Papers

The article arXiv:2410.21454 constructed a braided $\mathrm{W^*}$-tensor category of superselection sectors associated to a net of von Neumann algebras on a suitably geometric poset using localized and transportable endomorphisms. In this article, we construct an equivalent tensor product and braiding using the Connes fusion tensor product without localizing, using ideas from arXiv:1812.04470 for conformal nets. As an application, we prove that the category of superselection sectors associated to the 2D braided quantum spin system constructed from a unitary braided fusion category $\mathcal{B}$ in arXiv:2506.19969 is equivalent to $\mathsf{Hilb}(\mathcal{B})^{\rm op}$, the opposite of the unitary ind-completion of $\mathcal{B}$, as a braided $\mathrm{W^*}$-tensor category.

We give a short proof that $\kappa$-cocompact objects in a presentable category are subterminal. As our main result, we extend this to the setting of presentable $\infty$-categories. A consequence is that an $\infty$-category $\mathcal{C}$ such that both $\mathcal{C}$ and $\mathcal{C}^\mathsf{op}$ are presentable is a small complete lattice, extending a classical theorem of Gabriel-Ulmer. Along the way, we prove a nilpotence result for phantom maps in general pointed presentable $\infty$-categories. Additionally, we show that a strengthening of our main result is equivalent to the existence of a proper class of measurable cardinals.

We construct a symmetric operad $\mathfrak{C}$ on the class of all multivariate copulas, where operadic composition is given by Sklar substitution. We prove that the absolutely continuous subclass $\mathfrak{C}^{ac}$—which coincides with the $L^1$ class of copula densities—forms a suboperad; under composition, the density of the composite copula is given by the explicit Sklar substitution density formula $g(v)=\phi\big(\Psi_1(v^{(1)}),\dots,\Psi_n(v^{(n)})\big)\prod_{k=1}^{n}\psi_{k}(v^{(k)})$. Furthermore, we show that copulas with finite copula entropy—identified with the $L\log L$ class of copula densities—are closed under substitution and hence constitute a suboperad $\mathfrak{C}^{L\log L}$. On this suboperad, copula entropy is strictly additive: $H(\gamma(\Phi;\Psi_{1},\ldots,\Psi_{n}))=H(\Phi)+\sum_{k=1}^{n}H(\Psi_{k})$.

Run two process-discovery algorithms on the same event log and they return two different pictures of the same process. Petri nets, causal nets, process trees and BPMN each rely on routing machinery of their own, and their only common ground is the traces they generate. A trace lists activities one after another, so it discards the concurrency the notations exist to express, and two models with identical trace languages can describe genuinely different processes. Whether two discovered models mean the same thing therefore has no notation-independent answer. We show that all four notations admit one canonical presentation, a signature recording a model's activities and the typed interfaces along which they compose, and nothing of the routing machinery. Four construction theorems establish this presentation notation by notation, so two models are compared in a form that each notation determines on its own. Signature equality implies trace equivalence and is strictly finer, separating genuine concurrency from interleaved choice, which a trace comparison cannot, and signature inclusion implies trace inclusion. In a recovery study on one object-centric log, the discovered causal net has exactly the ground-truth signature, while the discovered Petri net's signature strictly contains it and locates every behaviour the Petri net adds in its silent structure. The gain is largest for object-centric data, where a single log supports several notations at once. The four notations become one calculus, in which a translation between them is a claim that can be checked.

We prove a multiplication formula for the Wang-Wei-Zhang cluster character on a Hom-finite $2$-Calabi–Yau Frobenius extriangulated category $\mathcal C$ with a cluster-tilting object under the assumption that its stable category has constructible cones with respect to the induced cluster-tilting object. This formula generalizes those obtained by Wang-Wei-Zhang for $\mathcal C$ in the one-dimensional case and by Keller-Plamondon-Qin for (stably) $2$-Calabi-Yau Frobenius or triangulated categories. As a consequence, we express the frozen term in an Auslander-Reiten mesh with the character of a minimal cosyzygy middle term. For acyclic quivers with principal coefficients, we compute these frozen multiplicities in Higgs categories and obtain explicit principal coefficient mesh relations.

