generalize fundamental notions of higher algebra, traditionally developed within the $\infty$-category of spectra, to the broader setting of $t$-structured tensor triangulated $\infty$-categories ($ttt$-$\infty$-categories). Under a natural structural condition, which we call "projective rigidity", we establish higher categorical analogues of Lazard's theorem and prove the existence and universal property of Cohn localizations. Furthermore, we generalize higher almost ring theory to the $ttt$-$\infty$-categorical setting, showing that $\pi_0$-epimorphic idempotent algebras are in natural bijection with idempotent ideals. By exploiting deformation theory, we establish a general \'etale rigidity theorem, proving that the $\infty$-category of \'etale algebras over a fixed connective base is completely determined by its discrete counterpart. Finally, we characterize the moduli of such projectively rigid $ttt$-$\infty$-categories, demonstrating that the presheaf $\infty$-category on the 1-dimensional framed cobordism $\infty$-category serves as the universal projectively rigid $ttt$-$\infty$-category.

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Crosslisted article(s) found for math.CT. https://arxiv.org/list/math.CT/new
[1/1]:
- Geometric Points in Tensor Triangular Geometry
Tobias Barthel, Logan Hyslop, Maxime Ramzi
https://arxiv.org/abs/2603.25664 @arXiv_mathAT_bot@mastoxiv.page

- Stone Duality for Monads
Richard Garner, Alyssa Renata, Nicolas Wu
https://arxiv.org/abs/2603.25710 @arXiv_csLO_bot@mastoxiv.page

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Directed path and Moore flow

Philippe Gaucher
https://arxiv.org/abs/2604.11347 https://arxiv.org/pdf/2604.11347 https://arxiv.org/html/2604.11347

arXiv:2604.11347v1 Announce Type: new
Abstract: This addendum extends prior work to the non-regular setting by introducing the tame realization of a precubical set as a multipointed $d$-space. Its execution paths are precisely the nonconstant tame $d$-paths in the geometric realization of the precubical set. The associated Moore flow induces a functor from precubical sets to Moore flows, which is naturally weakly equivalent, within the $h$-model structure, to a colimit-preserving functor whose image is included in the class of m-cofibrant Moore flows. For spatial (and thus proper) precubical sets, these functors coincide.

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[2026-04-14 Tue (UTC), 3 new articles found for math.CT Category Theory]

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Crosslisted article(s) found for math.CT. https://arxiv.org/list/math.CT/new
[1/1]:
- Global dimension of dg algebras via compact silting objects
Panagiotis Kostas
https://arxiv.org/abs/2604.13698 @arXiv_mathRT_bot@mastoxiv.page

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Crosslisted article(s) found for math.CT. https://arxiv.org/list/math.CT/new
[1/1]:
- Finiteness of homological dimensions in triangulated categories
Hongxing Chen, Xiaohu Chen, Jinbi Zhang
https://arxiv.org/abs/2604.10478 @arXiv_mathRT_bot@mastoxiv.page

- Semiprojective Banach lattices
Tomasz Kania, Mariusz Niwi\'nski
https://arxiv.org/abs/2604.10624 @arXiv_mathFA_bot@mastoxiv.page

- Finite Pre-Tensor Categories that are Morita Equivalent to Finite Tensor Categories
Thibault D. D\'ecoppet, Mateusz Stroi\'nski
https://arxiv.org/abs/2604.10753 @arXiv_mathQA_bot@mastoxiv.page

- Variable-Length Markov Chains on Finite Quivers: Boundary-Window Identifiability, Exact Depth, an...
Oleg Kiriukhin
https://arxiv.org/abs/2604.10792 @arXiv_mathPR_bot@mastoxiv.page

- Hopf substitutions in Species
Aaron Lauve, Anthony Lazzeroni
https://arxiv.org/abs/2604.10816 @arXiv_mathCO_bot@mastoxiv.page

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A Universal Quotient of Banking APIs

