Papers

We introduce gemstones in a pretriangulated category, which are used to perturb pretriangulated structures. For a triangulated category $\mathcal T$ whose stable category of coherent functors admits a gemstone, we construct pretriangulated structures on $\mathcal T$ which fail the octahedral axiom (TR4). This confirms a long-standing conjecture on triangulated categories.

We develop a general $\infty$-categorical framework for internal Eilenberg-MacLane objects, internal homotopy groups, and an internal notion of covering spaces, and study their behavior under suitable localizations. Applying this to the ordinary motivic localization, we identify internal $n$-Eilenberg-MacLane objects with strongly $\mathbb{A}^1$-invariant sheaves of (abelian when $n\geq 2$) groups, yielding a formal obstruction to the motivic homotopy category being an $\infty$-topos. We prove that taking $\mathbb{A}^1$-localizations induces an equivalence between classical $\mathbb{A}^1$-coverings of a Nisnevich local space and the internal motivic coverings of its motivic localization, providing a streamlined proof of a generalized motivic Van Kampen theorem. Along the way, we establish a Nisnevich-local-to-global $\mathbb{A}^1$-connectivity criterion: a $k$-scheme is $\mathbb{A}^1$-connected if and only if it admits a Nisnevich cover by $\mathbb{A}^1$-connected schemes such that each pairwise intersection has a ($k$-)point. Finally, we show that over a general Qcqs base, passing to the birational motivic homotopy category recovers certain essential topos-theoretic properties absent in the ordinary $\mathbb{A}^1$-setting. In fact, for a scheme with finitely many generic points, we show that the birational motivic homotopy category is a Postnikov-complete $\infty$-topos of cohomological dimension $0$.

We introduce the notion of R-full Schreier internal category, which is the monoid analogue of the notion of aspherical abelian groupoid. We associate with every R-full Schreier internal category a direction, which is a Schreier split extension with commutative and cancellative kernel. We show that this association is functorial and that such functor is a product preserving cofibration. Thanks to these properties, we equip the connected components of the fibres of such functor with canonical commutative monoid structures. Using the equivalence between Schreier internal categories and crossed semimodules of monoids, we describe these commutative monoids in terms of crossed Schreier extensions.

Given a category $\calC$ and a center, that is a collection of pairs $(d_i, N_i)$ consisting of a morphism $d_i$ and a sieve $N_i$ over its codomain, the dilatation of $\calC$ is a new category $\calC'$ in which every $n \in N_i$ factors, uniquely and functorially, through $d_i$. This paper presents the theory of dilatations of categories through a full formalization of the construction and its main theorems in the Lean~4 proof assistant, on top of the Mathlib library. An appendix collects a systematic dictionary between the mathematical statements and the Lean declarations that formalize them.

This paper studies the asymptotic growth of tensor powers in affine Hecke categories, or equivalently of powers of Kazhdan–Lusztig basis elements in affine Hecke algebras. We prove general bounds in arbitrary affine type, determine precise asymptotics in affine type A1, and establish corresponding results for several natural families in affine type A2.

For any symmetrizable generalized intersection matrix (GIM) $C$, we construct an acyclic valued quiver $(Q,\mathbf{d})$ endowed with an involution $\theta$. Let $\mathcal{D}$ be the bounded derived category of finite-dimensional representations of $(Q,\mathbf{d})$, and let $\Sigma$ stand for the suspension functor of $\mathcal{D}$. We show that the orbit category $\mathcal{D}/(\theta\circ\Sigma)$ carries a canonical triangulated structure and is $2$-periodic. Applying Peng–Xiao's construction to this orbit category, we prove that the GIM algebra $\operatorname{gim}(C)$ is isomorphic to the integral Ringel–Hall Lie algebra associated with $\mathcal{D}/(\theta\circ\Sigma)$. As a further application of the above machinery, we investigate elliptic Lie algebras of types $D_4^{(1,1)}$, $E_6^{(1,1)}$, $E_7^{(1,1)}$ and $E_8^{(1,1)}$. For each elliptic Dynkin diagram, we define a finite-dimensional algebra $A$ by taking an appropriate quotient of the acyclic quiver $Q$ attached to the GIM matrix $C$. From the resulting $2$-periodic triangulated categories, we build the corresponding Ringel–Hall Lie algebras, and establish a surjective Lie algebra homomorphism from each elliptic Lie algebra to its integral Ringel–Hall counterpart. This map is conjectured to be injective, and its injectivity on real root spaces is confirmed.

