pith. sign in

math.CT

Category Theory

Enriched categories, topoi, abelian categories, monoidal categories, homological algebra

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math.CT 2026-07-03

Directed univalence holds for simplicial objects in any ∞-topos

by Evan Cavallo, Emily Riehl +1 more

Directed univalence for simplicial objects in an infty-topos

Equivalence of hom types in the universal left fibration with function types validates the axiom in this semantic model.

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A fundamental component of homotopy type theory, a synthetic theory of $\infty$-groupoids, is Voevodsky's univalence axiom. Univalence characterizes the identity types in the universal fibration, a classifier for small type families: identity types in the universe are equivalent to types of equivalences. The directed univalence axiom plays a similar foundational role in simplicial type theory, a synthetic theory of $\infty$-categories. In its original form, which does not include universes or directed univalence, the simplicial type theory has semantics in categories of simplicial objects in an $\infty$-topos, with synthetic $\infty$-categories corresponding to internal $\infty$-categories. We verify that directed univalence holds in this semantic setting, constructing an equivalence between hom types in the universal left fibration and function types. In fact, we verify a higher version of this result, constructing an equivalence between homotopy coherent composites in the universal left fibration and composable sequences of functions between types. Using the technique of weighted limits, we reduce this theorem for simplicial objects in an arbitrary $\infty$-topos to calculations "on the left" with simplicial sets.
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math.LO 2026-07-03

Theory-topos duality defines conceptual completeness for logic fragments

by Ivan Di Liberti, Umberto Tarantino +1 more

Conceptual completeness for subgeometric logics

Fragments satisfying the duality embed conservatively into geometric logic; coherent, regular and disjunctive logics are shown to qualify.

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We explore the notion of conceptual completeness for a fragment of geometric logic in the framework developed by the first and third author. Unlike its traditional interpretation as a reconstruction of syntax from semantics, in this paper we characterise conceptual completeness of a fixed fragment in terms of a duality between theories and topoi. We then show that conceptually complete fragments are conservatively embedded in full geometric logic, thus casting conceptual completeness in a new proof-theoretic light. We give a new proof of conceptual completeness for coherent logic, and we also show that regular, disjunctive, and essentially algebraic logic with falsum are conceptually complete. Finally, we show that our notion is equivalent to a traditional reconstruction result under the assumption of completeness with respect to set-based models: in the coherent case, we thus recover Makkai's original reconstruction theorem via ultracategories.
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math.RT 2026-07-02

Silting mutation extended to infinite dimensions

by Diego Alberto Barceló Nieves

Large silting mutation in extriangulated categories

Theory defined for n-cosilting complexes over any ring and infinite n-tilting modules over finite-global-dimension rings.

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Silting mutation in triangulated categories, both at the level of objects and of subcategories, was introduced in arXiv:1009.3370, and later generalized to extriangulated categories in arXiv:2303.08125. It simultaneously encompasses the mutation theories of cluster-tilting objects in cluster theory and of compact 2-term silting complexes and support $\tau$-tilting modules in $\tau$-tilting theory. In this article, we develop an infinite-dimensional analog of silting mutation in extriangulated categories with set-indexed (co)products, which we then apply to obtain a theory of mutation for $n$-cosilting complexes over an arbitrary ring, as well as for infinite-dimensional $n$-(co)tilting modules over a ring of finite global dimension. The former theory is also shown to reinterpret the cosilting mutation introduced in arXiv:2201.02147.
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math.AT 2026-07-01

Six functors prove ANR homology manifolds are cohomologically smooth

by Markus Land, Marco Volpe

Homology manifolds via six functor formalisms

Compact cases are Poincaré duality complexes with Spivak tangent fibration matching the dualizing sheaf, and conical singularities force top

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We study homology manifolds through the eyes of the six functor formalism of spectral sheaves on locally compact Hausdorff spaces. As main results, we characterize cohomologically smooth objects by adapting an argument of Scholze, deduce that any hypercomplete locally compact ANR homology manifold is cohomologically smooth, show that compact ANR homology manifolds $X$ are Poincar\'e duality complexes whose Spivak tangent fibration identifies with the dualizing sheaf of $X$, and prove a generalization of Wilder's monotone mapping theorem about cell-like maps. Moreover, we introduce the notion of homotopy manifolds for which we prove an unstable analog of Wilder's orientability conjecture and show that hypercomplete ANR homology manifolds are homotopy manifolds. As a consequence, we show that for a compact $d$-dimensional ANR homology manifold, the Spivak tangent fibration of its associated Poincar\'e duality complex canonically destabilizes to a pointed $S^d$-fibration. Finally, we introduce homotopy manifolds with conical singularities, a generalization of Cohen's triangulated homotopy manifolds, and show that such objects are in fact topological manifolds, generalizing a result of Siebenmann. Along the way, we obtain comparisons between sheaf and singular cohomology and between the shape and the weak homotopy type of a topological space, explore the relation between various notions of cohomological dimension and hypercompleteness, and study six functor formalisms satisfying the K\"unneth formula.
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math.CT 2026-07-01

Category theory structures AI identity as a hierarchy of criteria

by Andrea Ferrario

A Category Theory Account of AI Identity

It replaces a single relation with synchronic and diachronic conditions based on trustworthiness-preserving paths, clarifying when governanc

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Artificial intelligence (AI) systems are routinely modified after deployment through retraining and changes in their environments. These transformations raise a metaphysical question: under what conditions does an AI system remain the same system over time or across deployments? Earlier work formulates synchronic and diachronic identity propositionally, by relating identity within a fixed AI system type to equality of trustworthiness levels. Such criteria specify when identity statements are true, but leave implicit the structure of the states compared, the transformations connecting them, and the temporal organization of persistence. We develop a category-theoretic formalization of AI identity. An AI system type is specified by a datum consisting of a techno-function, a trustworthiness profile, and a trustworthiness-level function. Profile-relative states are connected by admissible lifecycle paths, which are restricted to trustworthiness-level-preserving transformations and quotiented to obtain a reachability category. Temporally admissible functors represent AI system histories, while time-synchronous natural transformations compare realized histories. The formalization yields two categorical interpretations of the earlier AI identity criteria. A weak interpretation recovers identity as equality of trustworthiness level. A strong interpretation requires mutual trustworthiness-preserving reachability, expressed through state isomorphism or natural isomorphism of realized histories. Category theory therefore replaces a single AI identity relation with a structured hierarchy of diachronic and synchronic criteria. The resulting framework identifies identity-related preconditions for transferring responsible-AI claims, evidence, and governance procedures across versions or deployments, without treating categorical identity as sufficient by itself for such transfer.
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math.CT 2026-06-30

Signed measures uniquely extend additive maps from positive measures

by Evan Misshula

Signed Measures as the Linear Envelope of Positive Measures

The universal property shows they form the canonical group completion of positive measure theory.

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Signed measures are traditionally introduced as countably additive set functions that may take both positive and negative values. The classical Jordan decomposition theorem shows that every finite signed measure can be expressed uniquely as the difference of two mutually singular positive measures. While this theorem provides a structural description of signed measures, it does not characterize them by a universal property. We show that, for every measurable space, the abelian group of finite signed measures satisfies a universal property with respect to the commutative monoid of finite positive measures: every additive map from positive measures into an abelian group extends uniquely to a group homomorphism on signed measures. In this sense, signed measures are the canonical additive extension of positive measure theory. We compare this characterization with classical Grothendieck completion, clarifying both the analogy and the additional structure arising from countable additivity and Jordan decomposition. This places signed measures within the familiar framework of additive completion and linearization, providing a conceptual explanation for their role in analysis and probability.
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math.CT 2026-06-30

Marking generators invertible yields cofibrant ω-categories

by Thibaut Benjamin, Camil Champin +1 more

Computads with invertible generators for weak {ω}-categories

Generalised computads admit a coreflection to ordinary ones that preserves generated categories and form a presheaf topos.

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We extend the notion of computads for weak \(\omega\)-categories to allow marking certain generators as invertible, and describe inductively the free \(\omega\)-categories they generate. This gives a simple, finite description of the walking equivalences, the \(\omega\)-categories classifying invertible cells. We then construct a coreflection from generalised to ordinary computads, preserving the generated \(\omega\)-categories, and conclude that \(\omega\)-categories generated by generalised computads are cofibrant. Finally, we study the subcategory of generalised computads and generator-preserving morphisms, and show that it is a presheaf topos, similarly to the case of ordinary computads.
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math.RA 2026-06-30

Cartan subsemigroup yields Steinberg algebra via ultrafilter groupoid

by Tristan Bice, Malcolm Jones +1 more

Steinberg Algebras of Ample Semicategories and their Boolean-Cartan Restriction Semigroups

The reconstruction characterizes algebras with suitable restriction subsemigroups as Steinberg algebras of their ultrafilter groupoids, exte

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We extend the construction of Steinberg algebras of ample groupoids to \'etale semicategories. We also relate ample semicategories to Boolean restriction semigroups via a representation result extending previously known results for categories. Furthermore, we prove a reconstruction result which characterises an abstract algebra $A$ with a certain Cartan-like restriction subsemigroup $B$ (subject to conditions resembling those defining quasi-Cartan pairs) as the Steinberg algebra of the ultrafilter groupoid of $B$. In this way we obtain a twist-free extension of previous Steinberg algebra reconstruction results.
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math.CT 2026-06-30

Braided cogroupoids create monoidal equivalences on comodules

by Thi Hoa Nguyen (LMBP)

Braided cogroupoids

The structures generalize transmutation and bosonization from Hopf algebras and relate representation categories while preserving tensor pro

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We introduce and develop the theory of braided cogroupoids, a class of algebraic structures generalizing cogroupoids in a braided setting. We show that braided cogroupoids induce monoidal equivalences between the associated comodule categories, and we generalize Majid's transmutation and bosonization of braided Hopf algebras to the cogroupoid setting. Several examples are studied in detail, including the braided $SL_{n}$ cogroupoid and the braided bilinear cogroupoid.
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math.RT 2026-06-30

Ideal n-cotorsion pairs match exactly in Frobenius categories and their stables

by Yixia Zhang, Panyue Zhou

Ideal n-cotorsion pairs in Frobenius extriangulated categories

The quotient by projective-injective objects preserves the n-cotorsion conditions, transferring approximation results between the two settin

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Motivated by the correspondence between ideal cotorsion pairs in Frobenius exact categories and those in their stable categories, we introduce the notion of an ideal $n$-cotorsion pair in an extriangulated category. We study the relationship between ideal $n$-cotorsion pairs in a Frobenius extriangulated category $\mathcal C$ and those in its stable category $\underline{\mathcal C}=\mathcal C/\omega$. Our main result shows that $(\mathcal I,\mathcal J)$ is an ideal $n$-cotorsion pair in $\mathcal C$ if and only if $(\mathcal I/\omega,\mathcal J/\omega)$ is an ideal $n$-cotorsion pair in $\underline{\mathcal C}$. This provides a bridge between higher ideal approximation theory in Frobenius extriangulated categories and its counterpart in their stable categories. Additionally, in Krull--Schmidt exact categories, we establish a bijective correspondence between complete cotorsion pairs and complete ideal cotorsion pairs, answering a question of Fu, Guil Asensio, Herzog and Torrecillas.
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math.AT 2026-06-29

Oriented polytopes present (∞,∞)-categories as sheaves

by David Gepner, Hadrian Heine

An Oriented Street--Roberts Conjecture

The presentation generalizes the Street-Roberts conjecture and derives geometric formulae for operations such as the Gray tensor.

