904 papers
2604.04919Categorical Perspectives on Chemical Reaction Networks
We show that the Schur-complement reduction of a chemical reaction network (CRN) from Hirono et al. is the categorical complement of the stoichiometric arrow in the arrow category $[\mathbf{A}_2,\mathbf{Vect}]$. This identifies the ambient category in which topological reduction of chemical reaction networks is functorial and explains the reduced stoichiometric matrix as a universal diagrammatic construction. We further define a reconstruction functor from a restricted subcategory of $[\mathbf{A}_2, \mathbf{Vect}]$ back to CRNs and prove an adjunction with the stoichiometric functor.
2604.04919Apr 2026
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Kleisli semantics and hypergraph composition for Greimasian narrative programs
This article proposes a category-theoretic formalization of Greimasian narrative programs (NPs) that makes their compositional structure mathematically precise. Building on a reconstruction of the actantial model as a categorical schema, we introduce a refined typological schema of actants and derive Set-valued instances corresponding to role-indexed elements of a narrative. NPs are represented within a categorical schema whose morphisms are interpreted using monads on Set. In particular, the List monad provides a Kleisli semantics for modeling non-atomic, list-valued actantial configurations, while the Maybe monad encodes optional dependencies between programs. This yields a minimal representation of narrative programs as structured data with an intrinsic compositional interpretation. To account for the dynamics of narrative formation, we lift these constructions into a diagrammatic setting by freely generating a symmetric monoidal category, and subsequently a hypergraph category, from the set of actants. In this framework, narrative programs act as generators of morphisms, and their composition is realized through wiring diagrams. A narrative trajectory is thereby interpreted as a single composite morphism. This approach provides a unified mathematical framework for structural semiotics, connecting data-level representations of narrative elements with their compositional realization in discourse.
2604.04495Apr 2026
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Frobenius quotients, inflation categories and weighted projective lines
We propose the notion of Frobenius quotients between Frobenius exact categories. It turns out that any Frobenius quotient induces Frobenius quotients between the corresponding inflation categories. We obtain an explicit Frobenius quotient from the category of vector bundles on weighted projective lines with three weights to a certain category consisting of monomorphism grids.
2604.02626Apr 2026
View2604.01854Free rigid commutative algebras
We describe free rigid commutative algebras in $2$-presentably symmetric monoidal $(\infty,2)$-categories as oplax colimits over the $1$-dimensional framed cobordism category. The special case of the $(\infty,2)$-category $\mathrm{Pr}^\mathrm{L}$ itself provides a description of the free symmetric monoidal $(\infty,1)$-category with duals on a given $(\infty,1)$-category, while the case of $\mathrm{Mod}_{\mathcal{V}}(\mathrm{Pr}^\mathrm{L})$ provides a description of a similar object in the $\mathcal{V}$-enriched context, for $\mathcal{V}$ a presentably symmetric monoidal $(\infty,1)$-category. As a byproduct, we obtain new proofs of some results about rigidification of locally rigid categories, as well as a proof that any rigid category over $\mathrm{Sp}$ embeds into a compactly-rigidly generated one.
2604.01854Apr 2026
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Dioperads, Frobenius monoidal functors and duality
Motivated by duality phenomena for derived global sections on derived local systems on compact oriented manifolds, we introduce the notion of a $d$-duality context between symmetric monoidal enriched categories. In this setting, the right adjoint of a symmetric monoidal functor carries compatible lax and colax structures twisted by an invertible object $d$. For any enriched dioperad $\mathcal{P}$, we define a $d$-twist $\mathcal{P}\{d\}$ and prove that, in a $d$-duality context, the right adjoint sends $\mathcal{P}$-algebras to $\mathcal{P}\{-d\}$-algebras. To achieve this, the key conceptual result is that Frobenius monoidal functors between symmetric monoidal categories are precisely those functors inducing morphisms between the underlying dioperads. We also develop a dioperadic Day convolution, yielding an alternative proof of the main theorem and suggesting an $\infty$-categorical extension of the theory.
2604.01080Apr 2026
View2603.29815Day convolution for algebraic patterns
We characterize the exponentiable objects for a wide range of structures prevalent in $\infty$-categorical algebra, extending the construction of Day convolution to more general structures than $\infty$-operads. More precisely, we give a criterion that is both necessary and sufficient for many of these structures encountered in practice, such as (equivariant) $\infty$-operads and virtual double $\infty$-categories. We work within the framework of algebraic patterns of Chu-Haugseng that describe these structures in terms of weak Segal fibrations. As part of the proof, we give a new description of weak Segal fibrations in terms of generalized Segal spaces on certain "tree" categories. We also define the "underlying graph" of a weak Segal fibration, extending the notion of the underlying $\infty$-category for $\infty$-operads, and explicitly describe the underlying graph of exponential objects in weak Segal fibrations.