We introduce the notion of a unitary disk-like $n$-category, which is a disk-like $n$-category equipped with a reflection structure and a sphere trace inducing positive-definite pairings. Since a finite unitary disk-like $n$-category is defined to be the local field data of a fully extended $(n+1)$D unitary TQFT, we propose a complete finite unitary disk-like $n$-category as the definition of a finite $(n+1)$-Hilbert space for all $n$. For $n=1$ and $n=2$ we verify this proposal, proving that complete finite unitary disk-like 1- and 2-categories are isometrically equivalent to finite 2- and 3-Hilbert spaces respectively, and that these equivalences are functorial. For $n=1$ we recover a unitary refinement of Schommer-Pries' classification of oriented $(1+1)$D TQFTs in terms of $\mathrm{H}^*$-Morita equivalence classes of $\mathrm{H}^*$-algebras, and for $n=2$ we categorify this to classify oriented $(2+1)$D unitary TQFTs by $\mathrm{H}^*$-Morita equivalence classes of $\mathrm{H}^*$-multifusion categories. Along the way, we investigate fully incomplete 3-Hilbert spaces, prove a strictification result for pivotal dagger 2-categories, and define functors and higher transformations between disk-like $n$-categories.

Perverse schobers are a categorification of perverse sheaves, originally proposed by Kapranov and Schechtman. The purpose of this paper is to initiate a microlocal study of perverse schobers. We first categorify $\operatorname{Perv}(\mathbb{C},R)/\operatorname{Loc}(\mathbb{C})$, the category of perverse sheaves on a complex line with singular points at $R$, modulo local systems. We use this to propose a general definition for microlocal perverse schobers supported on the open conormal to a germ of a hypersurface, and we conjecture that this is invariant under Radon transform. Here we make the key observation that while the analogous quotient of perverse schobers $\mathsf{2Perv}(\mathbb{C},R)/\mathsf{2Loc}(\mathbb{C})$ is a reasonable categorification, the resulting theory fails to be invariant under Radon transform. In fact, our proposed categorification can be recovered by correcting an instance of this failure in a universal manner. We prove Radon invariance when our hypersurface is the curve $y^m=x^n$ in $\mathbb{C}^2_{x,y}$. Along the way, we explain how our theory relates to Fourier transforms of perverse schobers, periodic SODs, and spherical monads.

A variety $\mathcal{V}$ is called a Schreier variety if every subalgebra of a $\mathcal{V}$-free algebra is a $\mathcal{V}$-free algebra. We use ideas from Tame Congruence Theory to classify locally finite Schreier varieties. One version of the classification theorem states that a locally finite variety $\mathcal{V}$ is a Schreier variety if and only if (i) every finite algebra in $\mathcal{V}$ is a $\langle 0,1\rangle$-minimal algebra and (ii) if $\mathcal{V}$ has a constant $1$-ary term operation, then $\mathcal{V}$ also has a constant $0$-ary term operation.

This paper studies (co)products and additive structures under Gabriel–Zisman localization $Q:\mathcal A\longrightarrow\mathcal A[S^{-1}]$. Examples show that localization may destroy existing (co)products or fail to preserve (co)products that remain, even when $Q$ is additive. For a set-indexed family $(\mathcal A_i,S_i)_{i\in I}$, necessary and sufficient conditions are given for the canonical functor $\big(\prod_{i\in I}\mathcal A_i\big)\big[\big(\prod_{i\in I}S_i\big)^{-1}\big]\longrightarrow\prod_{i\in I}\mathcal A_i[S_i^{-1}]$ to be full or faithful. These criteria yield compatibility with $I$-indexed (co)products for localizations admitting a calculus of fractions or arising from model categories, provided the relevant (co)products exist and the designated classes of morphisms are closed under them.

We provide a universal characterisation of the Banach spaces of equivalence classes of $L^p$ Bochner integrable functions, under equality almost everywhere, as initial objects in appropriate categories of bifunctors, for all $p \in [1,\infty)$. As a by-product of our construction, we also obtain a unique characterisation of the Bochner integral.

We show that the universal rigid monoidal category on one object has an abelian envelope. Since it was proven by Coulembier-Etingof-Ostrik-Pauwels (2023) that this category cannot have an abelian envelope with the quotient property, this disproves the conjecture in [op.cit.] saying that every abelian envelope has the quotient property. Monoidal Ringel duality (Flake-Gruber arxiv:2512.19558) yields a candidate envelope as a lower finite highest weight category. Our proof that this is an abelian envelope relies on Coulembier-Etingof's (2024) continuants to show the required universal property. AI was used to find these results.