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We formulate a notion of oriented polytope, including Street's oriented simplices and Gray's oriented cubes, and use this to prove an oriented version of the Street--Roberts conjecture, presenting $(\infty,\infty)$-categories as sheaves on suitable families of oriented polytopes, generalizing work of Campion. This allows us to understand $(\infty, \infty)$-categories from a geometric perspective, as directed analogues of homotopy types. These familes of oriented polytopes induce basic operations in higher category theory: for instance, the join, Gray tensor, and bicone arise from the geometry of the orientals, cubes, and orthoplexes, respectively. We study the interaction of these operations and derive some geometric formulae, generalizing work of Ara--Maltsiniotis, Verity, and others.
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math.CT 2026-06-29

Operad unifies gradient descent and wave equation

by David I. Spivak

Compositional Dynamics in Learning and Mechanics

Two functors from arrangements of lenses and potentials recover learning and physical dynamics on graphs, with functorial composition.

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We give a single compositional setting in which gradient-based learning and Hamiltonian-style mechanics appear as functorial semantics. The syntax is an operad Arr whose objects are input-output interfaces (pairs of manifolds) and whose morphisms are *smooth adaptive arrangements*, which consist of a reactive parameter space, a lens given by smooth output and input maps, and a real-valued potential. The main technical result of the paper is what we call *lens internalization*, a lax symmetric monoidal functor Lens(C) $\to$ C associated to any symmetric monoidal closed category C. Using it, we provide two functors $\Phi_\text{phase}$, $\Phi_\text{conf}$: Arr $\to$ PC into the 2-category of polynomial coalgebras -- input-output discrete dynamical systems -- which we take as the semantics category. $\Phi_\text{phase}$ stores both position and momentum, whereas $\Phi_\text{conf}$ stores only position. When applied to a parameterized function, $\Phi_\text{conf}$ recovers the gradient descent training algorithm, with backpropagation as the lens' backward pass. When applied to harmonic particles wired together -- in series, or according to any finite directed graph -- one diagram yields two different regimes, both of which are governed by the graph Laplacian: $\Phi_\text{phase}$ gives the discrete wave equation, which is conservative and second-order, and $\Phi_\text{conf}$ gives the discrete heat equation, which is dissipative and first-order. They are two semantics of one adaptive arrangement, e.g. with the same potential in each case. And because Arr is an operad, such diagrams nest -- larger systems wired from smaller ones -- and each semantics assembles a system's dynamics functorially from its parts. These dynamics are moreover executable: a parameterized neural network and a graph of particles both compile, by the same construction, to explicit state machines one can run.
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math.CT 2026-06-29

Extensible doctrine morphisms coincide with ω-saturated ones

by Sam van Gool, Joshua L. Wrigley

The points of canonical extensions of doctrines

This yields a reconstruction of canonical extensions from presheaves of homogeneous models for any coherent theory with a countable saturate

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We analyse the space of points of the canonical extension of a coherent doctrine. We first give a full characterisation of doctrine morphisms that are extensible, and relate it to the existing notion of p-model of a coherent category. Through this characterisation, the extensible morphisms are shown to be exactly those which are {\omega}-saturated in the sense of coherent first-order logic. Next, we answer the question: when does a presheaf of models fully describe the canonical extension? We prove a characterisation theorem via two conditions, which are again natural from the perspective of coherent logic, namely, homogeneity and the realisation of all prime types in a strict sense. The characterisation theorem allows us to deduce a reconstruction result for any coherent theory with the property that all prime types can be realised in a countable, saturated model. For instance, {\omega}-stable coherent theories always have this property. We conclude by explaining how our results can be interpreted topos-theoretically, by relating them to the classifying topos and to the topos of types.
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math.CT 2026-06-29

SFC-categories fix sober spaces as adjunction fixed points

by Rui Prezado, Anna Laura Suarez

A general framework for the faithful pointfree representation of T₀-spaces

A general framework for pointfree representations of T0-spaces recovers the Banaschewski-Pultr sober and TD characterizations in a broader s

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We introduce a general framework for studying natural contravariant adjunctions that refine the adjunction between frames and spaces so that the fixpoints are $T_0$-spaces. Our objects of study are \textit{spatializable $\mathbf{Frm}$-concrete categories}, or \textit{SFC-categories}. These consist of a faithful functor $\mathcal O:\mathcal C\to \mathbf{Frm}$ equipped with an object $2_{\mathcal C} \in \mathcal C$, satisfying compatibility conditions that ensure that $(2_{\mathcal C},\mathbb{S})$ forms a dualizing object in the sense of Porst and Tholen, where $\mathbb{S}$ denotes the Sierpi\'nski space. Three important instances of pointfree $T_0$ spaces present in the literature fit into this framework: strictly zero-dimensional biframes, MT-algebras, and Raney extensions. We show SFC-categories are assembled in an ordered category -- a category enriched in preordered sets -- whose morphisms are suitable functors which preserve certain initial liftings. SFC-categories induce natural dual adjunctions, and morphisms between them will respectively induce suitable morphisms between these adjunctions. Motivated by the characterization of sober spaces as maximal objects in the fibers of $\Omega:\mathbf{Top}\to \mathbf{Frm}^{\mathsf{op}}$, and of $T_D$-spaces as the minimal ones, due to Banaschewski and Pultr, we study initial and terminal objects of fibers for an arbitrary SFC-category. We prove that the natural adjunction for fiber-initials has exactly the sober spaces as fixpoints, while for fiber-terminals contains at most $T_D$-spaces, recovering their results of in a much more general setting.
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quant-ph 2026-06-29

Quantum instruments compose by integrating channel-valued functions

by Robert I. Booth, Dominik Leichtle +2 more

Composing Quantum Instruments

The Okamura-Ozawa extension supplies monad multiplication, identifying quantum Markov kernels as the Kleisli category.

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We study the composition of classically-controlled quantum instruments--the natural quantum analogue of Markov kernels. Classically, Markov kernels compose by integrating one kernel against another. Defining this composition for quantum instruments with continuous outcomes requires an integral of quantum channel-valued functions with respect to a quantum instrument. We construct this integral in the Heisenberg picture using the Okamura-Ozawa normal extension to a von Neumann tensor product. This integral recovers the expected finite formula, preserves normal complete positivity and subunitality, and provides the multiplication for a monad governing the composition of quantum instruments. As an immediate consequence, we identify the category of quantum Markov kernels as the Kleisli category of this monad.
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cs.LO 2026-06-29

Quantum instrument monad generalizes state monad to quantum systems

by Tobias Fritz

The quantum instrument monad

It models computation-quantum interactions via a new integral on type I algebras and is shown to be a strong monad in both finitary and meas

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Monads are a ubiquitous structure in functional programming used for modelling computational effects. For example, the state monad models the effect of a computation interacting with a memory system. Here we introduce the quantum instrument monad $\mathcal{I}_\mathcal{A}$, which models the effect of a computation interacting with a quantum system with algebra of observables $\mathcal{A}$. It can be thought of as a noncommutative generalization of the state monad. We construct this quantum instrument monad in two versions: a finitary version on the category of sets and a measure-theoretic version on the category of measurable spaces (the latter under the assumption that $\mathcal{A}$ is a type I von Neumann algebra with separable predual). Both versions are strong monads. The construction of the measure-theoretic version is based on a new notion of integral of a quantum-operation-valued function against a state-valued measure.
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cs.LG 2026-06-29

Factor graph recovery shrinks probabilistic CRNs while keeping inference fixed points

by Mauricio Montes, Gregoire Sergeant-Perthuis

Reduction of Probabilistic Chemical Reaction Networks

Recovering hidden structure lets standard reduction methods cut network size without altering belief propagation results on surviving variab

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Programming adaptive behaviors at the cellular level is a long-standing goal that raises the question of how probabilistic computation can be implemented in biochemical systems. Chemical reaction networks (CRNs) provide such a substrate and have been shown to realize probabilistic models, including hidden Markov models and factor graphs, with dynamics reproducing Bayesian inference and belief propagation. However, encoding these algorithms typically requires prohibitively large reaction networks, and classical CRN reduction techniques do not directly apply. By recovering the factor graph structure encoded in Napp--Adams-compiled CRNs, we transport recent factor-graph reduction results to their chemical implementations, obtaining significantly smaller CRNs while preserving the belief-propagation fixed points on surviving variables.
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math.CT 2026-06-26

Lax grids let Gray tensor product arise by Day convolution

by Shai Keidar, Leor Neuhauser

The Gray Product of (infty, n)-Categories via Lax Grids

Segal sheaves on these pasting diagrams match Campion's construction and support Gray algebra on cobordism categories.

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We introduce a new model for $(\infty,n)$-categories as Segal sheaves on lax grids, which are pasting diagrams of lax cubes. This model allows for a direct construction of the Gray tensor product via Day convolution. We show that this agrees with Campion's construction of the Gray tensor product. These results will be applied in future work to equip the higher categories of cobordisms with a Gray-algebra structure given by the cartesian product of manifolds.
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math.CT 2026-06-26

Double-category monads are monadic over endomorphisms under colimit conditions

by Vasileios Aravantinos-Sotiropoulos, Theofilos Tsantilas +1 more

On categories of monads and comonads in double categories

Parallel and stable local colimits let the monads category inherit monadicity, cocompleteness and local presentability from the base double

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As is well known in the literature, the category Mon(V) of monoids in a monoidal category V satisfies various fundamental categorical properties, at least when the monoidal base V is correspondingly well-behaved. In particular, Mon(V) is monadic over V as soon as free monoids exist, while if V is cocomplete or locally presentable and its tensor b is sufficiently compatible with the appropriate colimits, then Mon(V) inherits the analogous property. In the present work, we extend such results to the context of double categories. More precisely, we identify conditions on a double category D under which one can show that the category Mnd(D) of monads in D is monadic over the category of endomorphisms End(D), is cocomplete or even locally presentable. We also tackle the issue of local presentability in the dual case Cmd(D) of comonads. In these results, our assumptions on the double category D revolve around notions of colimit, in particular those of parallel and stable local colimits, as well as a notion of local presentability of a double category which has been introduced in previous work.
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math.CT 2026-06-26

Isoregular theories yield accessible 2-categories with flexible limits

by Nicola Gambino, Giacomo Tendas

Isoregular theories, accessible 2-categories, and free constructions

This shows that many key 2-categories in category theory and logic admit flexible limits and new free constructions.

