Papers

We study the marked simplicial set obtained as the Roberts-Street nerve of the walking coinductive equivalence. We show that it is a non-contractible saturated complicial set for which all of its finite truncations are contractible. When regarding saturated complicial sets as a model for right $(\infty,\infty)$-categories, it represents a concrete and explicit example of a right $(\infty,\infty)$-category that is itself non-contractible, but whose reflection to a left $(\infty,\infty)$-category is contractible.

In this short note, making use of the theory of noncommutative motives, we prove that the Grothendieck classes of the twisted symplectic flag varieties are linearly independent and moreover homogeneous-quadratic independent in the Grothendieck ring of algebraic varieties.

The category TFS of thin flat surfaces, introduced by Khovanov, Qi, and Rozansky, is a strict symmetric monoidal category whose morphisms are compact oriented surfaces with corners arising as neighbourhoods of immersed graphs in a strip. We prove two main results. First, a normal form theorem: every connected viewable tf-cobordism is equal, in TFS, to a canonical composite determined by its topological type. Second, a sufficiency theorem: we establish a list of relations in the TFS category which are sufficient, meaning that any two composites of generators representing the same tf-cobordism are related by a finite sequence of the listed relations. Together, these results give a complete generators-and-relations presentation of TFS.

We study the singularity categories of simple singularities of the same dimension over an algebraically closed field of positive characteristic, and show that, as in characteristic zero, these categories are not equivalent as triangulated categories unless the underlying singularities are analytically isomorphic. In contrast to the characteristic zero case, simple singularities in positive characteristic cannot be distinguished solely from the Auslander-Reiten quivers of their singularity categories. To address this, we extend to positive characteristics a theorem by Hua and Keller, which asserts that the 0th Hochschild cohomology of the dg singularity category of an isolated hypersurface singularity in characteristic zero is isomorphic to the Tyurina algebra of the defining polynomial. Furthermore as an application, we determine the condition for the singularity category of a rational double point (i.e., a simple singularity of dimension two) to be standard. We prove that such a category is standard if and only if the defining polynomial is weighted homogeneous.

We describe a (combinatorial, monoidal, Cat-enriched) Quillen model structure on the category of Cartmell's generalized algebraic theories (gats); its homotopy bicategory consists essentially of Taylor's rooted display map categories. This allows us to compare two kinds of morphisms of gats: one where sort dependency and substitution are preserved strictly, thus directly matching the syntax, and one where the given structure is preserved up to isomorphism. We prove a strictification result for morphisms out of cofibrant theories, which are the retracts of theories without sort equality axioms. Along the way, we give a structural characterization of when a contextual category can be presented without sort equality axioms. Our results also imply that when restricted to cofibrant objects, the tensor product of gats has the expected semantic behaviour, namely, it corresponds to the tensor product of locally finitely presentable categories equipped with a cofibrantly generated weak factorization system. Strict and weak morphisms specialize, respectively, to two familiar concepts of model of a gat A: ones valued in iterated families of sets, with substitution interpreted as reindexing, and set-valued models of the contextual category C(A) viewed as a finite-limit sketch. We characterize strictifiability of a model of the latter kind via a loop freeness condition on a certain map of functors out of the category of context projections of A.

We prove an $(\infty,2)$-categorical version of Koszul duality for operads and cooperads in spectra by showing that there is an equivalence between a Morita category of operads and bimodules and a dual Morita category of cooperads and bicomodules. This result subsumes various forms of operadic Koszul duality present in the literature.

We show that power semigroups of groups, and more generally reduced finitary power monoids, arise naturally as convolution monoids in Kleisli categories of powerset monads: (1) for the non-empty powerset monad, the Kleisli Hom-space $\mathrm{Hom}_{\mathbf{Kl}(\mathscr P_+)}(1,G)$ is isomorphic to the power monoid $\mathcal P_+(G)$; (2) for the reduced finite powerset monad on pointed sets, the Kleisli Hom-space $\mathrm{Hom}_{\mathbf{Kl}(\mathscr P_{\mathrm{fin}})}$ $(\mathbb Z/2\mathbb Z,H)$ is isomorphic to the reduced finitary power monoid $\mathcal P_{\mathrm{fin},1}(H)$. This unifies several constructions in power semigroup theory: Kleisli convolution representations of semigroups, base change along surjective group homomorphisms, and rigidity of automorphism groups. As an application, we prove that for every proper numerical monoid $S$, the Kleisli Hom-monoid $\mathrm{Hom}_{\mathbf{Kl}(\mathscr P_{\mathrm{fin}})}(\mathbb Z/2\mathbb Z,S)$ is rigid. This gives a proof of the Tringali–Yan conjecture in the language of Kleisli categories. It should be mentioned that this conjecture was already proved by Bhowmik and Tringali in a preprint posted on arXiv on July 25, 2026.