We show how the direction functors can be used to develop a cohomology theory for Barr-exact and S-Maltsev categories, where S is a suitable class of split epimorphisms with a fixed section. Using the fact that, for any set B, the category of small categories with B as set of object is S-Maltsev with respect to the class of Schreier points, we show that the cohomology theory of small categories arising from the direction functors coincides with the one introduced by Hoff and Golasinski.

Topological defects in quantum field theories are believed to assemble into higher categories with extra structure. This has been made precise for defects in $2$- and $3$-dimensional oriented topological quantum field theories by Davydov, Kong, and Runkel and by Carqueville, Meusburger, and Schaumann, respectively. In this paper we study the extra structure present on these categories when the topological field theory is additionally reflection positive. We define reflection defect TQFTs as symmetric monoidal functors out of a defect bordism category that intertwine orientation reversal with complex conjugation; they are reflection positive if they satisfy an additional positivity condition. Our main result is that in two dimensions the bicategory of defects $\mathcal{T}_\mathcal{Z}$ associated to a reflection defect TQFT carries the natural structure of an $O(2)$-dagger bicategory, a structure we define explicitly. If the theory is reflection positive, $\mathcal{T}_\mathcal{Z}$ can be equipped with additional structure closely related to the definition of a 3-Hilbert space (the two agree up to some finiteness and completeness conditions).

In this largely expository paper we provide a deep and explicit exploration and exposition of the tangent structure on the category of schemes $\mathbf{Sch}_{/S}$ whose tangent functor $T(X) = T_{X/S}$ is the relative tangent scheme of Grothendieck described in \emphÉléments de Géométrie Algébrique} 4. In particular we provide explicit descriptions of the ways that the bifibration of quasicoherent sheaves and bifbration of quesicoherent sheaves of algebras over schemes may be built from the ways in which the bifibrations of modules and commutative algebras over commutative rings interact. We also show the ways in which these interactions give rise to an explicit description of the standard tangent structure on the category of schemes in terms of sheaves of Kähler differentials, properties of the relative spectrum functor, and more. Finally, we show that quasi-coherent sheaves can be reconstructed from their category of differential bundles by showing that for quasi-separated schemes $X$ and $Y$, there is an isomorphism $X \cong Y$ if and only if there is an equivalence of categories $\mathbf{DBun}(X) \simeq \mathbf{DBun}(Y)$.

We construct an explicit tilting equivalence between the incidence algebra of every rational Dyck staircase and a canonical idempotent corner of a higher Auslander algebra of type~$A$. In the coprime case, this corner identifies with the algebra $B_0$ introduced by Xing. The resulting Dyck-corner equivalence supplies the missing link in the previously known chain of equivalences and thereby proves the Chapoton-Ladkani-Rognerud conjecture for coprime positive integers. The Dyck-corner equivalence itself requires no coprimality hypothesis and is compatible with replicated algebras. Our main tool is a linear-categorical extension of the interval-tilting mechanism of Chapoton-Ladkani-Rognerud. The relative theorem applies to finite $\kk$-linear categories under finite-global-dimension assumptions on the total category and its fibers. In contrast with the incidence-category setting, it allows arbitrary finite-dimensional $\Hom$ spaces and zero composites of nonzero morphisms, and it does not require the diagonal endomorphism algebras to be semisimple. The tilting object is constructed from exact right Kan extensions of fiberwise representables. We compute its opposite indexed endomorphism category, including all forced-zero compositions, and hence its opposite endomorphism algebra. Iterating this construction one coordinate at a time yields an explicit derived equivalence between the incidence algebra of every finite coordinate staircase and an idempotent corner of a higher Auslander algebra of type~$A$. We further realize the resulting staircase derived categories as triangulated subcategories generated by product Lagrangians in partially wrapped Fukaya categories of stopped-disk symmetric products and, in the coprime Dyck case, as Fukaya-Seidel categories of symmetric Brieskorn–Pham singularities.