Christopher Doyle
https://arxiv.org/abs/2604.08833 https://arxiv.org/pdf/2604.08833 https://arxiv.org/html/2604.08833

arXiv:2604.08833v1 Announce Type: new
Abstract: Four axioms of immutable ledger, linear consent, payment irreversibility, and bounded credit manifest themselves as institutional facts codified by banking practice for the transfer of monetary value. These axioms certify the independence of 14 empirically observed and jurisdictionally invariant dimensions. Morphisms of the ambient category do not admit sections that would reconstruct one dimension from another, and every morphism admits epi-mono factorisation through the universal quotient Q_public. This factorisation is forced by definite causal order under classical realisation and echoes the factorisation theorem of Gogioso et al. Gaussian elimination across 4,590 endpoints from BIAN, CDR, and OBIE confirms rank 14 and witnesses the jurisdictional invariance of the quotient object. The axioms similarly constrain the monoidal structure. The information dominance preorder is a thin category; all five Szlachanyi conditions follow, establishing that Q_public carries left skew monoidal structure.

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[2026-03-30 Mon (UTC), no new articles found for math.CT Category Theory]

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[2026-03-31 Tue (UTC), 3 new articles found for math.CT Category Theory]

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Crosslisted article(s) found for math.CT. https://arxiv.org/list/math.CT/new
[1/1]:
- Unifying Koszul dualities via point-set models
Dan Petersen, Victor Roca i Lucio, Sinan Yalin
https://arxiv.org/abs/2603.29910 @arXiv_mathAT_bot@mastoxiv.page

- Quantale-Enriched Co-Design: Toward a Framework for Quantitative Heterogeneous System Design
Hans Riess, Yujun Huang, Matthew Klawonn, Gioele Zardini, Matthew Hale
https://arxiv.org/abs/2603.29921 @arXiv_eessSY_bot@mastoxiv.page

- The Homotopy 3-Type of Abelian C*-Algebras
Gregory Faurot, Giovanni Ferrer
https://arxiv.org/abs/2603.29985 @arXiv_mathOA_bot@mastoxiv.page

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Crosslisted article(s) found for math.CT. https://arxiv.org/list/math.CT/new
[1/1]:
- The Image of Functor Morphing
Ehud Meir
https://arxiv.org/abs/2603.26368 @arXiv_mathRT_bot@mastoxiv.page

- The motivic tt-geometry of real quadrics
Jean Paul Schemeil
https://arxiv.org/abs/2603.26492 @arXiv_mathAG_bot@mastoxiv.page

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[2026-04-02 Thu (UTC), 1 new article found for math.CT Category Theory]

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[2026-04-01 Wed (UTC), 2 new articles found for math.CT Category Theory]

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An Inductive Strategy Towards a Solution to the Generalized Homotopy Hypothesis

Johnathon Taylor
https://arxiv.org/abs/2604.09867 https://arxiv.org/pdf/2604.09867 https://arxiv.org/html/2604.09867

arXiv:2604.09867v1 Announce Type: new
Abstract: Using the theory of distributive series of monads, we construct an $(\infty,0)$-coherator called the \emph{inductive coherator}. The category of models out of the inductive coherator serve as a model for $\infty$-groupoids that possess an underlying globular set. Once we establish the construction for the inductive coherator, we provide the framework for an inductive strategy to prove the Generalized Homotopy Hypothesis obtained by transferring model structure off of the category of $n$-groupoids onto the category of $(n+1)$-groupoids. Moreover, we provide a necessary and sufficient condition for the transfer of model structure to be successful. We conclude by showing if the transfer of model structure may be completed successively, then the Generalized Homotopy Hypothesis is true.

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[2026-04-16 Thu (UTC), 2 new articles found for math.CT Category Theory]

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[2026-04-13 Mon (UTC), 2 new articles found for math.CT Category Theory]

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Replaced article(s) found for math.CT. https://arxiv.org/list/math.CT/new
[1/1]:
- Dimensional Type Systems and Deterministic Memory Management: Design-Time Semantic Preservation i...
Houston Haynes
https://arxiv.org/abs/2603.16437 @arXiv_csPL_bot@mastoxiv.page

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Replaced article(s) found for math.CT. https://arxiv.org/list/math.CT/new
[1/1]:
- An extension of Priestley duality to fuzzy topologies and positive MV-algebras
Marby Zuley Bola\~nos Ortiz, Ciro Russo
https://arxiv.org/abs/2508.19423 @arXiv_mathCT_bot@mastoxiv.page

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Topologically valued transition structures

Matthew Collinson
https://arxiv.org/abs/2604.14031 https://arxiv.org/pdf/2604.14031 https://arxiv.org/html/2604.14031

arXiv:2604.14031v1 Announce Type: new
Abstract: We investigate several categories related to transition structures, using a mixture of algebraic and topological methods. We show how two such categories are connected by a contravariant adjunction. This is the most detailed of a family of such results depending on topological restrictions on objects and morphisms.

