Papers

We give a short, conceptual account of Dwyer–Kan's hammock localization in the setting of $\infty$-categories. Starting from Mazel-Gee's formula for localization via the relative Rezk nerve, we show that its Segalification can be described explicitly as a Segal anima of zig-zags. We also compute its mapping animae and recover and generalize Dwyer–Kan's hammock formula. As an application, we show that, when a relative $\infty$-category supports fractions, the mapping animae of localization admit a simple description. This gives a unifying treatment for several formulas of this type in the existing literature.

In this survey article, we give an introduction to the relatively young subject of $2$-representation theory, which studies categorifications of important players in classical representation theory. In particular, we provide a streamlined exposition of the main results leading to the classification of simple $2$-representations for Soergel bimodules associated to finite Weyl groups in characteristic $0$, obtained in [M. Mackaay, V. Mazorchuk, V. Miemietz, D. Tubbenhauer and X. Zhang, Simple transitive $2$-representations of Soergel bimodules for finite Coxeter types. Proc. London Math. Soc. 126 (2023)].

These notes develop four interconnected ways of reading a quantum circuit. A circuit for us begins as an operational composition of gates; then, it becomes a diagram whose local equalities may be used as calculations; next, it becomes a protected process once errors, syndromes, and logical degrees of freedom are separated; and finally, it becomes geometric when its connectivity, topology, and boundary data are treated as physical design parameters. The development begins at the level of bits and qubits before appealing to Deutsch's and Grover's algorithms as basic examples of quantum circuits. With the basics in hand, we interpret quantum circuits diagramatically, leading us to compact closed string diagrams and the ZX-calculus. After that, we consider how to correct quantum circuits by introducing the Knill–Laflamme condition, homological surface codes, and related concepts with a view towards thinking of these as operations on diagrams. The lectures eventually arrive at the properties of hyperbolic quantum codes and the prospect of physical superconducting circuits emulating the negatively-curved lattices needed to support those codes. These mathematical ideas and physical experiments, taken together, represent one way to impart a geometric layer onto quantum circuits. By the very end, we bring the ideas nearly full circle by assessing the extent to which these device physics experiments operationalize the basic ZX diagrams encountered much earlier in the story. While the later material reports on original research, and while the discussion becomes increasingly mathematical as the sections progress, no prior knowledge of quantum information, quantum computing, or quantum error correction is actually assumed.

Let $A$ be a commutative simple algebra in a braided finite tensor category $\mathcal{B}$. We identify the largest transparent subalgebra of $A$ as the algebra induced by a central lift of the free-module functor. This identification gives formulas for the Frobenius-Perron dimension and the Müger center of the category of local $A$-modules. These formulas give criteria for nondegeneracy, symmetry, and modularity, together with sharp bounds on $\mathrm{FPdim}_{\mathcal{B}}(A)$. We also realize the Müger center of $\mathcal{B}$ as a category of local modules over an adjoint algebra. Finally, we prove a relative-center factorization and deduce that taking the category of local modules preserves the relative Witt class.

Working in the setting of enriched algebraic theories for a system of arities, we study several aspects of dual adjunctions between enriched algebraic categories, which we call algebraic dual adjunctions. Firstly, we generalize Freyd's theorem on contravariant algebra-valued right-adjoint functors to this setting. Secondly, we establish a biequivalence between a 2-category of algebraic dual adjunctions and a locally discrete 2-category of bifold algebras, in a sense defined in prior work of the author. Thirdly, we define special classes of algebraic dual adjunctions that we call (left- and right-)stable, in which certain free objects are reflexive, and we establish biequivalences between these and special classes of bifold algebras defined in terms of commutants in prior work of the author. Fourthly, we show that every algebra for an enriched algebraic theory canonically induces left- and right-stable algebraic dual adjunctions, and we establish a biequivalence between such algebras and left- (or right-)stable algebraic dual adjunctions, and also between saturated algebras and stable algebraic dual adjunctions. We also discuss examples of algebraic dual adjunctions, including dualization of internal modules, Pontryagin and Binz-Butzmann duality, dualization of internal affine spaces and convex spaces, dualization of semilattices, dualization of complete sup-lattices, and dualization of abelian groups with reference to a theorem of Ehrenfeucht and Łoś.

We give a systematic account of the shape theory of $\infty$-topoi, viewing the shape of an $\infty$-topos as its generalized homotopy type. We establish the basic functorial properties of the shape, including preservation of colimits, descent, and homotopy invariance. We then prove that shape preserves cofiltered limits under compactness and perfectness hypotheses and establish Künneth-type formulas for products. Finally, we give conditions under which the shape of an $\infty$-topos determines its cohomology and homology.