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We introduce isoregular theories, in which it is possible to express existential quantification up to unique isomorphism, as typically used to characterise category-theoretic universal constructions, such as limits. We then develop a functorial semantics for isoregular theories and prove that their 2-categories of models are accessible with flexible limits. We apply these results by showing that a number of 2-categories of interest in general category theory, categorical algebra, and categorical logic are models of isoregular theories, thereby establishing that they are accessible 2-categories with flexible limits and obtaining a number of new free constructions.
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math.CT 2026-06-26

Semantics fixes the geometry of categorified spectra

by Shih-Yu Chang

Intrinsic Geometry of Categorified Spectral Objects

Tangent complex, singular locus, inertia stack and curvature class arise canonically from duality and reconstruction.

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This paper develops the intrinsic geometry of the categorified spectral object $\mathfrak{Spec}(A)$ associated with an admissible operator-semantic system $A$ in the Categorified Spectral Duality (CSD) framework. We prove that the tangent complex, singular locus, inertia stack, and contextual curvature class are canonically determined by the duality adjunction between $\mathfrak{Spec}$ and the global sections functor, together with the reconstruction theorem identifying $A$ with the global sections of $\mathfrak{Spec}(A)$. The Canonical Geometry Theorem establishes that any CSD-compatible geometric structure is induced by the canonical datum consisting of $\mathfrak{Spec}(A)$, its tangent complex, its singular locus, its inertia stack, and its contextual curvature class; hence the geometry of $\mathfrak{Spec}(A)$ is intrinsic to the semantic structure of $A$. We prove that the tangent complex controls the deformation theory of $\mathfrak{Spec}(A)$ and satisfies a Hochschild realization, establishing a direct bridge between geometry and algebra. The assignment sending $A$ to its canonical geometric datum is functorial and Morita invariant, with explicit computations for the complex numbers and matrix algebras demonstrating that noncommutativity, detected by the inertia stack, is distinct from contextuality, which requires additional structures. Thus semantics determines geometry, and the intrinsic geometry of $\mathfrak{Spec}(A)$ provides a canonical geometric encoding of $A$ up to Morita equivalence.
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math.RT 2026-06-25

Cleft extension restriction functors match on singular equivalences under equivariance

by Miltiadis Karakikes

Equivariant Cleft Extensions and Singular Equivalences

The ordinary and equivariant versions induce singular equivalences together once the cleft extension lifts to the group action setting.

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We study the equivariant lifting of cleft extensions of abelian categories and its impact on singularity categories. Specifically, we establish the necessary framework for lifting a cleft extension to a G-equivariant cleft extension. Furthermore, we prove that a restriction functor associated to a cleft extension induces a singular equivalence if and only if its equivariant counterpart does. As a concrete application, we demonstrate that the skew group ring of a $G$-equivariant $\theta$-extension is isomorphic to a $\widehat{\theta}$-extension of the base skew group ring, allowing us to lift singular equivalences for these structures.
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cs.LG 2026-06-25

Framework transfers RL behavioral structures safely under abstraction

by Yivan Zhang, Ziyan Luo +1 more

Compositional Behavioral Semantics for State Abstraction in Reinforcement Learning

Local one-step dynamics enable sound definitions and transfer of value functions, bisimulations and metrics across state abstractions.

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State abstraction plays a key role in scaling reinforcement learning to complex but structured systems. In studying such systems, a wide range of behavioral structures have been studied in reinforcement learning, including value functions, invariants, bisimulation relations, and behavioral metrics. However, a general principle for determining what structures are provably preserved under state abstraction is still lacking. In this paper, we present a unified framework for defining and analyzing behavioral structures in reinforcement learning. Our framework provides a compositional way to specify behavioral semantics based on local, one-step descriptions of system dynamics. Using this framework, we establish results showing how behavioral structures can be safely transferred between abstract and concrete systems. We further show how to construct quantitative metrics from logical behavioral semantics with soundness guarantees. Together, these results provide a principled foundation for reasoning about behaviors under state abstraction in reinforcement learning and offer reusable definition and proof principles for a broad class of behavioral structures in reinforcement learning.
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math.CT 2026-06-24

Silting properties ascend via left adjoints under explicit conditions

by Simion Breaz, Andrei Marcuş +1 more

Migration of silting objects via adjoint pairs

Adjoint triples of triangle functors let silting and cosilting objects migrate between triangulated categories when the right conditions hol

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Consider an adjoint triple of triangle functors between two nice enough triangulated categories. In this paper, we are looking for conditions under which the silting, respectively, cosilting property ascends or descends via at most left, respectively, at most right adjoint.
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math.AT 2026-06-24

 is weakly equivalent to B haut(A)

by Jiahao Li

On the Classifying Space of Homogeneous Functors

The equivalence proves Tsopméné and Stanley's conjecture and extends Weiss classification to arbitrary simplicial model categories.

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Let $M$ be a manifold and let $\mathcal{M}$ be a simplicial model category. Given an object $A$ in $\mathcal{M}$, Tsopm\'en\'e and Stanley constructed a topological space $\hat{A}$ that classifies homogeneous functors of degree $k$ from the poset of open subsets of $M$ into $\mathcal{M}$. They showed that the set of weak equivalent classes of such functors that maps disjoint union of $k$ open balls to $A$ is in bijection with the set $[F_k(M), \hat{A}]$ of homotopy classes of maps out of $F_k(M)$, the unordered configuration space of $k$ points in $M$. In this paper, we begin a study of the space $\hat{A}$, and we prove that $\hat{A}$ is weakly equivalent to the classifying space $B\mathrm{haut}(A)$, where $\mathrm{haut}(A)$ is the simplicial monoid of self weak equivalences of $A$. This proves a conjecture of Tsopm\'en\'e and Stanley. Our result enables us to generalize the classification of homogeneous functors of Weiss for $\mathcal{M}=\mathcal{T}\mathrm{op}$ to any simplicial model category.
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math.CT 2026-06-24

Causal sufficiency defined by compatible Frobenius structures

by Sridhar Mahadevan

Infinitesimal Causality

Interventions deform copy/discard operations in Markov categories; Lie brackets check preservation of information flow.

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This paper introduces a categorical account of infinitesimal causality in Frobenius Markov categories equipped with tangent-bundle semantics. IDC captures the infinitesimal layer in which interventions act as tangent deformations of copy/discard structure. Two distinct Frobenius structures interact: (1) the categorical Frobenius algebra on classical variables encoding copying, comparing, and discarding; and (2) the geometric Frobenius integrability condition, namely involutive closure of the intervention distribution, distinct from the algebraic Frobenius structure. Categorical causal sufficiency is defined as the compatibility of these two notions. A key observation is that, for structural causal models, infinitesimal causality is most naturally formulated in the slice of deterministic mechanisms over exogenous variables, with visible stochastic kernels obtained only after pushforward. Interventions are tangent vectors that deform the Frobenius copy/discard operations; their Lie brackets measure whether this deformation preserves classical information-flow structure. Pearl's do-calculus is used as a guiding example of intervention identities: ignoring irrelevant interventions corresponds to counit invariance, action/observation exchange to coproduct compatibility with pushforward, and independence to involutive bracket closure of the visible intervention distribution.
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math.AG 2026-06-23

Tannakian duality unifies etale

by Loris De Vos

\'Etale Fundamental Groups -- a geometric and topological approach to fundamental groups in algebraic geometry

Fundamental groups arise as automorphisms of fibre functors, placing motivic Galois groups in the same framework.

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This thesis explores the notion of fundamental groups across three mathematical settings. We begin with the classical topological theory of covering spaces, highlighting its structural analogy with Galois theory. We then follow Grothendieck in transporting these ideas to algebraic geometry. The inadequacy of the Zariski topology motivates the \'etale topology, from which the \'etale fundamental group is constructed and compared to its topological counterpart via transcendental methods. Finally, we linearise the theory through Tannakian duality, where fundamental groups are recovered as automorphism groups of fibre functors on certain monoidal categories, a framework broad enough to encompass \'etale, topological, and motivic Galois groups alike.
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math.RT 2026-06-23

Singularity category of triangular matrix equals lower block

by Juan Andrés Orozco Gutiérrez, Valente Santiago Vargas

Recollements of Triangulated Categories and the Singularity Category of a Triangular Matrix Category

Recollement yields D_sg(Λ) ≃ D_sg(U) once homological conditions on T, M, U hold

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Introduced by Buchweitz, the singularity category of an algebra $A$ measures its homological singularity, vanishing if and only if $A$ has finite global dimension. This notion extends naturally to the context of $k$-categories. In this paper we study the singularity category of a triangular matrix category $\Lambda:=\left[ \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}\right]$. By utilizing the framework of recollements, we provide a characterization of this category, proving that when certain homological conditions are satisfied, there exists an equivalence of singularity categories $D_{sg}(\mathrm{Mod(\Lambda)})\simeq D_{sg}(\mathrm{Mod}(\mathcal{U}))$. This result generalizes the one obtained by Pin Liu and Ming Lu in [16].
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math.QA 2026-06-23

Virasoro c=1/2 module category equals Tambara-Yamagami over Z2

by Yuto Moriwaki

Conformal blocks, parenthesized braid operad, and c=1/2 Virasoro vertex operator algebra

Hypergeometric blocks and their analytic continuations fix the braiding that matches the known category.

Figure from the paper full image
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We review the construction of a pseudo-braided category structure on the $C_1$-cofinite module category of a vertex operator algebra using conformal blocks and analytic continuation along paths in configuration spaces. In the rational $C_2$-cofinite case, the pseudo-braided category is represented by tensor products and becomes a balanced braided tensor category. We then compute all four-point conformal blocks of the Virasoro vertex operator algebra of central charge $1/2$ in terms of hypergeometric functions. We explain how analytic continuation of these blocks determines the braiding and associator, and identify the resulting module category with the Tambara--Yamagami category over $\mathbb{Z}_2$ as a balanced braided tensor category.
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math.RT 2026-06-22

Codistinguished subcategories induce every exact structure

by Charley Cummings, David Nkansah

Codistinguished abelian subcategories

Abelian subcategories inside triangulated categories carry all possible exact structures as induced data.