Building on our previous work on 2-Morita equivalence for $E_2$-algebras using topological pictures, we develop a systematic framework for Morita equivalence of topological orders in different dimensions in terms of $n$-Morita categories $\mathrm{Mrt}_{E_n}(\mathcal{C})$. In this framework, various notions of $n$-Morita equivalence are unified as equivalences of objects in $\mathrm{Mrt}_{E_n}(\mathcal{C})$. We also compare the constructions of higher Morita categories due to Haugseng and to Gwilliam–Scheimbauer. For $n=1,2$, we prove that the functor $\mathrm{Mod}_n:\mathrm{Mrt}_{E_n}(\mathcal{C})\to \mathrm{Mrt}_{E_{n-1}}(\mathrm{LMod}^{\mathrm{rep}}(\mathcal{C}))$ is an equivalence, relating the algebraic description of higher Morita theory to its realization in terms of module categories. In this formulation, the notion of a bi-bimodule arises naturally and provides a common framework for local and confined modules. Its explicit orientation data, together with the corresponding fusion rules, clarifies the relations among defects arising from condensation in topological orders.

We answer the question whether all Heyting algebras can appear as the lattice of subterminal objects of an elementary topos in the negative. Concretely, we show that the free Heyting algebra on two generators, hence also on N generators for every N greater than 2, cannot be such a Heyting algebra. The mathematical results in this document were obtained with the help of ChatGPT 5.6 Sol, although the document itself was written entirely by us, and we take full responsibility for its contents.

Let $\mathcal C$ be an $(n+2)$-angulated category. Zhou proved that, when $n$ is odd, if the Auslander-Reiten $(n+2)$-angles generate the relations for the Grothendieck group of $\mathcal C$, then $\mathcal C$ is locally finite. Whether the corresponding statement remains valid for even $n$ is still open. In this paper, we give a partial affirmative answer to this problem by establishing a sufficient condition under which the same implication holds for even $n$. We further show that our sufficient condition is satisfied by a broad class of examples, thereby demonstrating that the result extends well beyond isolated cases.

We establish a new bridge connecting quantales, semigroups and the theory of formal concept analysis. By introducing residuated relations whose domains are semigroups, we show that every quantale arises as the formal concept lattice induced by such a relation. Furthermore, this construction yields an equivalence between the category of quantales and a category of such residuated relations whose morphisms are multiplicative bonds.

This paper studies finitistic dimensions in triangulated categories with a compact silting generator. We unify several notions of (big) finitistic dimension appearing in the literature, show that they are essentially equivalent, and relate them to the abelian heart when the generator is tilting. This yields a new categorical perspective on the finitistic dimension conjecture. Furthermore, we establish explicit inequalities for (big) finitistic and global dimensions under recollements, demonstrating that finiteness in the middle category is equivalent to finiteness in the outer categories. These results generalize classical ring-theoretic theorems and apply to triangular matrix rings, exact contexts and trivial extensions.

Esakia duality is the dual equivalence between Heyting algebras and Esakia spaces. However, the traditional proof uses the Prime Ideal Theorem to recover the algebra from its spectrum, a choice principle that is not constructively valid. We build on Townsend's localic Priestley duality to describe a fully constructive, localic Esakia duality. On the algebraic side, we use the recently introduced Heyting frames as the point-free version of Heyting algebras. On the spatial side, Esakia spaces are modeled by two point-free alternatives. First, Esakia locales are defined as a subclass of the ordered Stone locales introduced by Townsend, and it is shown directly that Townsend's equivalence restricts to a duality between Heyting frames and Esakia locales. Second, using the theory of conic frames, in which join-preserving closure operators on frames model localic preorders, we introduce Esakia frames as a frame-theoretic analogue of Esakia spaces. It is shown that Esakia frames are equivalent to Heyting frames and dually equivalent to Esakia locales, and that this factorises the restriction of Townsend's equivalence. This yields a fully constructive, localic Esakia duality.

There is an isomorphism between the frame of radical thick tensor ideals of an essentially small tt-category and the frame of open sets of the Hochster dual of its Balmer spectrum. This isomorphism allows us to compare topological properties of the latter with tt-geometric properties of the former. We use this to give an intrinsic definition of those tt-categories which satisfy a variant of the second De Morgan law, and furthermore give a construction for producing such tt-categories.

We give an example of a coherent theory, a topological space, and a continuous assignment of the (model-theoretic) types of the theory to the points of the space, such that there is no sheaf model of the theory over the space with the property that the fibre over a point, as a set-based model, realises the corresponding type; thus, we answer in the negative a problem posed by Kristóf Kanalas.