We prove that, under mild assumptions, the limit of a cofiltered diagram of $\infty$-categories is a reflective (resp. coreflective) localization of its oplax (resp. lax) limit. This gives rise to a number of exceptional 'stability' results for categorical properties under such limits, in particular one for adjunctions. As an application, we recover a push-pull formula describing certain filtered colimits in the $\infty$-category $\mathrm{Pr}^L$ of presentable $\infty$-categories.

Motivated by the need to reason about case statements quantitatively, we extend quantalic linear lambda-calculus with additive disjunction. We show that the resulting equational system is sound. We also show that when certain continuity properties (of the underlying quantale) are adopted, it is additionally (approximately) complete. We present several models of the extended calculus, involving for example meta-theoretical properties in categorical logic (gluing), probabilistic, and quantum computation. As a concrete application, we illustrate how a probabilistic model, based on Banach spaces, can be synergistically used with the calculus' equational system to reason about Cauchy sequences of random walks. This highlights the emergent shift from "program semantics as the science of program equivalence" to flexible, quantitative perspectives, involving functional analysis and beyond.

For each prime $p$ and integer $n \geq 1$, we construct a $p$-local spectrum $X_n$ with $X_n^{\otimes n} \not\simeq 0$ but $X_n^{\otimes (n+1)} \simeq 0$. This refutes the retract conjecture of Hovey-Palmieri. Our construction also allows us to determine the cardinality of the Bousfield lattice to be $2^{2^{\aleph_0}}$, answering a question of Dwyer-Palmieri.

A fusion category $\mathcal{C}$ is said to be $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$-quadratic if the group $G(\mathcal{C})$ of invertible objects is isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, and the remaining simple objects form an orbit under the action of $G(\mathcal{C})$. In this paper, we give a partial classification of $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$-quadratic fusion categories of rank six. More precisely, we show that its Grothendieck ring $\mathcal{K}_0(\mathcal{C})$ must be one of nine fusion rings if the fusion rule multiplicities are less than $20$, and the categorifications of five of them are previously known. We prove that one of the last four fusion rings can be realized as de-equivariantization of a near-group fusion category of type $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}+8$.

We show that every symmetric fusion category over $\mathbb{R}$ is equivalent to the category of finite-dimensional semi-linear representations of a $\mathbb{Z}_2$-graded finite super group. The proof uses Galois descent for tensor categories over $\mathbb{C}/\mathbb{R}$, reducing the classification to semi-linear $\mathbb{Z}_2$-actions on symmetric fusion categories over $\mathbb{C}$. As a further structural result, we establish a Tannaka-Krein type correspondence between symmetric fusion categories over $\mathbb{R}$ and finite groupoids with a $\mathbb{Z}_2 \times \mathrm{B} \mathbb{Z}_2$-action. This gives a complete real analogue of Deligne's classification result.