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We introduce codistinguished abelian subcategories of triangulated categories, which generalise Jorgensen's proper abelian subcategories and are dual to Linckelmann's distinguished abelian subcategories. We show that they come equipped with a non-trivial exact structure induced by the ambient triangulated category, and that every exact structure on an abelian category arises in this way. We also show that fully faithful adjunction triples produce new codistinguished abelian subcategories from old.
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math.ST 2026-06-22

Invariance under reformulation pins inference to the KL divergence

by Raphaël Trésor, Thijs van de Laar +1 more

Reformulation Invariance and the Axiomatic Foundations of Inference

A single requirement that equivalent problem statements yield the same answer narrows all classical divergences to one.

Figure from the paper full image
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Maximum entropy, Bayesian updating, and exponential-family estimation are all instances of a common inference principle: selecting the measure or distribution that minimizes a divergence subject to the available constraints. Which divergence to use is usually decided by analytic convenience, by empirical performance, or by a set of axioms chosen to single it out, leaving open a basic question: why one divergence and not another? We answer it from a single requirement: an inference method should return the same answer whenever the same problem is presented in an equivalent form, for instance, after simply renaming its parts. This requirement alone forces inference to be the minimisation of a classical divergence, and each further reformulation it must respect tightens the admissible family one notch, narrowing the broad f-divergences to the {\alpha}-divergences and finally to the single Kullback-Leibler (KL) divergence. Mathematically, inference is recast from minimising a numerical functional to selecting a least element under a preorder on positive measures, a divergence being merely one numerical scale that reproduces that preorder. The reformulations are the morphisms of a category of inference problems, and the invariance requirement says the inference operator is a covariant functor into the category of statistical models of Cencov, mirroring his characterisation of the Fisher metric. The representation is proved on finite spaces and lifted to general measurable spaces by an elementary closure, covering discrete and continuous spaces alike. Earlier axiomatisations, such as those of Shore-Johnson and Csiszar, postulate their consistency axioms directly and only on finite alphabets; here the axioms follow from reformulation invariance alone.
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math.CT 2026-06-22

Biased, unbiased and homotopy symmetric monoidal categories match

by Matteo Galbiati

A (not so) short note: the equivalence of various notions of symmetric monoidal category

Groupoid-enriched categories built from each notion are equivalent, so theorems transfer directly between definitions.

abstract click to expand
In this work, intended to be a companion note to a future preprint, we give a proof of the fact that the classical (biased) notion of symmetric monoidal category, the notion of unbiased symmetric monoidal category, and the notion of homotopy symmetric monoidal category are equivalent in a precise sense (in that suitably defined groupoid-enriched categories having, respectively, biased, unbiased, and homotopy symmetric monoidal categories as objects are equivalent as enriched categories).
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math.CT 2026-06-22

Double simplicial objects produce spectral sequences in semi-abelian categories

by Florent Afsa

Spectral Sequences in Semi-Abelian Categories

An exact couple is built directly from the simplicial data once the category meets the semi-abelian axioms.

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In this paper, we define the notion of a spectral sequence in the context of semi-abelian categories in the sense of Janelidze, M\'arki and Tholen. We show that from a double simplicial object, one can construct an exact couple, which gives rise to a spectral sequence. This extends Quillen's result on double simplicial groups to every semi-abelian category.
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math.CT 2026-06-22

Free internal frames constructed on presheaves of frames

by Vasileios Aravantinos-Sotiropoulos, Panagis Karazeris +1 more

Locales in presheaf toposes vs. presheaves of locales

The resulting left adjoint translates internal local compactness into conditions on sections and transition maps.

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By a well-known characterisation, in a presheaf topos every internal suplattice is a presheaf of suplattices, but not every presheaf of suplattices is an internal suplattice (and similarly for frames). In this paper, we construct the free internal suplattice/frame on an presheaf of suplattices/frames, yielding a left adjoint to the forgetful functor from the respective internal structures to presheaves of structures. The description of this left adjoint has also appeared in recent work of Henry and Townsend, in connection to a different universal property, namely that of turning a lax natural transformation between poset-enriched functors to a strict one. As an application of our construction, we investigate conditions on frames internal to a presheaf topos, such as being locally compact, compact, stably locally compact or Hausdorff, in terms of properties of their sections in the base topos. In the first three cases, it is necessary that all the sections have the respective properties, while the Hausdorff property is not transferred to the sections. Moreover for local compactness it is necessary that the transition maps preserve the way-below relation. Finally, for an internal locally compact frame in presheaves we analyse the connection of its way-below relation to the respective relations of its sections.
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math.CT 2026-06-22

First example separates right from left integrality

by Yaroslav Kopylov, Max Zinchenko

Some Remarks About Integral Categories

A concrete category meets right-integral rules but not left-integral rules, with new criteria to test each side separately.

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We consider integral categories, a class of categories that gained importance in the last three decades in the categorical foundations of algebra and functional analysis. We discuss some familiar criteria for integrality and prove new criteria for one- and two-sided integrality. For the first time in the literature, an example is given of a right but not left integral category.
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math.CT 2026-06-19

Cofibrant A-sets yield branching spaces independent of epsilon

by Philippe Gaucher

Branching spaces of transverse sets

Coend over c-direct category from thick cubes gives homotopy-invariant result for transverse sets.

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A c-direct category is a small category equipped with an ordinal degree function such that every morphism is level or degree-raising. Every c-direct category is c-Reedy. The c-Reedy model structure on any functor category from a c-direct category to a model category coincides with the projective model structure. In this framework, a realization functor is a colimit-preserving functor satisfying some mild homotopical conditions from the category of presheaves on a c-direct category with cofibrant representables to a model category. We prove that any two such realization functors are weakly equivalent on cofibrant presheaves. For categories of cubes, we prove that thick categories have cofibrant representables. As an application, we introduce the $\varepsilon$-branching space of an $\mathcal A$-set for any thick category of cubes $\mathcal A$. It is obtained as a coend over a c-direct category with cofibrant representables constructed from $\mathcal A$. We prove that, on free $\mathcal A$-sets generated by precubical sets, this new definition coincides with the earlier one. We prove that, for cofibrant $\mathcal A$-sets, the resulting space is independent of $\varepsilon$ up to homotopy.
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math.CT 2026-06-19

Fiber bundles over categories reduce to constant fibers

by Isaac Carcacía-Campos

Fiber bundles over small categories

Monodromy representations of the fundamental groupoid classify them up to isomorphism and set their gauge groups as centralizers.

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The theory of fiber bundles over small categories is developed, viewing them as locally constant functors to the category of small categories. The Grothendieck construction yields a total category equipped with a projection that is a bifibration. We show that, up to natural isomorphism, every such bundle admits a constant fiber, and that the monodromy gives a representation of the fundamental groupoid in the automorphism group of the fiber, which allows the classification of fiber bundles up to isomorphism. The gauge group of the bundle is proved to be isomorphic to the centralizer of the monodromy subgroup. We then give a precise analysis of sections and (lax) fixed points of the fiber bundle. Beat points for functors are introduced, and it is proved that every fiber bundle with some finiteness and acyclic conditions admits a minimal core, using a rigidity lemma for finite acyclic categories. These concepts are illustrated with explicit examples.
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math.RA 2026-06-19

Nijenhuis Lie 2-algebras equivalent to 2-term Nijenhuis L∞-algebras

by Apurba Das

Nijenhuis Lie 2-algebras

Semidirect products carry the structure and the two representation categories coincide.

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In this paper, we first introduce Nijenhuis Lie 2-algebras as the categorification of Nijenhuis Lie algebras. We prove that the category of Nijenhuis Lie 2-algebras is equivalent to the category of 2-term Nijenhuis $L_\infty$-algebras. Next, given a Nijenhuis Lie algebra, we introduce the notion of a 2-representation and show that the corresponding semidirect product inherits a Nijenhuis Lie 2-algebra structure. On the other hand, we consider a $2$-term representation up to homotopy of a Nijenhuis Lie algebra and obtain a $2$-term Nijenhuis $L_\infty$-algebra as the semidirect product. Finally, we show that the category of $2$-representations and the category of $2$-term representations up to homotopy of a Nijenhuis Lie algebra are equivalent.
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math.RT 2026-06-18

Hopfological algebra gains a higher-categorical generalization

by Juan Omar Gómez, Gustavo Jasso +1 more

Hopfological algebra, revisited

Module categories in monoidal infinity-categories refine the foundations and extend the theory to new monoidal settings.

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We propose an $\infty$-categorical approach to Khovanov--Qi's Hopfological algebra that, in particular, refines several foundational aspects of the theory by recasting the previous constructions in terms of $\infty$-categories of modules in monoidal $\infty$-categories. This perspective leads to a more general variant of Hopfological algebra that takes place over an arbitrary rigidly-compactly generated symmetric monoidal stable $\infty$-category, which we also outline in the article. In the appendix, we compare the construction of Hopfological derived categories to that of Holm--J{\o}rgensen's $Q$-shaped derived categories.
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0
cs.AI 2026-06-18

Monadic tensors embed neural symbols in unified truth semantics

by Daniel Romero Schellhorn, Till Mossakowski +1 more

NeSyCat Torch: A Differentiable Tensor Implementation of Categorical Semantics for Neurosymbolic Learning

The lazy log-tensor monad delivers differentiable training that beats LTN and DeepProbLog on MNIST addition while keeping the framework para

abstract click to expand
Neurosymbolic semantics is fragmented: classical, fuzzy, probabilistic and neural systems each define truth by their own inductive rules. NeSyCat, extending ULLER, subsumes them under a single inductive definition of truth, parametric in a strong monad and an aggregation structure on truth-values. NeSyCat has so far lacked an account of predicates and functions learned by neural networks. We provide NeSyCat Torch as the missing link and interpret computational symbols via neural networks, implementing the framework in probabilistic programming and tensor-based backends. We use the distribution monad for reference semantics and metric evaluation, and complement it by a monad for numerically stable, differentiable training: the lazy log-tensor monad over the log-semiring. For efficient training in batches, we furthermore employ a batch monad. The axioms are the source code: written once in monad-based do-notation, monadic bind performs marginalisation, lazily pruning unneeded branches. On MNIST addition, our HaskTorch, JAX, and PyTorch implementations outperform LTN and DeepProbLog in speed and accuracy, while achieving nearly the accuracy of DeepStochLog. However, unlike DeepStochLog, we stay in a uniform framework that applies to many first-order NeSy approaches. Namely, the construction is parametric in the monad; instantiating it with, e.g., the Giry monad extends the approach to continuous probability (working out a neural representation here is left for future work).
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math.CT 2026-06-18

Categorical methods produce dual Amplituhedron

by Julien Dalpayrat-Glutron

Brave new categorical spectral positive Schubert geometry and the categorical Dual Amplituhedron

The Amplituhedron viewed as a functor between higher categories gains a non-trivial dual with De Rham volume for amplitude calculations.