In this paper, we prove a product rule for Goodwillie derivatives: given a differentiable $\infty$-category $\mathcal{C}$ whose stabilization is equivalent to the $\infty$-category $\mathrm{Sp}$ of spectra, we show that the derivatives functor $\partial_* \colon \mathrm{Fun}^\omega(\mathcal{C}, \mathrm{Sp}) \to \mathrm{RMod}_{\partial_*\mathrm{id}_\mathcal{C}}(\mathrm{SSeq}(\mathrm{Sp}))$ is strong symmetric monoidal, where the source is equipped with the pointwise tensor product and the target with Day convolution. Since the Koszul dual of $\partial_*\mathrm{id}_\mathcal{C}$ can be recovered as a coendomorphism operad from Day convolution, this product rule is useful for calculating the operad $\partial_*\mathrm{id}_\mathcal{C}$ in examples. We derive the product rule as a consequence of the more general statement that taking derivatives preserves cartesian products on the $(\infty, 2)$-categorical level. In fact, the main theme of this paper is that the extraction of Goodwillie derivatives preserves a lot of structure when regarded as a functor of $(\infty, 2)$-categories: apart from products, it also preserves cotensors and certain pullbacks. We illustrate how our results can be used to calculate Goodwillie derivatives by determining the operad structure on the derivatives of the identity functor in pointed spaces, algebras over an operad and sheaves on a site.

In the context of enriched category theory, we give necessary and sufficient conditions for a module morphism $\alpha \dd M \to \CC(F,Z)$ to exhibit a functor $Z\dd \CA\to \CC$ as an absolute $M$-weighted colimit of a functor ${F\dd \CB\to \CC}$. We also review, with short proofs, the various criteria for the weight $M$ itself to be absolute, in the sense that any $M$-weighted colimit is absolute. Finally, we prove that any absolute $M$-weighted colimit can be viewed as a colimit weighted by an absolute weight $M'$.

We construct a categorical framework in which we can view the direct integral as a functor taking as input objects in a certain category of Abstract Banach bundles and outputting a Banach space. We do this by constructing a quasi-abelian category of abstract Banach bundles, and we prove certain foundational and theoretical results about this category and its objects. As a consequence, we are able to define a notion of direct integral of sheaves.

With the aid of the torsion-theoretical approach, several properties/characterizations of hereditary QF-$3^{+}$ rings are found. These characterizations are formulated in terms of the category of projective modules, the category of injective projective modules, and also in terms of the maximal left/right ring of quotients by Utumi. Moreover, it is shown that, for a hereditary QF-$3^{+}$ ring, the `largest stable submodule' radical (on the category of left/right modules) is permutable with injective envelopes. Besides, it is shown that there is a bimorphism (in the category of associative rings with identity) from such a ring to a semisimple left/right Artinian ring.

We study the cofree functor on the category of group actions and examine its categorical and combinatorial properties. Motivated by this construction, we introduce a family of functors associated with subgroups and develop their theory through a corresponding coreflective subcategory of group actions. Finally, we investigate the comonad in the category of sets induced by the cofree adjunction and prove that it classifies the groups.

We give an equivalent definition of a non-pointed torsion theory, also called a pretorsion theory in the literature. This uses a new, more general, notion of a near-torsion theory, and does not use non-pointed exactness. Furthermore, a general method of associating the `largest' (non-pointed) torsion theory to any given near-torsion theory is described. We also describe a near-torsion theory formed by the class of cokernel diagrams and the class of kernel diagrams in a pointed category with kernels and cokernels, and its associated torsion theory.

We construct the category of overt discrete quasi-Polish spaces and the category of compact Hausdorff quasi-Polish spaces (and some of their subcategories) as internal categories of the category of effective quasi-Polish spaces and computable maps. To demonstrate that the constructions are natural, we show that Stone duality is computable, in the sense that the dual contravariant functors and the natural transformations demonstrating their adjointness are computable.

Let $D(G)$ denote the derived category of smooth $G$-representations on $k$-vector spaces, where $G$ is a locally pro-$p$ group and $k$ is a field of characteristic $p$. In this paper we are primarily interested in the graded center of the subcategory of compact objects $Z^*(D(G)^c)$ and variants thereof. When $G$ is a $p$-adic Lie group, without proper open centralizers, we completely determine this center modulo locally nilpotent elements and give various applications.

We identify, functorially, the boundary limit of a simply defined presentation functor with the dimension quotient, both for groups and for Lie rings over integers.

We study semi-orthogonal decompositions of perfect derived categories of gentle algebras via marked ribbon surfaces. We characterize such decompositions in terms of suitable disjoint union decompositions of full formal arc systems, and relate this description to good cuts of the corresponding surfaces. For gentle algebras, rotations of curves induce fully faithful functors from extension-closed subcategories of module categories to the components of the associated semi-orthogonal decompositions. Under additional Abelian and extension-comparison conditions, these constructions give derived decompositions of the module categories.