We develop a comprehensive theory of dualizable additive categories. We provide several equivalent characterizations, notably identifying them as separated Grothendieck prestable categories satisfying the $\mathrm{AB4}^*$ and $\mathrm{AB6}$ axioms. We establish a connection to almost mathematics by demonstrating that they arise precisely as the categories of connective almost modules over connective $\mathbb{E}_1$-rings. Furthermore, we prove that dualizable additive categories are generated by flat objects, and that the passage to flat objects yields an equivalence between dualizable additive categories and compactly assembled additive categories. As a primary application within analytic geometry, we characterize the category $\mathrm{Nuc}(R)_{\geq 0}$ of connective nuclear modules (in the sense of Clausen–Scholze) over an adic $\mathbb{E}_\infty$-ring $R$ via a universal property, identifying it as the additive rigidification of the category of connective complete $R$-modules. Finally, we construct the universal finitary stable localizing invariant for dualizable additive categories, the presentable stable category $\mathcal{M}\mathrm{ot}_{\mathrm{pst}}$ of prestable motives, and demonstrate that its unit corepresents nonconnective algebraic $K$-theory. We prove that the motives of small additive categories and those of dualizable additive categories generate the same presentable stable subcategory.

We prove a number of Jacobson/solvable/nilpotent-radical (co)invariance results for finite-dimensional associative or Lie (co)module-algebras over Hopf algebras $H$, provided the characteristic, if positive, is large relative to the dimension of the algebra and the (co)actions satisfy trace-preservation conditions automatic when $H$ is involutory. This generalizes a number of such radical-invariance criteria in the literature, due to A. Gordienko, V. Linchenko, Pagon-Repovš-Zaicev and others, providing a uniform enriched-categorical framework for those results.

We promote the $n$-birational motivic homotopy category assignment $S\mapsto \mathcal{H}^n(S)$ to a $Pr^L$-valued presheaf on $Corr(\mathrm{Sch})_{uglt,sm}$. As a consequence, the birational motivic homotopy category $\mathcal{H}^{b\mathbb{A}^1}(X)$ of a scheme $X$ with finitely many generic points decomposes as the cartesian product of the birational motivic homotopy categories of those generic points; in particular, for a variety $V$, $\mathcal{H}^{b\mathbb{A}^1}(V) \simeq \mathcal{H}^{b\mathbb{A}^1}(k(V))$. This implies that birational equivalences of schemes in $Sm_X$ can be detected via the birational contractibility of their generic fibers. Finally, we show that stably birational morphisms and purely transcendental field extensions induce fully faithful embeddings of birational motivic homotopy categories.

This paper establishes three main results on the extension dimension of syzygy module categories:(1)we prove that excellent ring extensions preserve extension dimensions of syzygy module categories,including syzygy categories of modules of finite projective dimensions;(2) for cleft extensions, we investigate the behavior of extension dimensions under natural nilpotency and projective conditions;(3)for commutative Artin rings, we establish local-global characterisations for the extension dimension.

We record a general condition guaranteeing that the animation of a small $1$-category remains a $1$-category. The proof is extremely elementary universal algebra. This recovers as a very special case the striking observation of Antieau, \cite{Ant}, that the animation of $\fnt^{\mathrm{op}}$ is a $1$-category. Our result is a fair amount more general, and for example applies to $\fnt^{\mathrm{op}}_{\mathcal{G}}$ with $\mathcal{G}$ a groupoid, as well as beyond this.

Category theory is a mathematical theory of composition, widely used in logic, computing, and physics. Here we apply it to give a theory of secure composition. In particular, we provide a categorical treatment of Canetti's Universal Composability (UC) framework for systems with a static number of parties and sessions, often termed UC for static systems, yielding four benefits. First, we present our results graphically yet retain rigor by applying a standard categorical technique known as string diagrams. In particular, our formulation of the composition theorem can be graphically verified with a short sequence of diagrams, while remaining translatable to equations and amenable to formal verification. Second, categories let us generalize so that our results extend beyond interactive Turing machines to other forms of computation, such as quantum computation or domain-specific languages. Third, categories help us drop some unnecessary restrictions of UC (e.g., our adversary can be a computational network rather than a single Turing machine); we prove equivalence between our variant and the usual UC, showing no expressiveness is lost. Finally, the categorical perspective leads us to identify and correct some minor technical oversights in the standard formulation of simple UC.