Polygon equations form a family of equations generalizing the pentagon equation. In this paper, we construct semi-simplicial sets of permitted colorings associated with set-theoretic solutions of polygon equations and use them to define the corresponding (co)homology groups. We investigate several properties of these groups and establish an equivalence of categories between set-theoretic solutions of polygon equations and higher Segal semi-simplicial sets satisfying certain conditions. As a special case, our result recovers the correspondence between bijective set-theoretic solutions of the pentagon equation and $2$-Segal semi-simplicial sets proved by Dyckerhoff–Kapranov.

This paper is a continuation of arXiv:2606.17088, where I studied precartesian coextensions of commutative monoids by systems of abelian groups. In this paper, we generalise it to study coextensions by systems of commutative monoids. Among other things, we showcase that this version is able to simultaneously generalise Leech's version of coextensions and Redei's version, also called Schreier coextensions. We show that what going from systems of abelian groups to systems of commutative monoids costs us is quasi-inverses in $\mathsf{Pcoex}(M, \mathcal{L})$. Specifically, instead of a symmetric categorical group, they now only become symmetric monoidal groupoids. Moreover, though not explicitly stated in the body of the paper, another core difference is that for a monoid scheme $X$, regarded as a monoid functor, a precartesian coextension no longer need to be a monoid scheme. Thus, $\mathsf{Pcoex}(X, \mathcal{L})$ is, as stated, ineffective at studying monoid scheme coextensions. The rest of the theory goes through, but requires developing a new cohomological approach.

Arrow allegories provide a convenient abstract framework to work with lattice-valued relations, or more precisely, relations that use the elements of a given Heyting algebra as truth values. One characteristic of arrow allegories is that all relations of the given arrow allegory use the same Heyting algebra ${\mathcal H}$. In this paper we want to extend this approach to allegories where relations between different objects may use different lattices of truth values and even further to relations that use a different lattice of truth values for every pair in the relation. Therefore, we define three concrete allegories, $\mathrm{Rel}({\mathcal H})$, $\mathrm{Rel}^u({\mathcal H})$ and ${\mathcal H}{\rm-Rel}$, where the allegory listed later is a full suballegory of the previous ones. These three allegories capture the three different situations mentioned above. In particular, ${\mathcal H}{\rm-Rel}$ is the standard example of an arrow category. We investigate these allegories and provide suitable categorical definitions for these structures.

We introduce monoidal su-categories, an abstract categorical notion of single-input higher-order process over a monoidal category. The definition separates a base category C of lower-order processes from a monoidal category V of holes or supermaps and axiomatizes the compatibility needed for partial application to bipartite processes. For a fixed monoidal base C, monoidal su-categories, monoidal su-functors, and monoidal su-natural transformations form a 2-category MonSuCatC. We then show that the category Optic[C] of coend optics is 2-initial in this 2-category, giving an alternative universal-property characterisation of coend optics as the minimal monoidal theory of single-hole contexts.

We combine results of Rickard and Shaul with methods of intrinsic homological algebra and the theory of purity to prove that the finiteness of the big finitistic dimension of an Artin algebra is an intrinsic property of the singularity category of its opposite algebra, via its category of modules. This `object-free' approach complements work of Dey–Šťovíček and arrives at the same conclusion: the finiteness of the finitistic dimension of Artin algebras is a singular invariant (of the opposite algebras). In fact, our characterisation can be used to prove that finite finitistic dimension descends along certain fully faithful functors between singularity categories. We include an appendix where we explain how our methods can be used to prove non-existence of bounded t-structures on singularity categories.

We show that for a finite group $G$, the $G$-category $\underline{\mathrm{Sp}}_{G,\mathbb{Q}}$ of rational genuine $G$-spectra is parametrized semiadditive, in the sense of Cnossen–Lenz–Linskens, for morphisms of genuine $G$-spaces whose fixed point maps have $\pi$-finite fibers. In particular, this means that the canonical norm map between a colimit and a limit of rational genuine $G$-spectra indexed by such $G$-spaces is an equivalence. This extends the Wirthmüller isomorphism from $G$-orbits to a broader class of $G$-spaces and combines the theory of ambidexterity of Hopkins and Lurie with parametrized category theory.

We show that, over any algebraically closed field of any characteristic, the category of finite-dimensional projective modules over the preprojective algebra of generalized Dynkin type $\mathbb{L}_2$ has a pretriangulated category structure which is neither an algebraic nor a topological triangulated category structure.