It is an established idea in concurrency theory that every Petri net admits an unfolding semantics. This is a denotational object that represents its domain of possible executions. Unfoldings play an important role in practical analysis and verification. This paper is concerned with the following well-known problem: while the unfolding resembles a universal construction in the category of Petri nets, it generally fails to satisfy the expected universal property. This is because the unfolding construction overlooks the net's internal symmetries. There are two solutions: make these symmetries explicit to obtain a weak universal property (one that holds only ”up to symmetry”); or break the symmetries by assigning individual identities to components of the net. We review these two solutions and establish, in each case, a universal unfolding of Petri nets to event structures. This paper demonstrates a 2-categorical approach to Petri net unfoldings. We show that each unfolding semantics determines a 2-categorical relative adjunction involving Petri nets and event structures. Viewed in this way, the above two constructions can be related formally via an appropriate morphism of adjunctions. We exhibit a 2-density property of event structures which implies that unfolding functors are essentially unique.

In this work, we develop a fibrational approach to freely adjoining adjoints in an $(\infty,2)$-category. We construct the universal bifibration obtained from a cocartesian fibration of $(\infty,2)$-categories by adjoining cartesian lifts over a chosen class of $1$-morphisms in the base. We then use this construction to provide an explicit model for freely adjoining adjoints, together with a zig-zag formula for the resulting mapping $(\infty,1)$-categories. As applications, we give a model-independent proof of the universal property of the walking adjunction, providing an alternative proof of a theorem of Riehl–Verity, and establish the universal characterization of the simplex $2$-category conjectured by Dyckerhoff–Kapranov–Schechtman–Soibelman.

We construct an adjunction relating two approaches to modelling the planar lambda calculus: semi-closed operads and planar lambda-models. We use this to obtain a planar version of Scott's representation theorem.

We construct a complex spherical fusion category with cyclic Haagerup-Izumi fusion rules for every odd n >= 3, and deduce pseudo-unitary existence. The proof has four parts: explicit real coefficients and their quadratic identities; a contour calculation for a range of cubic Fourier coefficients; algebraic completion of all cubics; and categorical reconstruction. Matrix inversion supplies reflection, and an extension of the dimension-field automorphism supplies the positive-dimensional category.

We show that the formation of the product measure is the only natural way to assign to each tuple of probability measures on measurable spaces a probability measure on the product space. Here, naturality is meant in the sense of category theory and amounts to the condition that the assignment commutes with pushforward along measurable maps. In the standard Borel setting, we also prove an analogous result for relative products over a fixed base probability space. As a corollary, we conclude that the formation of relative products is a lax symmetric monoidal functor.

Finite support indicates that the dependence is limited, but not how each finite context controls what can be observed. For a data symmetry G \leq Sym(A) and a finitely supported G-set X, we study the finite-dependence profile X^S = {x : Fix_G(S) fixes x}, indexed by finite contexts S \subseteq A. For relations on A^n this yields Boolean algebras B^{(n)}_{G,S} = Inv_n(Fix_G(S)), forming the invariance hierarchy. The hierarchy separates several nominal principles. Its meet law is the relation-algebra form of the Bojanczyk-Klin-Lasota least-support criterion, while level injectivity is fungibility. The independent join law governs composition across unions of contexts; it holds for locally oligomorphic automorphism groups of ultrahomogeneous structures in purely relational languages of arity at most two, but can fail for unary observations over ternary data. Freshness remains valid sort by sort, and its unrestricted some/any form characterizes full equality symmetry among closed groups. With least supports, orbit-finiteness implies finite context levels and a uniform support bound; under local oligomorphicity the converse also holds. Pointwise closure does not change the profile, and standard atomic-site presentations show that suitable ultrahomogeneous structures with the same age have equivalent action topoi, even though their material FSS universes may retain different external cardinal data. Contextual orientability gives a complementary pointed invariant whose threshold is the least support size of choice from unordered pairs. For countable omega-categorical structures with degenerate algebraic closure, the meet law is equivalent to weak elimination of imaginaries.

