Papers

The category of all topological spaces and homotopy classes of maps is cartesian closed. In fact, the $(\infty, 1)$-category of all topological spaces and continuous maps, localised at homotopy equivalences, is cartesian closed. We model the exponential object using a topological space whose points are continuous maps equipped with putative neighbourhood filters. In general, this topological space will not even be $T_1$-separated, but the evaluation map induces a natural weak homotopy equivalence witnessing the universal property.

We establish a Kaplansky-Walker type lemma for idempotent complete additive categories with direct sums. It will be used in compactly generated triangulated categories to provide a version of Kaplansky's theorem on projective modules and an isomorphism-refinement result for direct sums of countable objects.

We calculate the chromatic spherical invariant $\mathscr{K}_{\mathrm{Rep}(H)}(L(p,q))$ of lens spaces $L(p, q)$ for arbitrary finite-dimensional unibalanced, unimodular Hopf algebras $H$. In the case of the index-2 Nichols Hopf algebra $\mathcal{K}_2$ and $q=1$, we find that the invariant of $L(p, 1)$ depends only on the parity of $p$ for $p\geq 1$. We also describe the duals $\mathcal{S_I}(M)^*$ of the admissible skein modules of handlebodies $M=H_g$ and of $M=L(p, q)$ for arbitrary tensor ideals $\mathcal{I}$ of a ribbon tensor category. In particular, $\mathcal{S_I}(L(p, q))^*$ corresponds with cyclic traces on $\mathcal{I}$ satisfying a property reminiscent of the modified trace condition.

This article surveys the main results on the representability of internal actions in varieties of non-associative algebras over a field. We begin by recalling the general framework of internal actions and split extensions in semi-abelian categories, before specialising to Orzech categories of interest and varieties of non-associative algebras. We discuss action representability, weak action representability, and action accessibility, with emphasis on the relationship between the latter two notions. We then describe the constructions of the universal strict general actor in Orzech categories of interest and the external weak actor in action accessible, operadic varieties of non-associative algebras over a field, as well as their role in the study of weak action representability. We also give a tensor-domain formulation of partial algebras and adapt the construction of the external weak actor to this setting. These notions are illustrated by examples arising from associative, Lie, Leibniz, Jacobi-Jordan, anti-commutative anti-associative, alternative, and $2$-nilpotent algebras. We conclude with a summary of the known results and some open questions for future research.

We give two types of examples where two non isomorphic finite quandles have isomorphic enveloping groups. This implies that two non-isomorphic finite quandles can share isomorphic representation categories (as additive and tensor categories simultaneously).

By equipping the category $\mathcal{M}$ of modules over a finite dimensional algebra with the action of a finite tensor category $\mathcal{C}$, we introduce a subcategory $\mathcal{PM}$ satisfying numerous remarkable properties: it is generating and cogenerating, functorially finite, closed under Auslander–Reiten translations (i.e. precluster tilting), and it computes the finitistic dimension. Through a functor category perspective, $\mathcal{PM}$ is identified with the $\mathcal{C}$-module functors that are projective, and we also prove a $\mathcal{C}$-equivariant version of the Auslander(–Solberg) correspondence. As examples, we show that $\mathcal{PM}$ contains only finitely many indecomposable modules for some tensor product algebras. For some family of 1-Iwanaga–Gorenstein algebras introduced by Geiss–Leclerc–Schröer, we show that $\mathcal{PM}$ is exactly the subcategory of $\tau$-locally free modules.

A tensor product is known for commutative Rota–Baxter algebras through a universal construction, but its underlying algebra need not be the ordinary tensor product $A\otimes B$ and can be substantially larger. We show that, after passing to augmented Rota–Baxter algebras, the ordinary tensor product itself carries a natural Rota–Baxter operator which makes the category monoidal with the usual associator and unit constraints. The resulting structure is intrinsically nonsymmetric: over a field of characteristic zero, this operator and its flip-conjugated counterpart are the only possibilities among linear combinations of five natural terms, while over any field the full category admits no braiding. Its comonoid objects are precisely Rota–Baxter cocycle bialgebras. Moreover, the free Rota–Baxter adjunction yields a bimonad and free cocycle bialgebras on ordinary bialgebras, with compatible Hopf structures for connected filtered inputs, in both the associative and commutative settings.