We establish a duality between quotients and comprehensions in relational doctrines, a class of indexed posets modelling (a minimal fragment of) the calculus of relations. We show that every relational doctrine determines an “opposite" relational doctrine and that one has quotients if and only if the other has comprehensions. This duality extends to the 2-categorical level, yielding a 2-dual isomorphism between the 2-categories of relational doctrines with quotients and that with comprehensions, allowing to derive results on relational doctrines with comprehensions from their dual counterparts for quotients, and viceversa. We also study the interaction between quotients and comprehensions and their relative completions, giving sufficient conditions for the induced 2-monads to be composable, \ie sufficient conditions ensuring that applying both the completions, the second one preserves the properties added by the first one.

We introduce the category $\mathbf{SSS}$ of sample-smooth spaces over a mixed site. The test objects are the products $\Omega_n := \mathbb{R}^n \times \Omega$ of a Cartesian space with the universal Hilbert cube $\Omega$ carrying all universally measurable sets, and a space is a set with a family of admissible plots $\Omega_n \to \mathcal{X}$ closed under precomposition. Smoothness and measurability are then not two structures glued along an axiom, but one structure over one site. The site has finite non-empty products, because $\Omega$ absorbs its own square; its Karoubi envelope contains every $\mathbb{R}^n$; and it has mixed morphisms $\omega \mapsto (W(\omega),\Phi(\omega))$, which turn measurability of a smooth family from an axiom into a consequence. $\mathbf{SSS}$ is a concrete quasitopos: complete, cocomplete, cartesian closed and locally cartesian closed, with a classifier for embeddings. Morphisms of Cartesian spaces are exactly the $C^\infty$ maps and manifolds embed full and faithfully, both without Boman's theorem. Every object has tangent and cotangent spaces, every morphism a differential. The modalities sit in an adjoint string $\Pi \dashv \flat \dashv \natural \dashv \sharp \dashv \Lambda$, making $\mathbf{SSS}$ cohesive over quasi-universal spaces. The point is the probability monad. Defining the plots of $\mathsf{P}(\mathcal{X})$ as push-forwards of $\mathcal{X}$-plots at every test object, $\mathsf{P}$ is an unconditional strong commutative affine monad on all of $\mathbf{SSS}$ – functor, unit, product of kernels, multiplication and the monad laws are each one line of seed splitting – and its Kleisli category, of differentiable simulators, is a Markov category. The reparametrisation trick holds by construction: every Kleisli morphism is plot-wise a sampler, stably under composition. A reflection theorem locates the whole gain in a single plot family.

We introduce cleft extensions for Hopf algebroids. We prove the equivalence between cleft extensions, $\sigma$-twisted crossed products, and Hopf-Galois extensions with the normal basis property, thereby generalizing the theory of cleft extensions for Hopf algebroids developed by Böhm and Brzeziński, and fitting in with the general theory of Galois and biGalois extensions over Hopf algebroids developed by the authors. We investigate the Ehresmann Hopf algebroid associated with a cleft extension and show that it is isomorphic to a generalized version of the Connes-Moscovici Hopf algebroid. A special case of the Connes-Moscovici Hopf algebroid, namely the case where the coinvariants of the cleft extension coincide with the base of the Hopf algebroid, is a Drinfeld twist of a Hopf algebroid by a two-cocycle, generalizing work of Böhm, Han and Majid.

This article shows that the exact completion as a regular category of the homological category of Polish groups and continuous group homomorphisms is the category of groups with a Polish cover and Borel-definable group homomorphisms. We also obtain several equivalent characterizations for the morphisms in the exact completion. An analogous description is deduced for several important subcategories of the category of Polish groups.

Given a complete first-order theory $T$, we characterise elimination of hyperimaginaries in $T$ in terms of the exactness of the pro-completion $\textrm{Pro}(\textrm{def}(T))$ of the syntactic category $\textrm{def}(T)$ of $T$, thus extending Makkai's result connecting elimination of imaginaries with the exactness of $\textrm{def}(T)$. Likewise, we characterise the heq construction in terms of the exact completion of $\textrm{Pro}(\textrm{def}(T))$.

We introduce reparametrization categories with cuts. For every such category $\mathcal{P}$, the tensor product of two objectwise weak homotopy equivalences of $\mathcal{P}$-spaces is an objectwise weak homotopy equivalence, and the q-model structure on $\mathcal{P}$-flows is left proper. The reparametrization categories $\mathcal{G}$ and $\mathcal{M}$, as well as the terminal category $\mathbf{1}$, have cuts. The terminal category recovers the left properness of the q-model structure on ordinary flows. We also give a symmetric reparametrization category with cuts and a strict unit. This category is not equivalent to $\mathbf{1}$.