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On the (algebraic) notion of 2-ring

Josep Elgueta
https://arxiv.org/abs/2604.10154 https://arxiv.org/pdf/2604.10154 https://arxiv.org/html/2604.10154

arXiv:2604.10154v1 Announce Type: new
Abstract: By a 2-ring we mean a groupoid with a structure analogous to that of a ring, up to coherent isomorphisms. Two different notions of 2-ring appear in the literature: the notion of {\em Ann-category}, due to Quang, and the notion of {\em categorical ring}, due to Jibladze and Pirashvili. The underlying data are the same in both cases, but the required axioms differ. In this note, we clarify the relationship between these notions by explaining why an additional axiom must be imposed for the two notions to be equivalent. Essential to this analysis is an equivalent description of a symmetric monoidal category.

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Replaced article(s) found for math.CT. https://arxiv.org/list/math.CT/new
[1/1]:
- Revisiting colimits in $\mathbf{Cat}$ and homotopy category
Varinderjit Mann
https://arxiv.org/abs/2603.07773 @arXiv_mathCT_bot@mastoxiv.page

- Enriched coalgebras are sometimes comonadic
Ois\'in Flynn-Connolly
https://arxiv.org/abs/2604.09354 @arXiv_mathCT_bot@mastoxiv.page

- Chevalley property and discriminant ideals of Cayley-Hamilton Hopf Algebras
Yimin Huang, Zhongkai Mi, Tiancheng Qi, Quanshui Wu
https://arxiv.org/abs/2506.21879 @arXiv_mathQA_bot@mastoxiv.page

- Homotopy Posets, Postnikov Towers, and Hypercompletions of $\infty$-Categories
David Gepner, Hadrian Heine
https://arxiv.org/abs/2603.09903 @arXiv_mathAT_bot@mastoxiv.page

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Crosslisted article(s) found for math.CT. https://arxiv.org/list/math.CT/new
[1/1]:
- Recollements of Cohen-Macaulay Auslander algebras for gentle algebras
Jiacheng Xu, Yu-Zhe Liu, Xin Ma, Guiqi Shi
https://arxiv.org/abs/2604.00109 @arXiv_mathRT_bot@mastoxiv.page

- Deformations of mixed associators in module categories
Matthieu Faitg, Azat M. Gainutdinov, Christoph Schweigert
https://arxiv.org/abs/2604.00837 @arXiv_mathQA_bot@mastoxiv.page

- A Categorification of Subword Complexes and Its Hall Algebra
Mikhail Gorsky, Zijun Li
https://arxiv.org/abs/2604.00879 @arXiv_mathRT_bot@mastoxiv.page

- Makkai's lost proof of projectivity of N in the free topos
Henrik Forssell, Peter LeFanu Lumsdaine, Andrew W. Swan
https://arxiv.org/abs/2604.01139 @arXiv_mathLO_bot@mastoxiv.page

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Replaced article(s) found for math.CT. https://arxiv.org/list/math.CT/new
[1/1]:
- Bundles of metric structures as left ultrafunctors
Ali Hamad
https://arxiv.org/abs/2406.11076 @arXiv_mathCT_bot@mastoxiv.page

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Internal structures in the category of right-preordered groups

Aubril Ony
https://arxiv.org/abs/2604.14105 https://arxiv.org/pdf/2604.14105 https://arxiv.org/html/2604.14105

arXiv:2604.14105v1 Announce Type: new
Abstract: We show that the category of (right-)preordered groups is a quasivariety of universal algebras by giving explicit axioms. We then look at lattices of effective equivalence relations, which turn out to be similar to the lattices of equivalence relations in the category of groups. We study internal structures in the category of right-preordered groups, and we especially consider the class $\sS$ of Schreier split epimorphisms. The category of right-preordered groups turns out to be action representable when we restrict our attention to split epimorphisms in $\sS$. Relatively to this class of split epimorphisms, we define the notion of $\sS\mhyphen$precrossed modules, and then of $\sS\mhyphen$crossed modules that correspond exactly to Schreier internal reflexive graphs and Schreier internal categories, respectively. Lastly, we characterize groupoids among Schreier internal categories and give some examples.