We study the functorial and categorical structure of the canonical rigid direct-system representation of finite local-unit-aligned totally ordered monoids. The local-unit map $\tau$ induces a canonical $\tau$-multiplication-coherent decomposition into component monoids, and the associated representation reconstructs the original ordered monoid from a finite chain-indexed rigid direct system whose proper transition maps are unit-constant. The present paper identifies the morphism classes for which this representation is categorical. First, we prove an equivalence between finite local-unit-aligned totally ordered monoids with strict block morphisms and rigid direct systems with directed-order-compatible system morphisms. Second, we prove an intrinsic equivalence for $\tau$-compatible homomorphisms, that is, isotone unital homomorphisms commuting with the local-unit map. In this second setting, distinct positive idempotents and hence distinct canonical components may collapse to a single target component; on the direct-system side this is represented by non-injective isotone index maps together with component maps satisfying the corresponding collapse and absorption axioms. Thus the canonical rigid direct-system representation is functorial both for strict block morphisms and for intrinsic $\tau$-compatible homomorphisms.

The heart of a $t$-structure and the ideal quotient by a cluster tilting subcategory are classical constructions that produce abelian categories from triangulated categories. Their higher analogues, namely $n$-extended hearts and ideal quotient categories by $(n+1)$-cluster tilting subcategories, are generally no longer abelian, but are known to carry both pretriangulated and extriangulated structures when the underlying triangulated category is algebraic. In this article, we introduce the notion of an abelian $n$-truncated category as a common framework for such higher constructions. We extend the heart construction for cotorsion pairs to $n$-cotorsion pairs on arbitrary triangulated categories, and prove that the resulting extended heart naturally carries compatible pretriangulated and extriangulated structures forming an abelian $n$-truncated category. This construction simultaneously generalizes the $n$-extended heart of a $t$-structure and the ideal quotient by an $(n+1)$-cluster tilting subcategory. It may also be regarded as a higher-dimensional generalization of the general heart construction for cotorsion pairs on triangulated categories. Finally, we show that the heart can be realized as an extriangulated localization of a suitable relative extriangulated structure on the ambient triangulated category.

In higher category theory, lax colimits are often understood to be a more useful and powerful generalization of usual (homotopy) colimits, which can be recovered from the lax colimit by a suitable localization. However, lax colimits do not provide the correct notion of gluing from the geometric perspective. Indeed, they are incompatible with the notion of categorical dimension, the Gray tensor product, and other basic geometric operations. In this paper, we develop the theory of oriented colimits, which correct the defects of lax colimits, and agree with lax colimits in dimension less than or equal to one. In order to study oriented colimits, we introduce a version of the Grothendieck construction which is compatible with enrichment in the Gray tensor product. We prove that the Grothendieck construction induces an equivalence between cartesian fibrations and presheaves of $(\infty,\infty)$-categories, which is enriched in the Gray tensor product of $(\infty,\infty)$-categories. Oriented colimits simultaneously generalize the concept of lax colimits and the Gray tensor product, and differ from lax colimits in much the same way in which the Gray tensor product differs from the cartesian product. We demonstrate the necessity of oriented colimits by showing that various fundamental constructions in higher category theory fail to be lax colimits but are instances of oriented colimits. As applications, we classify higher-categorical principal bundles, represent higher dimensional adjunctions by bicartesian fibrations of $(\infty,\infty)$-categories, and obtain higher categorical versions of Quillen's Theorems A and B, which admit very natural formulations in our framework.

We show that the methods of rewriting theory to establish coherence theorems can applied at the level of quasicategories. More precisely, for any category C which admits a presentation by a convergent rewrite system, we show that the corresponding weak (infinity,1)-category admits an Amick-Groves-Squier style presentation whose generators correspond to (some of) the critical branchings of the rewrite system. This follows entirely from reinterpreting K. Brown's simplicial proof of the usual homological Anick-Groves-Squier presentation in terms of the Joyal model structure instead of the Kan-Quillen model structure. We give several applications of this to the theory of quasicategories, including a relatively general coherence theorem for loop-free planar category theoretic diagrams and a new proof that pushout of Dwyer maps are homotopy pushouts.