2603.29815Mar 2026
View2603.29573Clock systems for stochastic and non-deterministic categorical systems theories
One of the characteristic features of categorical systems theory is that the behavior of systems can be characterized by certain morphisms into them. In other words, behaviors form a representable covariant functor to Set. And more generally, in the compositional setting, behaviors form a representable double functor to Span. Clock systems are convenient because behavior functors represented by clock systems are automatically well-behaved. It was previously not known whether stochastic and non-deterministic systems theories have clock systems. In this paper, we show that indeed they do have clock systems. Moreover, the clock systems for non-deterministic systems point to generalized notions of behavior for non-linear time.
2603.29573Mar 2026
View2603.28688A synthetic construction of universal cocartesian fibrations
We give a model-independent construction of directed univalent cocartesian fibrations of $(\infty,1)$-categories, and prove a straightening equivalence against such fibrations. The key step is showing that cocartesian fibrations descend along localisations, which we accomplish by analysing mapping spaces of localisations. Along the way we introduce a directed version of the join construction, giving a sequential colimit description of the full image of any functor.
2603.28688Mar 2026
View2603.27786Higher algebra in $t$-structured tensor triangulated $\infty$-categories
We generalize fundamental notions of higher algebra, traditionally developed within the $\infty$-category of spectra, to the broader setting of $t$-structured tensor triangulated $\infty$-categories ($ttt$-$\infty$-categories). Under a natural structural condition, which we call "projective rigidity", we establish higher categorical analogues of Lazard's theorem and prove the existence and universal property of Cohn localizations. Furthermore, we generalize higher almost ring theory to the $ttt$-$\infty$-categorical setting, showing that $π_0$-epimorphic idempotent algebras are in natural bijection with idempotent ideals. By exploiting deformation theory, we establish a general étale rigidity theorem, proving that the $\infty$-category of étale algebras over a fixed connective base is completely determined by its discrete counterpart. Finally, we characterize the moduli of such projectively rigid $ttt$-$\infty$-categories, demonstrating that the presheaf $\infty$-category on the 1-dimensional framed cobordism $\infty$-category serves as the universal projectively rigid $ttt$-$\infty$-category.
2603.27786Mar 2026
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Unitary, Inner product, and Dagger categories
This article provides an alternate characterization of dagger categories, which are central to the study of categorical quantum mechanics, in terms of inner product categories. An inner product category is an "achiral involutive" category with an inner product combinator. Inner product categories are, in turn, precisely the same as unitary categories, which are a weaker form of dagger categories. In unitary categories, there is an isomorphism between an object and its dagger, instead of the identity function as in the case of dagger categories. Every unitary category is equipped with a global inner product structure, which allows one to strictify the involutive structure on the unitary category to obtain a dagger category, making unitary categories 2-categorically equivalent to dagger categories. By regarding the inner product as an abstract metric on an (achiral) involutive category, one can define metric-preserving maps (isometries) in inner product categories, and also develop other notions of special maps -- unitary, Hermitian, positive, and normal maps -- in this setting.
2603.27614Mar 2026
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Introducing pixelation with applications
Motivated by the desire for a new kind of approximation, we define a type of localization called pixelation. We present how pixelation manifests in representation theory and in the study of sites and sheaves. A path category is constructed from a set, a collection of "paths" into the set, and an equivalence relation on the paths. A screen is a partition of the set that respects the paths and equivalence relation. For a commutative ring, we also enrich the path category over its modules (=linearize the category with respect to the ring) and quotient by an ideal generated by paths (possibly 0). The pixelation is the localization of a path category, or the enriched quotient, with respect to a screen. The localization has useful properties and serves as an approximation of the original category. As applications, we use pixelations to provide a new point of view of the Zariski topology of localized ring spectra, provide a parallel story to a ringed space and sheaves of modules, and construct a categorical generalization of higher Auslander algebras of type $A$.