Figure from the paper full image
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This ArXiv preprint of my doctoral dissertation, which, at this stage, has not yet been accepted by the doctoral thesis committee, is intended both to lay the groundwork for a series of papers, and to confirm the existence of a first proposed solution to the "Dual Amplituhedron" problem posed in 2014 by Princeton physicists N. Arkani-Hamed and J. Trnka, a mathematical object which encapsulate the calculation of scattering amplitudes in high-energy particle colliders. The first part of this thesis deals with a rewriting of the positive real Grassmannian and performing the singular gluing of its positroid varieties in a new way via Spectral Algebraic Geometry of J. Lurie on "structured" spaces and a categorification of his "Tannaka Duality for Quasi-coherent Stacks", finding in this formal moduli problem, a compact yet holistic formulation via perverse intersection complexes of P. Deligne. This algebraico-geometric perspective paired with a synthetic differential-geometric perspective, namely the Differential Cohesion of B. Lawvere and U. Schreiber, subsumes infinitesimal thickenings, crystalline cohomology of De Rham stacks, the "Modalities of Structured Geometries" and the unification of their cohomologies of the underlying concrete topological etale algebraic space. The second part uses this rewriting of positive Schubert geometry and "The Cohomology of Brauer-Grothendieck Spaces" of B. To\"en and B. Antieau, to show that, instead of the Grassmannian spectral Deligne-Mumford stack which is autodual in the infinity-category of prestable infinity-categories of modules on E\infty-ring spectra, the Amplituhedron, categorified in a functor between infinity topo\"i, possesses a concrete non-trivial dual, and has a De Rham volume, facts of interest for the expected Duality between the Standard Model of particles and String Theory. This new construction yields the Dual Amplituhedron.
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cs.LO 2026-06-18

Complete axioms for probabilistic Boolean circuits as Markov kernels

by Filippo Bonchi, Cipriano Junior Cioffo

Completeness for Probabilistic Boolean Tapes

Partial-circuit and rig-category-tape completeness results yield a full calculus for finite probabilistic program equivalence.

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Probabilistic Boolean circuits have recently been proposed as a string-diagrammatic foundation for finite probabilistic programming. In this paper, we present a complete set of axioms for their semantics in terms of Markov kernels. Our approach is based on two intermediate results: completeness for \emph{partial} Boolean circuits and completeness for probabilistic Boolean tapes, a diagrammatic language for rig categories.
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math.FA 2026-06-17

Cross-connection semigroup equals finite-rank operator algebra

by A. Anju, P. G. Romeo

Cross-connections of the normed algebra of finite rank bounded operators on a Hilbert space

Cones from the normal category of subspaces yield a normed algebra isomorphic to bounded finite-rank operators on Hilbert space.

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In this article we examine the cross-connections of the normal category of finite dimensional subspaces of a Hilbert space and it's dual space. Further, we describe the cross-connection semigroup of cones, which is a normed algebra isomrphic to the normed algebra of finite rank bounded operators on a Hilbert space. We also characterize compact operators and their spectrum by the normal cones in the normal category of proper subspaces of a Hilbert space.
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math.CT 2026-06-15

Coextensions of monoid schemes form stacks of categorical groups

by Ilia Pirashvili

Deformation Theory of Monoid Schemes I

The construction uses systems of abelian groups to replace naive exactness and produces a classification via cohomology for sheaves before l

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The aim of this paper is to develop a deformation theory of monoid schemes, generalising the approach developed by Grillet. The core idea of this approach is to introduce the notion of a system of abelian groups, as the naive approach to exactness does not work for monoids. We first study the case of monoid sheaves (functors over a poset into the category of monoids) and prove a classification theorem in this setting, showing that the coextensions of a monoid functor with a system of abelian groups is a symmetric categorical group and equivalent to the one obtained by the abelian group homomorphism $[\mathcal{C}^0 \to \mathsf{ker}\partial^1]$, thereby linking with cohomology of certain types of complexes, as expected. We then move towards monoid schemes, which are a type of a monoid sheaf, but where localisations now allow us to develop our most noteworthy result: We show that coextensions can be seen in a natural way as a stack of symmetric categorical groups. We will mention a few mild implications of this, but leave the deeper uses of stack theory in this setting for later papers.
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cs.CL 2026-06-12

Operads model LLM question decomposition as algebras

by Nathaniel Bottman, Kyle Richardson

Operads for compositional reasoning in LLMs

A questions operad turns decomposition into substitution; consistency across tree collapses tracks accuracy on multi-hop tasks.

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Question decomposition, i.e. breaking a complex query into simpler sub-queries whose answers are composed to produce a final answer, is a widely used strategy for improving LLM reasoning, yet it currently lacks a rigorous mathematical foundation. In this paper, we propose operads, mathematical structures that model many-in, one-out operations and compositions thereof, as a natural framework for describing question decomposition. We define the questions operad $Q$, in which operations correspond to question templates and composition corresponds to substitution of sub-answers, and show how QA models can be interpreted as algebras over $Q$. Beyond reframing existing practice, this operadic perspective points toward new methods, in particular a notion of operadic consistency, which measures whether a QA model's answers agree across the partial collapses of a question decomposition tree. Empirical evaluation of operadic consistency is reported in our companion paper (Bottman, Liu, and Richardson, 2026), which finds it strongly correlated with accuracy across twelve LLMs and four multi-hop QA datasets and outperforming standard temperature-based self-consistency baselines. We argue that operads are the natural mathematical home for question decomposition, and that invariants such as operadic consistency open new directions for analyzing and improving the reliability of multi-step reasoning.
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math.RT 2026-06-12

Poset isomorphisms link silting subcategories to s-torsion pairs

by Liangwei Huang, Haicheng Zhang

Silting subcategories and (co)torsion pairs associated to extended hearts

The bijections also equate hereditary cotorsion pairs and extend to tau-tilting pairs and silting complexes over dg algebras.

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We establish the poset isomorphisms between $(d+1)$-term silting subcategories, functorially finite $s$-torsion pairs in the $d$-extended heart, and hereditary complete cotorsion pairs in a suitable subcategory. As an application, we also give dg algebra versions of these bijections, which establish the poset isomorphisms between $\tau$-tilting pairs, $(d+1)$-term silting complexes, and functorially finite $s$-torsion pairs.
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math.CT 2026-06-11

Three monoidal structures force braidings to symmetries

by Eugenia Cheng, Alexander S. Corner

A higher-order Eckmann-Hilton argument

Pairwise interchange on every pair makes derived braidings into symmetries, so hom-categories of n-degenerate higher categories are symmetri

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We give a higher-order higher-dimensional Eckmann-Hilton argument that is entirely algebraic. First we give an explicit argument showing that if we have two monoidal structures on a category with suitable interchange, we can derive a braiding on either of the monoidal structures. Then we show that given third monoidal structure, with suitable pairwise interchange on any pair of monoidal structures, each canonical braiding is forced to be a symmetry. As a motivating example, we show that for $n \geq 3$ any $n$-degenerate semi-strict $(n + 1)$-category has three suitably coherent monoidal structures on its single hom-category, thus the hom-category has the structure of a symmetric monoidal category.
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math.AT 2026-06-11

Relative dendroidal Rezk nerve ties to operad localization

by Kensuke Arakawa, Victor Carmona +1 more

Relative dendroidal Rezk nerve and applications

The relation generalizes an earlier theorem and supplies proofs for cyclic operads plus factorization algebras on spheres.

Figure from the paper full image
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We extend the dendroidal Rezk nerve to the setting of relative $\infty$-operads. Our main theorem relates it to localization of $\infty$-operads, generalizing a theorem of Mazel-Gee. By exploiting the relation, we obtain a surprisingly effective tool to prove localization results in operadic contexts. As applications, we obtain a number of new results on operadic localizations, including a generalization of Willwacher's recent result on cyclic operads and operadic modules, and a description of locally constant factorization algebras on spheres in terms of discrete geometry.
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math.CT 2026-06-11

Categorical Hopf map factors through Hopf fibration and S3 gerbe

by Ali Khalili Samani

Categorical Hopf map

New bundle over S2 uses Ganter's categorical circle as fiber and recovers String(3) symmetries by conjecture

Figure from the paper full image
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We introduce the categorical Hopf map as a categorical principal bundle over the two-dimensional sphere with fibre the categorical circle of Nora Ganter. We investigate its connection to the Hopf map. We present a factorisation of the categorical Hopf map through the Hopf map and the basic bundle gerbe over the three-dimensional sphere. We discuss three equivalent constructions for the basic bundle gerbe over the three-dimensional sphere. Finally, we conjecture that the categorical group String(3) is equivalent to the categorical group of symmetries of the categorical Hopf map.
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math.RT 2026-06-11

Hochschild homology matches singular version for symmetric algebras

by Yu Wang, Xiaozhuan Liang

Singular Hochschild complex and Cartan matrix

For basic algebras the Cartan matrix is symmetric precisely when the dual mixed complex shifts by -1; counterexamples exist for general Frob

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If A is a symmetric algebra, then Hochschild homology of the dg enhancement of the singularity category of A agrees with singular Hochschild homology of A. For a basic finite dimensional k algebra A, the Cartan matrix of A is symmetric if and only if the k dual of the mixed complex of the dg enhancement of its singularity category is isomorphic to its shift by -1. We provide two counterexamples to show that neither result holds for general Frobenius algebras.
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math.QA 2026-06-10

O(2) subgroups label distinct 2-categorical Hilbert spaces

by Giovanni Ferrer, Lukas Müller +2 more

The many faces of higher Hilbert spaces

Fixed points under an O(2) action on 2-vector spaces recover the module categories of C*, W*, and H*-algebras via different choices of G.