In this paper, we introduce a new symmetric monoidal structure on $\mathbf{Cat}$, called the graph tensor product, with unit given by the terminal category. This tensor product falls in the middle of a factorization between the funny tensor product and the Cartesian product, giving a factorization connecting these two classical monoidal structures. We extend this construction to a symmetric monoidal structure on $2\mathbf{Cat}$, again with unit $D^0$, which provides an analogous factorization between the funny tensor product and the Cartesian product of $2$-categories. Using the $(\mathrm{bo},\mathrm{lff})$ factorization system on $2\mathbf{Cat}$, we construct a new symmetric monoidal closed model structure on $2\mathbf{Cat}$ whose tensor product restricts to the Cartesian product on the subcategory of flexible $2$-categories. Finally, we prove that this symmetric monoidal model structure fits into a square of weak symmetric monoidal Quillen equivalences relating the Gray tensor product and the flexible tensor product.

The extensional quotient completion of relational doctrines provides a common generalization of both the exact completion of categories with weak finite limits and the elementary quotient completion of existential elementary doctrines. In this paper, we study projective objects in relational doctrines with quotients, characterizing those obtained through the extensional quotient completion as those admitting a projective cover. We apply this result to doctrines of algebras for monads on relational doctrines with quotients, describing in which cases these arise as the extensional quotient completion of their restriction to (appropriate subcategories of) free algebras. This extends a similar result for monadic categories over exact ones, covering also more examples such as monads over the category of metric spaces giving rise to variants of quantitative algebras.

Let $k$ be an algebraically closed field of characteristic $p\geq 5$, and let $\mathrm{Ver}_p^+$ be the even part of the Verlinde fusion category $\mathrm{Ver}_p$, the semisimplification of $\mathrm{Rep}_k(\mathbb Z/p)$. Let $\mathfrak g$ be a linearly reductive Lie algebra in $\mathrm{Ver}_p^+$, i.e., one whose finite-dimensional representations are semisimple. A basic class of examples is obtained by semisimplifying a simple Lie algebra over $k$ equipped with the action of $\mathbb Z/p$ by a principal unipotent element, when $p$ exceeds its Coxeter number. We prove that $\mathfrak g$ is invariantless, i.e., that the unit object is not a summand of $\mathfrak g$. For odd $m$ with $3\leq m\leq p-2$, set $\mathfrak{g}_m:=\operatorname{Hom}_{\mathrm{Ver}_p^+}(L_m,\mathfrak g)$ and $E_{\mathfrak g}:=\bigoplus_{3\leq m\leq p-2,\ m\ {\rm odd}}\mathfrak g_m^{(1)}[m]$, where $(1)$ denotes Frobenius twist. Our main result is an isomorphism of graded algebras $H^\bullet_{\mathrm{CE}}(\mathfrak g)\cong\bigwedge^\bullet E_{\mathfrak g}^*$. We also identify this algebra with the de Rham cohomology $H^\bullet_{\mathrm{dR}}(G)$ of the group scheme $G=\exp(\mathfrak g)$ and show that the induced graded Hopf algebra structure agrees with the standard one on the exterior algebra. Moreover, if $V$ is a simple $\mathfrak g$-module on which $\mathfrak g$ acts nontrivially, then $H^\bullet_{\mathrm{CE}}(\mathfrak g,V)=0$. Hence for every finite-dimensional $\mathfrak g$-module $V$ one has $H^\bullet_{\mathrm{CE}}(\mathfrak g,V)\cong\bigwedge^\bullet E_{\mathfrak g}^*\otimes V^{\mathfrak g}$. This recovers the theorem of Borel and Chevalley on the cohomology of complex semisimple Lie algebras and its analogue in sufficiently large positive characteristic. We also prove similar results for relative cohomology.