2603.25432Mar 2026
View2603.25117New examples of non-unique enhancements for triangulated categories
We present a general procedure for constructing triangulated categories, linear over a field, with distinct enhancements. Some of our examples can be equipped with a (non-degenerate) t-structure, thereby showing that the existence of a t-structure does not imply uniqueness of enhancements, whether in the strong or weak sense (depending on the example).
2603.25117Mar 2026
View2603.24300Enhanced left triangulated categories
In this short note, we study dg categories with homotopy kernels, whose homotopy categories are known to admit a natural left triangulated structure. Prototypical examples of such dg categories arise as dg quotients of exact dg categories. We demonstrate that the stablization of the homotopy category of such a dg category admits a canonical dg enhancement via its bounded derived dg category.
2603.24300Mar 2026
View2603.23982A pretorsion theory for right groups
Let $S$ be a right group. Then there exist two congruences $\sim$ and $\equiv$ on $S$ such that $S$ is the product of its quotient semigroups $S/{\sim}$ and $S/{\equiv}$, where $S/{\sim}$ is a group and $S/{\equiv}$ is a right zero semigroup. If $E$ is the set of all idempotents of $S$ and we fix an element $e_0\in E$, then the pointed right group $(S,e_0)$ is the coproduct of its pointed subsemigroups $(Se_0,e_0)$ and $(E,e_0)$ in the category of pointed right groups. In general, there is a pretorsion theory in the category of right groups in which the torsion objects are right zero semigroups and the torsion-free objects are groups.
2603.23982Mar 2026
View2603.23018On the equivalence of two approaches to multiplicative homotopy theories
We study the relation of two frameworks for multiplicative homotopy theories: Presentably symmetric monoidal $\infty$-categories and combinatorial symmetric monoidal model categories. Our main theorem establishes an equivalence of their homotopy theories. As consequences, we solve Pavlov's conjecture and obtain a solution to a special case of Hovey's 10th problem. We also prove several variations of the main theorem, such as an analog for non-symmetric monoidal semi-model categories.
2603.23018Mar 2026
View2603.22242A strengthened $(\infty, n)$-categorical pasting theorem
We extend Campion's pasting theorem for $(\infty, n)$-categories to a larger class of polygraphs, called the directed complexes with frame-acyclic molecules. It follows, for instance, that this pasting theorem applies to any polygraph presented by a semi-simplicial set. We also set up a comparison between directed complexes and Henry's regular polygraphs, and show that they coincide up to dimension $3$. As a corollary of our main results, the pasting theorem also applies to the class of regular $3$-polygraphs.
2603.22242Mar 2026
View2603.22200On the semi-abelianness of cocommutative Hopf monoids
By providing a suitable generalization of Newman's bijective correspondence known for cocommutative Hopf algebras, we prove that the category of cocommutative Hopf monoids in any abelian symmetric monoidal category is semi-abelian, once faithful (co)flatness conditions are satisfied. This result unifies and generalizes the semi-abelianness of cocommutative Hopf algebras and of cocommutative color Hopf algebras known up to now. As a consequence of the semi-abelianness, the category of cocommutative Hopf monoids is also action representable. Finally, we prove that abelian objects in the category of cocommutative Hopf monoids coincide exactly with commutative and cocommutative Hopf monoids, which form so an abelian category.
2603.22200Mar 2026
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Groupoid G-spans and matrices over group rings
When G is a finite abelian group, we define G-spans of groupoids and their associated matrices with entries in the group ring QG and show that composition of spans corresponds to multiplication of matrices.
2603.20787Mar 2026
View2603.19854Operads and equivariance
Operads were originally defined by May to have right actions of the symmetric groups, but later formulations have also used no groups actions at all or group actions by such families as the braid groups. We call such families action operads, as they are the algebraic objects that encode parametrized group actions on operads. In Part I of this paper, we study the basic algebra of action operads $Λ$ and the $Λ$-operads they act upon. In Part II, we study $Λ$-operads in the 2-category of small categories.
2603.19854Mar 2026
View2603.19553Induced structures of operated algebras with applications to multi-Novikov algebras
We provide a general notion of induced structures of operated algebras in the context of unary-binary operads. This notion fully captures the binary quadratic relations encoded by a unary-binary operad, thereby unifying and formalizing the various constructions that have appeared in the literature under the informal term of ``induced structures''. As an application, we show that the Novikov algebra and the recently introduced multi-Novikov algebra are the induced structures of the differential commutative algebra and the multi-differential commutative algebra respectively. We also explicitly determine the induced structure of the noncommuting multi-differential commutative algebra.
2603.19553Mar 2026
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