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Finite-dimensional operator algebras can be viewed as $\mathrm{C}^*$, $\mathrm{W}^*$, or $\mathrm{H}^*$-algebras, leading to different notions for their categories of modules and correspondence 2-categories. In this article, we show how these differences can be understood systematically using the notion of $G$-dagger category from arXiv:2403.01651 for different subgroups $G\leq O(2)$. To do so, we first introduce $G$-Hermitian $2$-vector spaces using fixed points of a certain $O(2)$-action on $2\mathsf{Vect}$. We then propose criteria for when such pairings are `positive', generalizing the passage from Hermitian vector spaces to Hilbert spaces. Finally, we outline an inductive approach to defining higher Hilbert spaces in arbitrary dimension, suggesting an extension of these ideas beyond the 2-categorical setting.
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math.CT 2026-06-10

Cellular generation equals almost everywhere quasieffectiveness

by Sean Cox, Mark Kamsma +1 more

Cellular generation revisited

In locally presentable categories the smallness property holds precisely when restrictions to almost all subuniverses produce quasieffective

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Cellular generation, which generalises cofibrant generation, is an important categorical smallness condition on a class of morphisms. A general challenge is to determine whether a given class of morphisms $\mathcal{M}$ is cellularly generated, in which $\mathcal{M}$-effective squares are often useful. These are commuting squares consisting of morphisms in $\mathcal{M}$, so that the induced morphism from the pushout square is also in $\mathcal{M}$. When we drop the requirement that the vertical morphisms in the square are in $\mathcal{M}$ we obtain the weaker notion of $\mathcal{M}$-quasieffective square. We prove that, in a locally presentable category, $\mathcal{M}$ is cellularly generated if and only if $\mathcal{M}$ is almost everywhere quasieffective. The latter is a set-theoretic condition stating that for almost every partial elementary set-theoretic subuniverse $\mathfrak{N}$, we have that restricting any morphism in $\mathcal{M}$ to $\mathfrak{N}$ yields an $\mathcal{M}$-quasieffective square. For locally finitely presentable categories this yields an additional categorical characterisation in terms of filtrations of $\mathcal{M}$-quasieffective squares. If we additionally assume that $\mathcal{M}$ is continuous (i.e., the corresponding wide subcategory is closed under directed colimits) then we obtain a stronger characterisation of cellular generation in terms of accessibility of the category of $\mathcal{M}$-effective squares. This improves on a theorem by Lieberman, Vasey, and the third author.
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math.CO 2026-06-10

Orthomodular posets form full coreflective subcategory of strong orthoposets

by John Harding, Gejza Jenda +1 more

From orthoposets to orthomodular posets

Order modification on any strong orthoposet yields an orthomodular poset on the same set; the functor is right adjoint when restricted to or

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We show that the category of orthomodular posets is a full coreflective subcategory of the category of strong orthoposets, those orthoposets in which any two orthogonal elements have a join. This coreflection is obtained by building from a strong orthoposet $P$, an orthomodular poset with the same underlying set and same orthocomplementation as $P$, but with modified order. This coreflector restricts to a functor from the category of ortholattices to the category of orthomodular posets, and this functor is right adjoint.
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math.CT 2026-06-10

Representability links perfect objects to coherent objects

by Giovanni Rossanigo

Kernel theorems for rigidly-compactly generated infty-categories

Two theorems express linear functionals on these objects via the objects themselves inside rigidly-compactly generated ∞-categories.

Figure from the paper full image
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We prove two representability results for rigidly-compactly generated $\infty$-categories and functors between them. The first one represents contravariant linear functionals out of a category of perfect objects with values in a category of (pseudo)-coherent objects in terms of (pseudo)-coherent objects. The second one represents covariant functionals out of coherent objects with values in a category of coherent objects in terms of perfect objects. The techniques used belong to the realm of "functional analysis" of presentable stable categories and ultimately depend on the interaction between three notion of finiteness, namely compactness, dualizability and coherence. These results apply to $\mathbb{E}_\infty$-ring spectra, quasi-proper maps of quasi-compact quasi-separated schemes and certain spectral algebraic spaces. We also reformulate Grothendieck duality in terms of internal left adjoints.
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math.AT 2026-06-10

Sheaves are the unique six-functor formalism on LCH spaces

by Ulrich Bunke, Marco Volpe

A characterization of sheaves among six functor formalisms on LCH

A list of natural properties isolates sheaves, so every continuous formalism reduces to Shv with coefficients from its value at a point.

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Let $\mathcal{C}$ be any stable presentably symmetric monoidal $\infty$-category. In this paper, we characterize $\mathrm{Shv}(-,\mathcal{C})$ on locally compact Hausdorff spaces as the unique six functor formalism satisfying a list of very natural properties. As a consequence, we deduce that every continuous six functor formalism $D$ in the sense of Zhu is equivalent to $\mathrm{Shv}(-, D(\mathrm{pt}))$.
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math.CT 2026-06-09

Span functor from double ∞-categories admits squares right adjoint

by George Raptis, Wolfgang Steimle

The span-squares adjunction

The adjunction supplies new proofs that Q-, S-, cobordism and squares constructions give equivalent algebraic K-theory.

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We show a universal property of the span $\infty$-category that yields a description of functors defined on this category. For this, we view the span construction as a functor from double $\infty$-categories to $\infty$-categories, and show that this functor admits a right adjoint defined by the double $\infty$-categories of squares. Using this adjunction, we obtain new proofs of the equivalences between different models of algebraic $K$-theory, given by the $Q$-, the $S$-, the cobordism model, and the squares construction.
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math.CT 2026-06-09

Monoid action equivalences become adjunctions in category fibers

by Stefano Ambra

Actions, semidirect products and crossed semimodules in the category of small categories with a fixed set of objects

The equivalence between actions and split extensions holds only for single-object cases; crossed structures follow a parallel pattern in eac

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We generalize to the fibres of the fibration $\mathcal{O}\colon\mathbf{Cat}\rightarrow\mathbf{Set},$ defined by mapping a small category $\mathbb{X}$ to its set of objects $X_0=ob(\mathbb{X}),$ the classical notions of action and semidirect product of monoids. We prove that the equivalence between monoid actions of a monoid $Y$ and Schreier split extensions on $Y,$ which is well known to generalize the equivalence between actions and split extensions for groups, is an instance of a broader adjunction between Schreier points and actions in the fibres $\mathcal{O}^{-1}(B).$ This adjunction is an equivalence if and only if $B=1,$ i.e., for the category $\mathbf{Mon}$ of monoids. Similarly, we prove that there is an adjunction (which, in the case of monoids, results in a known equivalence due to Patchkoria) between Schreier internal categories in the fibres $\mathcal{O}^{-1}(B)$ and the category of crossed semimodules in $\mathcal{O}^{-1}(B).$ The latter are defined by translating in $\mathcal{O}^{-1}(B)$ the notion of crossed semimodule in $\mathbf{Mon}.$ Eventually, we prove that, by defining crossed modules appropriately, this last adjunction yields an equivalence between crossed modules and Schreier internal groupoids in the fibres of $\mathcal{O}.$
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math.AT 2026-06-09

Magnitude-path spectral sequence gives homotopy theories at every page

by Muriel Livernet, Emily Roff +1 more

Homotopy theories via the magnitude-path spectral sequence

r-quasi-isomorphisms and r-cofibrations turn each page into a metric homology theory with Mayer-Vietoris and explicit colimits via Brown cat

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We introduce a family of homotopy theories for generalized metric spaces with natural number distances, via the magnitude-path spectral sequence (MPSS). The first page of the MPSS is known as magnitude homology; the second page is known as bigraded path homology, and contains GLMY path homology as its top row. For each natural number r, we define a class of maps of metric spaces called r-quasi-isomorphisms: those maps that induce a quasi-isomorphism at page r of the MPSS. We show that every page of the spectral sequence satisfies a suitable metric analogue of each of the Eilenberg-Steenrod axioms. In particular, we introduce the notion of r-cofibration and prove a Mayer-Vietoris theorem for page r with respect to r-cofibrations. We establish a family of Brown category structures on generalized metric spaces which allow us to explicitly compute homotopy colimits. We apply this to describe r-suspension and r-spheres of dimension n, and compute their spectral sequences. Finally we prove that for r = 1 the entire theory restricts to directed graphs.
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math.CT 2026-06-09

Centre comonad empties state spaces of non-commutative algebras

by Joey Woo

The Degeneracy of the Centre Comonad Model and the Precomposition Obstruction for Quantum Modalities on Presheaf Topoi

It sends their representables to the empty presheaf, makes Day convolution cartesian, and collapses linear logic to classical logic.

abstract click to expand
The centre comonad model provided the first concrete cohesive linear $\infty$-topos, settling an open problem of Schreiber. However, the model is degenerate: the quantum modality annihilates all non-commutative algebras, and the associated linear logic collapses to classical cartesian logic. In this paper we give a complete mathematical diagnosis of this degeneracy. We prove that the centre comonad sends the representable sheaf of a simple non-commutative algebra to the empty presheaf, and that the state space of any such algebra is empty. We then prove that the Day convolution on the classical core is cartesian, forcing the Seely isomorphism to hold trivially and collapsing the linear logic. We isolate the structural reason behind this collapse: whenever the opposite of the classical core is monoidally equivalent to a cartesian monoidal category, any coreflective precomposition comonad will exhibit the same degeneracy. We conclude that a non-degenerate quantum modality must be constructed without precomposition, and we briefly discuss possible directions.
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math.CT 2026-06-09

Residue field generation classifies DG subcategories by spectrum subsets

by Leovigildo Alonso, Ana Jeremías +1 more

Colocalizing subcategories on differentially graded algebras

Localizing and colocalizing subcategories of D(A) biject with subsets of Spec(H^0(A)) when the generation condition holds.

abstract click to expand
Let $A$ be a bounded non positive commutative differential graded algebra $A$. Let $\mathbf{D}(A)$ its derived category of DG-modules. If $\mathbf{D}(A)$ is generated by the DG-modules corresponding to the residue fields of the ordinary ring $H^0(A)$ then its localizing subcategories and its colocalizing subcategories are in bijection with the subsets of $\textrm{Spec}(H^0(A))$. These results generalize well-known theorems by A. Neeman (from 1992 and 2011, respectively), because any Noetherian ring satisfies this condition.
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math.AT 2026-06-08