We characterize the central extensions, in the sense of the categorical theory of central extensions of Janelidze and Kelly, relative to the Birkhoff subcategory of $n$-nilpotent objects of a semi-abelian and algebraically coherent category. An extension $f \colon A \to B$ with kernel $K$ turns out to be central precisely when the iterated Huq commutator $[A,[A,\dots,[A,K]\dots]]$, with $n$ occurrences of $A$, vanishes. For $n=2$ this condition reads $[A,[K,A]] = 0$, that is, $[K,A]$ is a central subobject of $A$. In the special case of groups this description was obtained by Everaert and the first author. The present proof is categorical, and applies in particular to Lie algebras, to crossed modules, to cocommutative Hopf algebras over a field, and to any category of interest.

We investigate Eduard Hanslicks formalist aesthetics through the prism of Category Theory

We identify the $(\infty,1)$-category presented by the dagger Bergner, dagger Joyal, and dagger Rezk models of Parts I and II with that of univalent flagged dagger $(\infty,1)$-categories. We first construct a realization–nerve adjunction between reversal simplicial anima and Hermitian flags. We prove that its nerve restricts to an equivalence between flagged dagger $(\infty,1)$-categories and dagger Segal anima, under which univalence corresponds to unitary completeness. We then construct the adjunctions for univalentization and Hermitian completion and identify univalentization with unitary completion.

We develop a support theory for wide subcategories of essentially small Krull–Schmidt \(n\)-abelian categories. We construct spectra associated with families of wide subcategories and establish the corresponding support and radical theories. In particular, we characterize prime wide subcategories as the completely meet-irreducible elements of the lattice of wide subcategories and prove that every wide subcategory is recovered from the prime wide subcategories containing it. We further show that any classifying support datum is recovered, together with its object supports, from the canonical prime spectrum. Finally, we provide a \(2\)-abelian example admitting no classifying support datum.

Spectral duality structures (SDS) explain why a distinguished family of initial distributions makes the absorption probabilities of resetting processes insensitive to the resetting mechanism. Here we study their axioms. We prove that the six defining conditions are mutually independent and determine what each one contributes. The key tool is a structure theorem valid for an arbitrary permutation and an arbitrary positive weight: every insensitive distribution is supported on a union of cycles balanced at a common level, and the level of a cycle is fixed by the geometric mean of the weight on it. From it, positivity selects the admissible root, the constant orbit product is equivalent to the existence of an insensitive distribution of full support, and the involution is equivalent to the response rank being equal to the number of free weight parameters. Without the duality relation, insensitivity of a single distribution is cheap and generically isolated; duality turns it into structural neutrality of the separatrix. The full-rank condition splits into two layers: a weaker condition is equivalent to maximal response rank, while full rank also excludes insensitivity at any other level. We then show that a single SDS is internally rigid: compatible semigroup structures force a constant weight, compatible abelian group structures obey a precise arithmetic condition, congruences form a simple lattice, and the automorphism group is computed explicitly. Finally, the axioms do not determine the orientation of the response; in the two-site case we give the exact condition under which the geometric and analytic sign criteria agree.

Using the formalism of presheaf automata, we introduce new classes of automata for modelling concurrency that extend higher-dimensional automata and symmetric higher-dimensional automata. We study their languages and characterise these languages in terms of posetal cospans.

We give negative answers to two questions of Farjoun about localizations of groups. For every odd prime $p$, we construct a finite $p$-group $G$, an infinite non-nilpotent group $H$, and a local embedding $f\colon G\hookrightarrow H$. The associated localization satisfies $L_f(G)\cong H$, showing that localizations need not preserve nilpotence and that the localization map $G\to L(G)$ need not be surjective when $G$ is a finite $p$-group.

We investigate set-theoretic solutions of the Yang–Baxter equation from a categorical perspective. The paper has two main objectives. On the one hand, several existing constructions and classification results in the literature are recast in categorical terms, thereby showing that certain hypotheses previously imposed on such solutions are in fact superfluous. On the other hand, the categorical approach leads naturally to new classes of Yang-Baxter solutions. Our approach consists in identifying suitable subcategories of the category $\mathcal{YB}$ of Yang-Baxter solutions and providing categorical characterisations of the solutions they encode. As an application, we prove that the category of Płonka bi-magmas is isomorphic to the subcategory of $\mathcal{YB}$ consisting of solutions of the Baaj-Long-Skandalis equation. Along the way, we introduce and study a new class of Lyubashenko type solutions and examine the extent to which non-degenerate bijective solutions of the braid equation arise from skew left braces.