Let $(\mathscr{C}, \mathbb{E}, \mathfrak{s})$ be a weakly idempotent complete extriangulated category, and let $\xi_1\subseteq\xi_2$ be proper classes of $\mathbb{E}$-triangles. We prove an analogue of Cole's mixing theorem in this setting: two compatible admissible model structures, relative to $\xi_1$ and $\xi_2$ respectively, give rise to a mixed admissible model structure $\mathcal{M}_m$ relative to $\xi_1$, which has the trivial objects of the second model structure and the fibrant (or cofibrant) objects of the first. We also describe the cofibrant objects of $\mathcal{M}_m$ explicitly in terms of the two original model structures. For exact categories this recovers the mixing theorem for exact model structures, and for triangulated categories it applies to proper classes of triangles; we give a non-degenerate example on the homotopy category of a quasi-Frobenius ring.

The problem of determining whether a category is balanced (i.e., every morphism that is both an epimorphism and a monomorphism is an isomorphism) has been widely investigated, especially in algebra. For complete categories with an extremal projective generator, we characterise the property of being balanced, and the stronger property that every monomorphism is regular, in terms of implicit operations. These characterisations apply, for example, to any quasiequational class of algebras, possibly with a proper class of function symbols and unbounded arities, that admits free algebras.

We study stylic monoids from the viewpoint of higher-dimensional rewriting. A coherent presentation records not only the defining relations of a monoid, but also the relations among them. Using the $N$-tableau model, we construct a finite coherent presentation for every stylic monoid of finite rank. We start from the known finite convergent row presentation. By adjoining one generating confluence for each critical branching, Squier's coherence theorem yields a finite coherent extension. We then simplify the resulted coherent presentation by two successive homotopical reductions adapted to the row combinatorics of $N$-tableaux. Finally, coherent Tietze transformations and a further homotopical reduction transfer the construction from the auxiliary row generators to the standard letter generators. This gives an effectively computable finite homotopy basis for the standard letter presentation. The construction is illustrated in detail in rank $2$.

Inspired by Grothendieck's foundational work, we establish that the sequence of functors $(Ext^n_{pBan}(E,-))_{n \ge 0}$ on the category of $p$-Banach spaces ($0 < p < 1$) forms a universal $\delta$-functor. To this end, for any pair of $p$-Banach spaces $E$ and $X$, we construct a $p$-Banach space $\mathcal N_E^X$ that renders each functor $Ext^n_{pBan}(E,-)$ effaceable. Furthermore, we show that this space serves as a relative injective object for $E$ and $X$, enabling the construction of relative injective presentations and proving that the functors $Ext^n_{pBan}(E,-)$ are the right derived functors of $\mathcal{L}(E,-)$. We will also examine the situation in the category of quasi-Banach spaces.

A Grothendieck topology on a category determines both a subform of covering sieves and a quotient form obtained by identifying locally equivalent sieves. We place these two constructions in a single short exact sequence. For this purpose we introduce distributivity forms: indexed meet-semilattices equipped with distinguished indexed joins over which finite indexed meets distribute, and a strong version in which these joins also satisfy Beck–Chevalley. Over a fixed base, the resulting categories have all small limits and have kernels and cokernels relative to the closed ideal of fibrewise constant-top morphisms. Their monomorphisms are the fibrewise injective morphisms, and their relative cokernels, together with the top-reflecting morphisms, form an orthogonal factorization system. Cokernels compose but need not be stable under pullback, whereas kernels need not compose. We call a short exact sequence with prescribed middle term an extensional presentation, and prove that Grothendieck topologies on a category are precisely the extensional presentations of its maximally distributive form of sieves. Further applications recover universal productive closure operators and Lawvere–Tierney topologies, functorial non-Archimedean group topologies, and functorial linear topologies on commutative rings.

We give a counterexample to completeness descent from ideal cotorsion pairs to their objects. A radical-square-zero algebra on the two-cycle gives a finite-dimensional example. The construction is intrinsically non-weakly-idempotent-complete.