Weak split extensions match continuous sum structures on B × A

by María V. Ferrer, Salvador Hernández-Muñoz +1 more

Weak split extensions of topological Abelian groups

The group of such extensions equals the set of continuous ways to add elements while keeping B a subgroup and A a quotient, with a cocycle d

abstract click to expand
In the category of topological Abelian groups, we consider the usual notion of an extension $E=(B \to X \to A)$ of $B$ by $A$, together with the notion of a weakly split extension, i.e., an extension for which the projection $X \to A$ admits a continuous section $A \to X$. Given a weakly split extension $E$, the topological Abelian group $X$ is homeomorphic to $B \times A$, although in general it is not algebraically isomorphic to $B \times A$. For two topological Abelian groups $A$ and $B$, we study the Abelian group $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ of weakly split extensions of $B$ by $A$, modulo extension isomorphisms. We show that $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ can be described as the group of all continuous sum structures defined on the product space $B \times A$ (up to topological isomorphism), with $B$ as a topological subgroup and $A$ as a topological quotient. We also provide an alternative description of $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ as a quotient $Z_c(A,B)/B_c(A,B)$, where $Z_c(A,B)$ consists of cocycles given by continuous maps $A \times A \to B$, and $B_c(A,B)$ denotes the corresponding coboundaries. Furthermore, we compare $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ with the group of standard extensions $E_A(A,B)$, where $A$ and $B$ denote the underlying Abelian groups, and relate these constructions by means of a six-term exact sequence. Although the Bohr topology of discrete Abelian groups has been investigated by many workers, there still remain many parts that are not well understood. Here, as an application of the methods developed in the paper, new examples of nontrivial $ws$-extensions for discrete Abelian groups equipped with the Bohr topology are provided and some related open questions are also proposed.
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math.QA 2026-06-08

Braided structure on vertex modules extends to colimit completion

by Robert McRae, Cris Negron

Cocompletions for non-abelian vertex tensor categories

The unique extension holds inside generalized modules without assuming abelianness or compactness, supporting applications to VOA extensions

abstract click to expand
It was recently shown by Huang that the category of $C_1$-cofinite modules for any vertex operator algebra $V$ admits a natural braided monoidal structure. Here, we show that this structure extends uniquely to a vertex algebraically natural braided monoidal structure on the completion of the category of $C_1$-cofinite $V$-modules under filtered colimits, within the ambient category of all generalized $V$-modules. A critical point here is that we do not assume the category of $C_1$-cofinite $V$-modules is abelian or that $C_1$-cofinite modules are compact in the cocompletion, since these properties are not known to hold in general. Our results have many applications in the representation theory of vertex operator algebra extensions, since many vertex operator algebras can be realized as objects in the filtered colimit completion of the category of $C_1$-cofinite modules for a vertex operator subalgebra. Generalizing from the specific vertex algebraic setting, we also establish existence and uniqueness for extensions of monoidal structures along a dense inclusion $\mathscr{C}_0 \to \mathscr{C}$ from an abstract, essentially small monoidal category into a well-structured cocomplete target.
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math.GN 2026-06-08

Convex powerdomains reflect continuity and quasicontinuity but do not preserve the latter

by Yuxu Chen

Directed Convex Powerspaces and Convex Powerdomains

Upper and convex cases now complete the full preservation and reflection profile across all powerdomains on dcpos.

Figure from the paper full image
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It is known that lower powerdomains preserve and reflect both continuity and quasicontinuity, while the preservation of quasicontinuity by upper and convex powerdomains had long been open. Directed spaces provide a topological extension framework for dcpos. Powerdomains of dcpos can be characterized as the $D$-completions of the corresponding directed powerspaces. Using this observation, the authors proved in 2024 that upper powerdomains do not preserve quasicontinuity. In this paper, we prove that directed convex powerspaces and convex powerdomains do not preserve quasicontinuity, but they do reflect both continuity and quasicontinuity. We also prove that upper powerspaces and upper powerdomains reflect quasicontinuity. Together with the known results for lower and upper powerdomains, these arguments give the complete preservation and reflection profile of the lower, upper and convex powerdomains for continuity and quasicontinuity.
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math.CT 2026-06-08

Ehresmann connections defined via tangent-bundle splittings

by Geoffrey Cruttwell, Marcello Lanfranchi

Ehresmann connections in tangent categories

The same splittings produce parallel transport and curvature obeying the Bianchi identity in any tangent category.

abstract click to expand
The theory of connections is at the very core of differential geometry. Discovered by Levi-Civita and Christoffel and later studied by Cartan, Koszul, and others, connections appear in their most general form under the name of Ehresmann connections. An Ehresmann connection consists of a splitting of the tangent bundle of a submersion into the vertical sub-bundle and a given horizontal distribution. In this paper, we generalize Ehresmann connection to a categorical setting called tangent categories. Initially introduced by Rosick\'y in 1984 and later generalized by Cockett and the first author in 2014, tangent categories provide a categorical framework to study geometry that extends well beyond smooth manifolds, including algebraic geometry and non-commutative geometry. In this paper we introduce and study Ehresmann connections in the context of tangent categories. We give various equivalent formulations in term of full and abstract connections and prove that they generalize Koszul connections. We also define parallel transport and curvature for such connections, and prove the structural equation and the Bianchi identity for the curvature.
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math.CT 2026-06-05

Planar higher-rank trees have rank at most four

by David Pask

Finite connected singly connected locally convex non-degenerate cases cannot exceed rank four, via K5 subdivision

abstract click to expand
We prove that a finite, connected, singly connected, locally convex higher-rank tree whose $1$-skeleton is planar and which is \emph{non-degenerate}, in the sense that every edge of each colour forms a commuting square with every other colour, has rank at most four. Under these hypotheses this establishes the planarity conjecture stated in \cite{Pask}. The obstruction side of the argument uses only the non-planarity of $K_5$; it makes no appeal to the four-colour theorem. The engine is a monotonicity property of the set of colours emitted at a vertex (``backward propagation''), which forces, in any finite singly connected non-degenerate $k$-graph, a single vertex emitting all $k$ colours; once $k\ge 5$, local convexity manufactures a subdivision of $K_5$ at such a vertex.
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math.CT 2026-06-05

Three invariants fix every spectral propagation rule

by Shih-Yu Chang

A Universal Theory of Spectral Propagation for Compositional Operator Networks

Operadic spectra, derivatives and residues alone determine how spectra combine in any compositional system.

abstract click to expand
Classical spectral theory lacks a framework for understanding how spectra propagate through compositional systems like deep neural networks, feedback control loops, and quantum circuits. This paper develops a universal theory governed by three invariants: the operadic spectrum (local spectral data), spectral derivatives (perturbation sensitivity), and interaction residue (emergent interface-generated content). We prove three main theorems: the Spectral Propagation Theorem decomposes global output into propagated local spectra, residues, and derivative corrections; the Stability Theorem introduces the SOC stability radius and condition number; and the Universality Theorem shows any reasonable propagation rule is uniquely determined by the three invariants. These results provide a coordinate-free, representation-invariant language for spectral analysis of compositional operator systems.
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math.AT 2026-06-03

Hochschild homology computed for Reedy categories

by Alexandra Ballow

The Hochschild Homology of Reedy Categories

Explicit results obtained for the simplex category, finite sets, finite-dimensional categories, and operad PROPs.

Figure from the paper full image
abstract click to expand
We calculate the Hochschild homology of generalized Reedy categories, such as the simplex category, the category of finite sets, the category of finite-dimensional categories, and the PROP associated to an operad.
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math.AT 2026-06-03

Tidy maps generate both Goodwillie and Weiss towers

by Mathieu Anel, Georg Biedermann +2 more

Left exact monoidal localizations from tidy maps

A unified framework shows the Weiss tower as a completion tower of left exact localizations in symmetric monoidal categories.

abstract click to expand
We put Goodwillie's calculus of functors and Weiss' orthogonal calculus in a unified framework. We do so in two ways. On the one hand, the relevant categories are all symmetric monoidal and controlled by their compact objects. We introduce the notion of tidy map as a means to generate symmetric monoidal localizations in this setting. These localizations are always left exact. Then we show that both the Goodwillie and Weiss towers are generated by such maps. On the other hand, the relevant categories are also topoi, for which there is a general theory of completion towers of left exact localizations. We had shown in a previous work that the Goodwillie tower is an instance a such a tower. We show here that the Weiss tower is a completion tower as well, and therefore that the general theory applies to orthogonal calculus.
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quant-ph 2026-06-03

Essential unitarity uniquely extends to higher-order quantum maps

by Samson Abramsky, Radha Jagadeesan

Essential Unitarity for Higher-Order Quantum Computation

It is the only predicate compatible with dagger-monoidal structure and currying that reduces to standard unitarity at first order.

Figure from the paper full image
abstract click to expand
We develop a semantic framework for higher-order quantum computation based on a boundary-centric presentation of compact closed categories, building on Kelly--Laplaza and Abramsky.Morphisms are polarized boundary linkings composed by execution, with a unit-free monoidal sum providing reversible control and branching. We identify a notion of \emph{essential unitarity} generalizing unitarity from first-order processes to higher-order interfaces;at first order it coincides with standard unitarity, and at higher order it characterizes when information is preserved relative tothe boundary. Essential unitarity is the unique predicate compatible with dagger-monoidal structure, coherence reindexing, and currying, and reducing to ordinary unitarity at first order. Every morphism of the quantum core is essentially unitary. The framework realizes the coherent quantum switch and other one-slot, equal-ratio, purity-preserving supermaps as coherent pure-comb dilations. Extended Abstract appears in QPL 2026
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math.LO 2026-06-03

Étale-finite Heyting algebras realized as topos truth lattices

by Marco Abbadini, Rodrigo Nicolau Almeida +1 more

A topos for \'etale-finite Heyting algebras

Esakia duality yields an elementary topos for each such algebra and shows they are exactly those appearing in finitely propositional toposes

Figure from the paper full image
abstract click to expand
A longstanding open problem is whether every Heyting algebra is the lattice of truth values (i.e., of subterminal objects) of some elementary topos. A positive answer is known for complete Heyting algebras (i.e., locales) via sheaves, and for Boolean algebras via a construction due to Peter Freyd. We extend Freyd's construction to all \'etale-finite Heyting algebras, in the sense of Evgeny Kuznetsov. These are the Heyting algebras satisfying a generalisation of the law of excluded middle relative to some finite Heyting subalgebra. For every \'etale-finite Heyting algebra $H$, we use Esakia duality to construct an elementary topos whose lattice of truth values is isomorphic to $H$, thereby extending the class of Heyting algebras for which a positive answer to the Heyting-to-topos problem is known. The toposes we construct are categories of certain compact \'etale spaces. As a consequence, they are finitely propositional: every object has a finite cover by subterminal objects. We show that a Heyting algebra occurs as the lattice of truth values of some finitely propositional topos if and only if it is \'etale-finite. This exhibits an obstruction to extending the use of compact \'etale spaces beyond the \'etale-finite case.
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cs.DB 2026-06-03

Formalizing all published math benchmarks general reasoning

by A. Mayeux

Formalizing all indexed mathematics as a benchmark for general reasoning, with the example of implementing dilatations of categories

A continuously updated verifiable corpus tests whether theorem provers can manage mathematics' full scale and interdependence.