We present a typed proof-term calculus for intuitionistic intensional first-order logic with Hilbert's epsilon-operator and prove its soundness and completeness with respect to hyperdoctrinal models with thin bicartesian closed setoid categories (BiCCCs) as fibres and formula-indexed witness selections. The calculus provides an explicit internal language for these epsilon-hyperdoctrines: for each fixed typed signature, its syntactic term model is initial for syntax-indexed, interpretation-preserving model morphisms, with uniqueness up to setoid equality. The nested epsilon syntax is organized by a well-founded dependency relation on formula labels. Concrete countermodels exhibit failure of excluded middle and show that formula conversion need not identify the selected witnesses. We also compare the full calculus with its epsilon-free and epsilon-only fragments. The principal constructions and metatheorems are formalized in Rocq. With setoid BiCCCs, proof irrelevance is a choice of equality, not the erasure of proofs: the proofs and their categorical operations remain available as the basis for a future proof-relevant categorical semantics.

We present DisCo algebra—a compositional framework for joint probability built from a symmetric cooperad (disintegration) and a symmetric operad (combination) on the bare family $\mathcal{P}_n=\mathrm{Prob}([0,1]^n)$. No probability integral transform, no external summary functions, no enrichment of objects with conditional kernels. The disintegration side uses block-wise joint CDFs as intrinsic, measure-determined 1D summaries, yielding a lossy compression. The combination side reads the same summaries and reconstructs via regular conditional measures, yielding a lossless embedding. We prove that combination followed by disintegration is the identity on admissible inputs, but disintegration followed by combination is not—the information loss is exactly the conditional KL divergence between the true conditional distribution and the conditionally independent product. This asymmetry reveals a hard limit on how much cross-block dependence can be captured by scalar block summaries.

We study how local coherenAAce of abelian categories behaves under gluing. For AB5 locally finitely presentable abelian categories, we give conditions on the gluing functors under which local coherence is transferred along a gluing $(\mathcal{A}, \mathcal{B}, \mathcal{C})$: from $\mathcal{A}$ and $\mathcal{C}$ to $\mathcal{B}$, and conversely from $\mathcal{B}$ to $\mathcal{A}$ or $\mathcal{C}$. The coherence theorem for augmented homotopes of Bondal and Zhdanovskiy is recovered as a special case, the coherence theorem for trivial extensions of Fossum, Griffith, and Reiten is a companion result in the same direction. We also introduce quasiprojective pairs of adjoint functors and prove a coherence criterion that generalises the criterion of Gomez Pardo and Martinez Hernandez to arbitrary Grothendieck categories. Finally, we extend Stenström's characterisation of coherence via FP-injective modules to compactly assembled Grothendieck categories: such a category is coherently assembled if and only if the class of CM-injective objects, defined through compact morphisms, is closed under filtered colimits. Along the way we revisit the various equivalent conditions for the coherence of a ring, collecting them in a single place. The proofs make use of the compactly assembled categories of Efimov.

Let $A=\mathbb{Q}[t^{\frac{1}{2^n}}:n\geq 0]$ and consider $A[u,u^{-1};\sigma]$, a skew Laurent polynomial ring, where $\sigma(t^q)=t^{2q}$. We study the kernel of derived extension of scalars along $ A[u,u^{-1};\sigma]\longrightarrow\mathbb{Q}[u,u^{-1}], $ the quotient by $\mathfrak{m}A[u,u^{-1};\sigma]$, with $\mathfrak{m}=(t^{\frac{1}{2^n}}:n\geq 0)$. A finite free resolution of $\mathfrak{m}A[u,u^{-1};\sigma]$ over $ A[u,u^{-1};\sigma]\otimes_{\mathbb{Q}} \bigl(A[u,u^{-1};\sigma]\bigr)^{\mathrm{op}} $ proves smoothness. A specialization inequality for matrix ranks supplies a nonzero exact colimit-preserving functor that vanishes on every compact object, proving failure of compact generation.

Let $\mathcal{T}$ be a tensor triangulated category with a compact tilting generator $M$ and abelian heart $\mathcal{H}_M = M^{\bot_{\neq0}}$. We define a triangulated flat dimension $\mathrm{flatdim}_{\mathcal{T}}X$ for $X\in\mathcal{T}$, and give criteria in terms of tensor vanishing and resolutions by the class of pure quotients of $\mathrm{Add}(M)$. We further introduce $M$-ghost length and $M$-ghost dimension, and show that the global dimension of $\mathcal{H}_M$ controls the $M$-ghost length of all objects in $\mathcal{T}$, and that the $M$-ghost dimension of $\mathcal{T}$ coincides with the weak global dimension of $\mathcal{H}_M$ and the supremum of $\mathrm{flatdim}_{\mathcal{T}}X+\mathrm{inf}X$ of bounded objects, and the relative triangulated dimension of $\mathcal{T}^c$ with respect to $\mathrm{add}(M)$. This extends the classical Hovey-Lockridge theorem, which is recovered for derived categories of rings.