For any category, there exists both a free dagger category and a cofree dagger category over it. A natural question to ask is: given a dagger category, how can we tell if it is free or cofree without specifying an external base category? In this paper, we provide characterizations of both free dagger categories and cofree dagger categories via internal dagger category structure. To characterize cofree dagger categories, we use rectangular bands and show that a dagger category is cofree if and only if it is enriched over rectangular bands. For free dagger categories, we define the notion of a zigzag dagger category, and then show that a dagger category is free if and only if it is a zigzag dagger category. We also show that free dagger categories can be characterized as the coalgebras of the induced comonad from the free dagger category adjunction, and similarly that cofree free dagger categories can be characterized as the algebras of the induced monad from the cofree dagger category adjunction.

We introduce a notion of nilpotency for surjective quandle homomorphisms, which relativizes the nilpotency of quandles recently defined by Darné. A surjective homomorphism is called nilpotent if its relative inner automorphism group is nilpotent relative to the inner automorphism group of the source. We show that this condition can equally be described through the relative transvection group and the relative adjoint group, and that it is equivalent to being a finite composite of covering homomorphisms. Extending the relative lower central series to transfinite ordinals, we then introduce hypocentral homomorphisms and characterize them as the surjections that are right orthogonal to the strongly connected homomorphisms. As a consequence, the strongly connected and the hypocentral homomorphisms form an orthogonal factorization system for the surjections in the category of quandles. Finally, we also introduce reduced and operator-reduced homomorphisms, which relativize reduced quandles, and give sufficient conditions for such a homomorphism to be nilpotent.

We construct a categorification of the functor of grouplike elements of a Hopf algebra in the setting of braided monoidal categories. Given a braided bistrong monoidal functor admitting a right adjoint, we define a functor assigning to each Hopf monoid a group object obtained as a subobject of its underlying object by means of finite limits. We prove that this construction is functorial and provides a right adjoint to the induced free Hopf monoid functor, thereby extending the classical correspondence between groups and grouplike elements of Hopf algebras to a categorical setting.

Let $Q$ be a commutative unital quantale and $\QRel$ the ordered category of sets and $Q$-relations. We develop a categorical foundation for formal and property-oriented concept lattices of $Q$-relations via the diagonal category $\DQRel$ and back-diagonal category $\BQRel$. Our main results are: (1) the formal concept lattice construction is induced by a representable functor on $\BQRel$; (2) the property-oriented concept lattice construction corresponds to a representable functor on $\DQRel$; and (3) for regular $Q$-relations, the property-oriented concept lattice of their pointwise tensor product is naturally isomorphic to the tensor product of the respective property-oriented concept lattices.

In recent years, higher categorical variants of idempotents have become an object of interest. Notably, they appear when categorifying algebraic structures motivated by mathematical physics. Working in the setting of bicategories, we give a rigorous definition of the bicategorical idempotent completion as outlined by Gaiotto and Johnson-Freyd in arXiv:1905.09566. We show that this agrees with the Cauchy completion of a locally idempotent complete bicategory in the sense of Lawvere.

Let $K$ be a field, and $\mathcal{T}$ a $K$-linear essentially small triangulated category equipped with an extendable $t$-structure $(\mathcal{T}^{\leq0}, \mathcal{T}^{\geq0})$ with respect to a good metric $\mathfrak{B}$. Given a torsion class $\mathcal{X}$ in the heart of $(\mathcal{T}^{\leq0}, \mathcal{T}^{\geq0})$, we prove that lifting the HRS-tilt of $(\mathcal{T}^{\leq0}, \mathcal{T}^{\geq0})$ at $\mathcal{X}$ along the completion of $\mathcal{T}$ coincides with the HRS-tilt of the lifted $t$-structure at the completion of $\mathcal{X}$. As an application, we provide the compatibility result between left silting mutation in $\mathcal{T}$ and HRS tilting in the ambient completion.

Evidence suggests a tight connection between the existence of antipode-like maps on bialgebra-like structures, and the Frobenius property for the associated free Hopf module functor. In this paper, we prove that in any braided monoidal category satisfying few mild assumptions, a bimonoid is a one-sided Hopf monoid and the antipode is a bimonoid anti-homomorphism, if and only if the free Hopf module functor is Frobenius. Our interest in these structures stems from having found genuine examples of one-sided Hopf monoids in contexts where the braiding is non-trivial. In fact, we provide a construction of the free one-sided Hopf monoid over a comonoid in any symmetric monoidal category with some non-restrictive additional assumptions. We present several examples of this construction, and describe the resulting one-sided (sometimes two-sided) Hopf monoids.