abstract click to expand
Formal rigor distinguishes mathematics from other disciplines, in the sense that mathematical statements are derived from explicit axioms by logically verifiable steps. Interactive theorem provers support this by expressing definitions, theorems, and proofs in a fully formal language and verifying them mechanically. We consider the benchmark problem of formalizing all published mathematics as a machine verifiable and continuously updated corpus of mathematical knowledge. This viewpoint treats mathematics as a structured database of interdependent results and raises questions about scalability and organization of large formal libraries. As a case study, we present an ongoing formalization in categorical algebra, namely dilatations of categories, extending classical localizations and illustrating what such an implementation looks like in practice.
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cs.LO 2026-06-03

Orthogonality types match Isbell nucleus fixed points

by Juan Luis Gastaldi (D-GESS, SPHERE UMR 7219) +5 more

A calculus of types in Isbell nuclei

Minimal execution and measurement generate a noncommutative Lambek calculus with associative product and residuals.

abstract click to expand
We identify two constructions from different mathematical traditions. In linear logic and realisability, logical types are generated rather than fixed in advance: one begins with a universe of realisers equipped with execution, uses orthogonality to test their interactions, and takes types to be the biorthogonally closed subsets. In enriched Isbell duality, a quantitative relation induces an adjunction whose fixed points form a category, its nucleus. These constructions proceed by different means; we show that, in the present setting, they produce the same objects. The shared datum is minimal: an associative product, called execution, and a real-valued measurement, with no compatibility assumed between them. The failure of the measurement to be additive is at once the relation defining orthogonality and the quantitative relation whose Isbell nucleus we form, and the types cut out by orthogonality are exactly the fixed points of the associated adjunction. The identification pays off in both directions. The most natural product of types fails to be associative; repairing this failure forces a different notion of type, sensitive to both sides of a composite, on which the induced product is associative and, when execution has units, carries two residuals. What emerges is a noncommutative Lambek calculus, derived directly from execution and orthogonality rather than imposed. In the reverse direction, each such type, read on the categorical side, generates a quantitative relation of its own, and with it a derived adjunction and a further generation of types; these derived types are again types of the original situation, computed by the residuals of the Lambek calculus. We also prove a coherence theorem for the threefold arrangements of this construction and, in the finite-dimensional case, give explicit formulas for the product.
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math-ph 2026-06-03

BF theory on BG encodes continuous symmetries via Lagrangian boundaries

by Hao Xu

Classical Symmetry TFTs for Continuous Symmetries via Higher Symplectic Geometry

The (n+1)-dimensional bulk for G-actions on n-dimensional sigma models is the AKSZ theory on T^*[n](BG), with gauging realized by domain wal

abstract click to expand
We propose a shifted-symplectic formulation of a classical continuous analogue of the symmetry TFT paradigm. Let $G$ be an algebraic or Lie group acting by topological defects on an $n$-dimensional classical topological sigma model with target an $(n-1)$-shifted symplectic derived stack $(X,\omega)$ via the AKSZ construction. We argue that the corresponding $(n+1)$-dimensional bulk theory should be the AKSZ theory with target the shifted cotangent stack $T^*[n] (\mathrm B G)$, equivalently the $(n+1)$-dimensional BF theory for $G$. We characterize the Dirichlet and Neumann boundary conditions, and more general topological boundaries, in terms of shifted Lagrangians in $T^*[n] (\mathrm B G)$. We realize the gauging of the $G$-symmetry in the original theory as inserting a topological domain wall between the corresponding topological boundaries in the BF bulk, and introduce the notion of Hamiltonian reduction, syplectic reduction, and Lagrangian reduction in the shifted symplectic setting. We also discuss prequantum refinements of continuous SymTFTs. In this refinement, higher gerbes on $\mathrm B G$ encode classical analogues of 't Hooft anomaly data by decorating the shifted cotangent bulk and its Lagrangian boundary conditions. Finally, in dimension three we compare the infinitesimal BF model $\mathrm B(\mathfrak g\ltimes\mathfrak g^\vee)$ with the factorizable double $\mathrm B(\mathfrak g\oplus \mathfrak g)$. The resulting topological boundaries are described by Lagrangian Lie subalgebras, and the factorizable case relates the SymTFT dictionary to $r$-matrices and Belavin--Drinfeld data.
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math.CT 2026-06-03

Flexible 2-categories form model category with all objects cofibrant

by Alexander Campbell

A convenient model category for bicategories

The structure is Quillen equivalent to Lack's and its fibrant objects match bicategories with normal pseudofunctors.

Figure from the paper full image
abstract click to expand
We introduce and study the model category of flexible $2$-categories, which is Quillen equivalent to Lack's model category of $2$-categories, but enjoys several excellent properties not shared by the latter. In particular, every object of this model category is cofibrant, it is a monoidal model category with respect to its cartesian closed structure, and its full subcategory of fibrant objects is equivalent to the category of bicategories and normal pseudofunctors.
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math.CT 2026-06-02

Abstract slopes framework explains shared weight filtrations

by Carolyn Echter

Weight filtrations via slopes

Mixed structures from separate contexts obey the same filtration rules inside one neutral setting.

abstract click to expand
Mixed structures and their weight filtrations appear in various contexts, prominently Hodge theory and the theory of Galois representations. In the setting of Andr\'e's formalisation of slopes, we propose an abstract framework explaining why mixed structures have shared characteristics.
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cs.PL 2026-06-02

Fixed-point combinator preserves structure through Clef lowering

by Houston Haynes

Fixed-Point Scaffolding in the Clef Programming Language

A functor from the compilation poset supplies verified code transformation to MLIR while keeping dimensional and numeric properties intact.

Figure from the paper full image
abstract click to expand
For fans of Gabriel's "Worse is Better" it may be ironic that C++, by way of MLIR, serves as the scaffold for compiling an ML-family language whose correctness properties are structural. A crucial intersection in our Composer compiler initiates its lowering with a fixed-point combinator that preserves the dimensional, grade, escape, and numeric-representation structure from the Program Semantic Graph. And the MLIR that's witnessed from the PSG is no passive host. Its use of static single assignment, attribute system and dialects carry that structure materially. We show that our compiler middle end uses categorical construction for lowering code with companion verification to that strata: a functor from the compilation poset to a target category, subject to the compositionality equation. The grounding of our approach comes from three sources, each on its own algebraic object: Ohori's machine-code proof theory grounds the compilation axis, parametricity grounds the content at the base, and adjoint mode logic grounds the traversal between our verification tiers. To extend the thesis we introduce compact-closed negative and fractional types, and show the type machinery can be carried with preserved structure and realized through tooling MLIR provides. More broadly, the same fixed-point primitive that preserves types through compilation also supplies proof terms that can continue to be exercised in MLIR to verify its integrity as lowering proceeds through the pipeline. We argue that this foundation is a unique additional point anticipated by our framework that includes dimensional types, Tarau's groupoid, and cellular sheaves. Throughout, the formalism is instrumented as an internal scaffold: the abstractions support the compiler's mechanics, where a developer is never required to reach for category theory in order to rely on the guarantees the compiler provides.
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math.DG 2026-06-02

Lie groupoid CNNs equivalent to algebroid versions

by Michael Astwood

Theoretical Aspects of Lie Groupoid and Lie Algebroid Equivariant Convolutional Neural Networks

Layers define continuous natural transformations between feature functors and generalize group pooling.

Figure from the paper full image
abstract click to expand
We introduce Lie groupoid equivariant neural networks as a specialization of recently proposed topological category-equivariant neural networks to the differentiable setting. Lie groupoid equivariant neural networks are composed from Lie groupoid lifting convolutions and Lie groupoid convolution layers, and we show how for suitable Lie groupoids they are equivalent to certain Lie algebroid-equivariant neural networks. We additionally describe groupoid invariant global pooling as a generalization of group invariant global pooling. Furthermore, we show that each of the aforementioned layers is a special case of recently introduced admissible category-equivariant layers by demonstrating that they define continuous natural transformations between continuous feature functors.
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math.AT 2026-06-02

Regular clock maps form a locally presentable category

by Philippe Gaucher

Regular clock map and trace space

The canonical quotient from directed paths to traces is a homotopy equivalence on their underlying spaces.

abstract click to expand
A regular clock map is a regular map of directed spaces from a saturated directed space to the directed circle. We prove that the category of regular clock maps is a small-orthogonality class of the category of clock maps. Hence it is locally presentable. Any geometric realization of precubical sets and of transverse sets gives rise to a regular clock map. Finally, we prove that for the underlying directed space of a regular clock map, the canonical quotient from directed paths to traces is always a homotopy equivalence.
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math.CT 2026-06-02

C*-algebra centers define quantum modality in cohesive infinity-topos

by Joey Woo

A Cohesive infty-Topos with a Quantum Modality from Finite-Dimensional C^(*)-Algebras

The centre functor yields a comonad whose coalgebras are classical field theories and enables a synthetic no-cloning theorem.

abstract click to expand
We construct a cohesive $\infty$-topos $\mathbf{H}_{\mathbb{Q}}$ equipped with a \emph{quantum modality} -- an idempotent product-preserving comonad $Q^{\diamond}$ with right adjoint $Q_{\bullet}$ satisfying the Beck--Chevalley compatibility conditions with the cohesive structure $(\Pi,\flat,\sharp)$. The model is the functor $\infty$-topos $\operatorname{Fun}(\mathbf{C}^{*}\mathbf{Alg}_{\mathrm{fd}},\; \mathbf{H}_{\mathrm{sm}})$, where $\mathbf{H}_{\mathrm{sm}}$ is the smooth cohesive $\infty$-topos and $\mathbf{C}^{*}\mathbf{Alg}_{\mathrm{fd}}$ is the category of finite-dimensional $C^{*}$-algebras with centre-preserving $*$-homomorphisms. Cohesion is lifted pointwise from $\mathbf{H}_{\mathrm{sm}}$; the quantum comonad is precomposition with the centre functor. We endow the topos with the Day convolution monoidal structure $\otimes_{\mathrm{Day}}$ induced by the tensor product of $C^{*}$-algebras and prove that $Q^{\diamond}$ is a strong monoidal comonad. The category of $Q^{\diamond}$-coalgebras is equivalent, via Gelfand duality, to the topos $\operatorname{Fun}(\mathbf{FinSet}^{\mathrm{op}},\mathbf{H}_{\mathrm{sm}})$ of discrete classical field theories. The comonad is interpreted as decoherence. This yields a cohesive linear $\infty$-topos in which the cartesian linear-logic structure degenerates, while the Day convolution provides a non-degenerate affine model of multiplicative intuitionistic linear logic. We also prove a synthetic no-cloning theorem and discuss the limits of the centre modality for representing quantum channels. This work provides the first rigorous instance of the cohesive linear framework and settles the open problem of finding a concrete model for cohesive linear homotopy type theory.
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