We develop a homotopical framework for understanding the space of quantum spin systems based on topological $n$-nets, which assign associative unital algebras to local regions in an $n$-dimensional space in a manner that captures the locality structure of the underlying lattice theories. The net-theoretic perspective admits a natural categorification, in which the category of topological $n$-nets carries the structure of a higher Morita category. Moreover, the nets of greatest physical interest form a Karoubi-complete subcategory that shares many of the characteristic structural features of the category of finite TQFTs. Our main result identifies the group of locality-preserving symmetries of (3+1)-dimensional quantum spin systems (i.e. quantum cellular automata modulo stabilization, finite-depth quantum circuits, and translations) with the full Witt group of nondegenerate braided fusion categories, thereby answering a question posed by Haah.

We announce an explicit construction of a unitary, strongly rational vertex operator algebra $U$ of central charge $8$ inside $V_{E_6\oplus A_2}$, with a ribbon equivalence $\mathrm{Rep}(U)\simeq Z(\mathcal H_3)$ realizing the Haagerup double. The vertex algebra is generated by six affine currents and three specified weight-two fields; its weight-one Lie algebra is $\mathfrak{sl}_2\oplus\mathfrak{sl}_2$, at levels $1$ and $39$. We outline the argument, which combines exact lattice calculations, an all-weight strong-generation induction and $C_2$-cofiniteness, semisimple descent for a full corner of Zhu's algebra, modular Galois symmetry, and holomorphic $E_8$ extensions.

Let $f$, $g\colon A\to B$ be two A$_{\infty}$-morphisms between two A$_{\infty}$-algebras (linear over a commutative ring $\mathbb{K}$). In this paper, we compare different notions of (right) homotopy between $f$ and $g$. We note that the A$_{\infty}$-algebra $B\otimes R$, where $R:=\mathbb{K}[t,dt]$, is a model for the interval $[0,1]\times B$ in the sense of Fukaya-Oh-Ohta-Ono, and we give a short alternative proof, in the unfiltered case, of the existence of comparison morphisms between such models. We use it to prove that two A$_{\infty}$-morphisms are homotopic à la Seidel if and only if, considered as morphisms of 2-colored DG operads, they are elementary homotopic à la Markl-Shnider-Stasheff through a homotopy which is constant on the two monochromatic parts. We also provide an example of two A$_{\infty}$-morphisms which are elementary homotopic but not weakly equivalent (and so homotopic à la Seidel). Moreover, over a field, we observe that the models for the interval are good path objects in the category of A$_{\infty}$-algebras with (non strict) A$_{\infty}$-morphisms, and we explain why weakly equivalent A$_{\infty}$-morphisms need not be homotopic.

In this work, we propose a mathematical description of the critical point of boundary phase transitions in 2+1D topological orders. Given two boundary phases, $\mathcal{C}_A$ and $\mathcal{C}_B$, with corresponding Lagrangian algebras $A$ and $B$ in a 2+1D topological order $\mathcal{C}$, respectively, we propose that the critical point of the phase transition between these two phases, if it exists, corresponds to $D = A \times_{M} B$, which is the fiber product of $A$ and $B$ over $A$-$B$-algebra $M$. We prove that $D$ is also a condensable algebra. Additionally, we present an alternative classification of condensable algebras in $Z(Vec_G^\omega)$ and reveal splitting phenomenon that arises during the condensation of certain algebras. We also perform explicit computations for the condensable algebras within several examples of $Z(Vec_G^\omega)$, covering both Abelian and non-Abelian groups $G$.

In this paper, we study the tensor triangular geometry of band algebras $RB$, where $B$ is a finite left regular band and $R$ is a commutative Noetherian ring. We show that the Balmer spectrum of $\Perf(RB)$ is homeomorphic to $\Spec R\times L_B$, where $L_B$ is the support lattice of $B$. We also show that there is a bijection between the localizing ideals of $D(RB)$ and the stable subsets of $\Spec R\times L_B$. As a consequence, the telescope conjecture holds for $D(RB)$. Finally, we construct a band $B$ such that the generalized Nerves of Steel Conjecture fails for the non-rigid tensor triangulated category $\Perf(RB)$.