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Introducing pixelation with applications

J. Daisie Rock
https://arxiv.org/abs/2603.25432 https://arxiv.org/pdf/2603.25432 https://arxiv.org/html/2603.25432

arXiv:2603.25432v1 Announce Type: new
Abstract: Motivated by the desire for a new kind of approximation, we define a type of localization called pixelation. We present how pixelation manifests in representation theory and in the study of sites and sheaves. A path category is constructed from a set, a collection of "paths" into the set, and an equivalence relation on the paths. A screen is a partition of the set that respects the paths and equivalence relation. For a commutative ring, we also enrich the path category over its modules (=linearize the category with respect to the ring) and quotient by an ideal generated by paths (possibly 0). The pixelation is the localization of a path category, or the enriched quotient, with respect to a screen. The localization has useful properties and serves as an approximation of the original category. As applications, we use pixelations to provide a new point of view of the Zariski topology of localized ring spectra, provide a parallel story to a ringed space and sheaves of modules, and construct a categorical generalization of higher Auslander algebras of type $A$.

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Replaced article(s) found for math.CT. https://arxiv.org/list/math.CT/new
[1/1]:
- Profunctorial algebras
Quentin Aristote, Umberto Tarantino
https://arxiv.org/abs/2601.22721 @arXiv_mathCT_bot@mastoxiv.page

- Mapping spaces between operads in relation to bimodules
Hoang Truong
https://arxiv.org/abs/2312.07906 @arXiv_mathAT_bot@mastoxiv.page

- Mixed Hodge Modules and Canonical Perverse Extensions for Multi-Node Conifold Degenerations
Abdul Rahman
https://arxiv.org/abs/2604.05367 @arXiv_mathAG_bot@mastoxiv.page

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[2026-04-15 Wed (UTC), no new articles found for math.CT Category Theory]

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Crosslisted article(s) found for math.CT. https://arxiv.org/list/math.CT/new
[1/1]:
- Contraherent cosheaves of contramodules on Noetherian formal schemes
Leonid Positselski
https://arxiv.org/abs/2603.27732 @arXiv_mathAG_bot@mastoxiv.page

- Ribbon categories from ind-exact algebras: simple current case
Kenichi Shimizu, Harshit Yadav
https://arxiv.org/abs/2603.28215 @arXiv_mathQA_bot@mastoxiv.page

- Anick Resolution for Lawvere Theories from Algebraic Discrete Morse Theory
Mirai Ikebuchi
https://arxiv.org/abs/2603.28382 @arXiv_mathKT_bot@mastoxiv.page

- Categorical Time-Reversal Symmetries
Rui Wen, Sakura Schafer-Nameki
https://arxiv.org/abs/2603.28720 @arXiv_condmatstrel_bot@mastoxiv.page

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Enriched coalgebras are sometimes comonadic

Ois\'in Flynn-Connolly
https://arxiv.org/abs/2604.09354 https://arxiv.org/pdf/2604.09354 https://arxiv.org/html/2604.09354

arXiv:2604.09354v1 Announce Type: new
Abstract: We introduce an enriched notion of coalgebras over $\V$-operad $\P$ in a symmetric monoidal V-category C. When C is semicartesian, we construct an endofunctor on C associated to P and give conditions under which it is a comonad with co-Eilenberg--Moore category equivalent to the category of enriched P-algebras in V. In many cases, this permits easy computation of V-categories of coalgebras. We give several simple examples and show that our theorem generalises one direction of a well-known theorem of Fox.

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[2026-04-17 Fri (UTC), 2 new articles found for math.CT Category Theory]

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