Let $G$ be a finite group. In this note, we construct an element ${\sigma}^G$ in the Burnside ring of $G$ over $\mathbb{Q}$, which gives a rational number $m(G)$ that is the sum of the inverses of the element orders in $G$, by using the deflation map ${\mathrm{Def}}^G_{G/N}$ induced by the $(G/N,G)$-biset $G/N$ for a normal subgroup $N$ of $G$. As a corollary to Theorem, we obtain another expression for the rank of the crossed Burnside ring $B^{\rm c}(G)$ of $G$ and the determinant of the Cartan matrix of $p$-local Mackey algebra of $G$ over a field $k$ of characteristic $p>0$ with big enough.

The notion of a base of computability $\mathscr{C}$ in a category $\mathscr{C}$ was introduced as a tool to generate computability models, in the sense of Longley and Normann, from categories. In this paper we introduce the category $\mathsf{CatBaseComp}$ of categories with a base of computability, and we show that $\mathsf{CatBaseComp}$ has all pie limits. We prove that a Grothendieck fibration lifts a base of computability in the base category to a base of computability in the total category of the fibration, and conversely, a pullback-preserving Grothendieck fibration maps a base of computability in the total category to a base of computability in the base category of the fibration. Connecting $\mathsf{CatBaseComp}$ with the semantics of dependent type theory, we show that $\mathsf{CatBaseComp}$ is a type-category, or a (fam, $\Sigma$)-category with a terminal object. Moreover, we prove that CatBaseComp is a (2-fam, $\Sigma$)-category, a 2-categorical generalisation of a (fam, $\Sigma$)-category. Finally, we describe the canonical (2-dep, $\Sigma$)-structure of CatBaseComp, i.e., the canonical dependent arrows of CatBaseComp that are compatible with its (2-fam, $\Sigma$)-structure.

We provide some information about the Drinfeld center $\Z(\Ver_p)$ of the Verlinde category $\Ver_p$. We compute explicitly the graded Cartan matrix, bound quiver and cohomology of $\Z(\Ver_p^+)$, and prove that the category $\mathscr{Z}(\Ver_p^+)$ is wild for every $p\ge 7$. In the special case $p=5$, we show that $\Z(\Ver_5^+)$ has exactly $10$ non-isomorphic indecomposable objects, classify them and describe the Green ring of $\Z(\Ver_5^+)$, and compute the semisimplification of $\Z(\Ver_5^+)$.

Retrodiction is the act of inferring a cause from its effects, the most common example of which is Bayesian inference. Retrodiction can be defined by its structural process-theoretic properties, which are mathematically captured by category theory. This categorical definition of retrodiction has recently been shown to potentially isolate the Petz recovery map as a unique universal candidate for quantum Bayesian inference. This paper extends these results to the infinite-dimensional setting on von Neumann algebras. In the process, we provide a pedagogical review of the Petz recovery map in infinite dimensions and its relation to the more commonly used expression in the finite-dimensional setting. We formalize the open question as to whether these categorical axioms for retrodiction do in fact uniquely characterize the Petz recovery map. If such a characterization holds, this would show that Bayesian inversion and the Petz recovery map are structural necessities and not simply useful algorithms for classical and quantum inference.

Let $\mathsf{P}$ be a finite graded poset and $\Delta_a^{\mathsf{P}}$ the height-$a$ thickening of its diagonal, with projections $q_1,q_2$ to $\mathsf{P}$. We study the window convolution $C_a=\operatorname{Lan}_{q_1}\circ q_2^\ast$ on $\mathrm{Shv}(\mathsf{P};k)$, a discrete analogue of convolution against a thickening kernel. An interleaving distance needs the homotopy window convolution $\mathbb{C}_a$ to compose as a flow, $\mathbb{C}_a\mathbb{C}_b\simeq\mathbb{C}_{a+b}$; where the meet assignment $\Phi$ is total it is a functor and carries a comparison map. Finality is sufficient, and necessary at every minimal apex and wherever the finality defect of $\Phi$ is essential; where $\Phi$ is total at a minimal apex with unit windows, it is the failure of a length-two interval above the apex to have a single interior element. The flow fails at every branching length-two interval with minimal bottom element, and with it on the face poset of every finite regular cell complex of dimension $\ge2$. It survives on tame posets, where $\mathrm{id}\Rightarrow\mathbb{C}_a$ gives a canonical extended interleaving pseudometric on $\operatorname{D^{b}}(\mathrm{Shv}(\mathsf{P};k))$; in the saturation cases computed here it takes no finite value above the length of $\mathsf{P}$, and is finite if and only if the derived colimits agree.