For $n\ge1$, we study representations of a simplicial group $G$ in the full $\infty$-subcategory of simplicial vector spaces whose normalized homology is concentrated in degrees $0,\ldots,n-1$, retaining its full mapping spaces. The resulting functor category depends only on the $n$-truncated homotopy type of $\mathbf{B}G$ and admits projectively bifibrant strict diagram and simplicial group-algebra module models. If restriction along a specified map $f:G\to H$ is an equivalence, then $\pi_0(f)$ is a group isomorphism; the proof uses ordinary extension and restriction of scalars. Yet for every $n\ge2$ and prime $q$ invertible in $k$, the nontrivial source $K(\mathbb Z/q,n-1)$ has the same $n$-representation category with vertex evaluation as the trivial group. The group-algebra augmentation is a weak equivalence, so the constant functor is also an equivalence for untruncated representations. Thus $n$-representations depend only on the $n$-truncation of $\mathbf{B}G$, but need not determine that truncation, even together with vertex evaluation.

We develop a semisimple obstruction theory for finite-length abelian categories. To each object \(X\) we associate a canonical obstruction subobject \(\mathcal O(X)\), consisting of the part invisible to all morphisms into semisimple objects. The quotient \(X/\mathcal O(X)\) is semisimple and universal among such morphisms, yielding a canonical semisimplification reflector and characterizing semisimplicity by the vanishing of obstruction. Iterating the construction gives an obstruction filtration which, in finite-length module categories, recovers the classical radical filtration and Loewy theory. Obstruction is also heart-dependent: under Happel–Reiten–Smalø tilting, extension data can appear as new obstruction layers, leading to obstruction shadows and visible torsion. Thus classical radical theory arises as the fixed-heart module-theoretic instance of a categorical framework for semisimple reconstruction, persistence, and variation of heart.

We introduce a systematic way to use the subobject functor on any category to set up a framework for generalized topological and measure-theoretic notions and inferences. Concretely, we develop two connected calculi: 1. a fourfold calculus derived from notions of smallness and density that generalizes, among other things, topological density and measure-theoretic negligibility, 2. a twofold calculus that generalizes topological and measure-theoretic inference procedures. The resulting theory is distinguished in that it is inter-categorical in character: its definitions can be applied not just to describe smallness and density of subobjects in any category and classify morphisms by how they handle them, but also functors and natural transformations by how they transform them, somewhat like the theory of limits but with a simple and intuitive conceptual meaning that applies to each incarnation. We provide examples of these notions from many areas of mathematics and outline their unfolding throughout category theory as we add assumptions that bring them closer to classical settings, which we complete by generalizing the topological duality of open and closed subsets and outlining a possible framework for a categorical measure theory. Central to this interaction is the subobject 2-functor and the 2-category of upward-bounded preorders, so in the last part of the text we provide a study of their function. Concretely we derive representation theorems for this 2-functor and 2-category and a conjecture about how to generalize these theorems to n-categories.

Let $\mathcal{G}(G,A,\alpha)$ be a discrete 2-group, given as the {categorical} extension of an ordinary group $G$ by an abelian group $A$. We define principal $\mathcal{G}$-bundles over a manifold in terms of Čech data over a good open cover. In the case of the manifold $S^2$, we calculate this bicategory explicitly, and identify its 2-skeleton. We compare with the holonomy bicategory $\mathrm{Fun}(*/\!\!/ \pi_{\le 2}(S^2), * /\!\!/ \mathcal{G})$, and provide a concrete equivalence between the two.

We construct effective finite group actions on smooth projective complex curves for which Conjecture A of Polishchuk and Van den Bergh fails. For every $h\geq1$, an explicit $S_3$ action on the smooth projective model of $y^2=t^{12h-5}+t$ has quotient of genus $h$ and exactly one ramification orbit, with inertia of order three. The fixed quotient for the three-cycle class consists of two points, but its derived category admits no $\C$-linear exact fully faithful embedding into the equivariant derived category. More generally, when the coarse quotient has positive genus, we prove that the maximum number of pairwise completely orthogonal exceptional objects is $\sum_i\lfloor e_i/2\rfloor$, where $e_i$ are the inertia orders. We also characterize conjugacy-class decompositions with embeddings linear over the coarse quotient: such a decomposition exists if and only if $|[g]\cap H_i|\leq\lfloor e_i/2\rfloor$ for every nonidentity conjugacy class and every branch inertia group $H_i$. The sufficiency proof uses coloured paths of type $A$ and derived reflection functors, and produces kernels on the natural fibre products. The known ordered curve decomposition still yields a decomposition indexed by the connected components of the fixed quotients.