We prove that the q-model category of $\mathcal P$-multipointed d-spaces for $\mathcal P\in\{\mathcal G,\mathcal M\}$ is not left proper by constructing a weak equivalence whose pushout along a q-cofibration obtained by attaching a single 1-dimensional globular cell is not a weak equivalence. The counterexample is constructed in the category of $\Delta$-Hausdorff $\Delta$-generated spaces. For $\mathcal P=\mathcal G$, all objects are automatically saturated, and for $\mathcal P=\mathcal M$, the original weak equivalence is between saturated objects. The failure is caused by a family of nonconstant execution paths that converges in the ambient mapping space to a constant map which is not an execution path; after the globular cell is attached, this degeneration causes two previously distinct path components to merge.

In this note, we show the category of $\Gamma$-sets with the sphere spectrum, the smash product, and the substitution product gives rise to an isomix linearly distributive category, which is a categorical semantics of the multiplicative fragment of linear logic. We also show Lydakis' assembly map between these two monoidal products corresponds to the mixor.

This work proposes a semantic environment for arithmetic geometry in which passage between presentations carries an explicit weight. Morphism costs induce a Lawvere generalized distance by taking infima over transports, and heights satisfy the corresponding transport bounds. A labeled quantitative $(2,1)$-category retains invertible $2$-morphisms as coherent comparisons between transports. Its vertical groupoids admit a transport interpretation inspired by the identity types of Voevodsky's C-systems. If a height factors through the underlying object, then a vertical comparison has equal endpoints, whatever its transport cost. We formulate an intensional Szpiro inequality for elliptic packages, where the discriminant height transports along a morphism with a controlled model defect, and we relate a bound of this form to the $abc$ conjecture. The defect remains visible between non-minimal and normalized presentations, while conductor complexity stays attached to the underlying curve. We then treat the Birch and Swinnerton-Dyer conjecture in its pre-modularity form, using strong point-count asymptotics with an explicit convergence requirement. The refined constant retains separately typed arithmetic factors. A finite differential correction makes the period independent of the chosen differential and links its normalization to model transport. Within the declared finite-descent theory, Tate–Shafarevich finiteness reduces to finitely many closed prime columns of Selmer shadows and a uniform cutoff. For $y^2=x^3-q^2x$ with $q\equiv3\pmod8$ prime, two-isogeny descent closes the two-primary column uniformly. The statements admit expression in $\mathsf{ACA}_0$ relative to certified descent and Mordell–Weil data, with certified Cauchy names for the period and regulator.

Recently Batanin and Markl developed a theory of Koszul duality for operads over a general operadic category in the spirit of Ginzburg-Kapranov. However, their approach doesn't apply directly to the operadic category of n-trees, which are used to describe categorical structures. In this paper we present a category of (locally constant) 2-nets which addresses the main difficulty of extending their theory to the case of 2-trees. Specifically, our category of 2-nets is an operadic category containing 2-trees as a full subcategory, and every map between 2-nets admits a factorization into a chain of elementary maps.

We introduce $C$-lifting functors, which axiomatize lifting properties of non-abelian tensor and exterior products with respect to prescribed classes of group extensions. For a homomorphism $f\colon\Gamma\to G$, we associate to every $C$-lifting functor $F$ a relative quotient $F_f(G)$. This quotient maps epimorphically onto the subgroup determined by $F$ in every $f$-extension belonging to $C$. We show that the construction is functorial in $f$ and that, for an $f$-extension $p\colon\widetilde G\to G$, it gives a morphism of natural exact sequences. In degree two this yields an epimorphism $ \frac{H_2(G;\mathbb Z)}{f_*H_2(\Gamma;\mathbb Z)} \longrightarrow \ker p\cap[\widetilde G,\widetilde G]. $ Applying the construction to iterated tensor and exterior powers, we obtain relative Schur-Baer theorems for the lower central and derived series. We also compare the relative tensor and exterior squares, relate the exterior construction to the relative Schur multiplier $H_2(G,\Gamma;\mathbb Z)$, and derive applications to orderability. As a consequence, we prove that a virtual knot group is left-orderable if and only if it is circularly orderable.

In our previous article [arxiv:2410.05120], we introduced the notion of a finite dimensional 3-Hilbert space, categorifying Baez's 2-Hilbert spaces. In this article, by further categorifying Baez's higher linear algebra, we provide useful tools for working with 3-Hilbert spaces, including, generalized scalar multiplication, orthonormal bases, and unitary adjoints for operators. We use these tools to endow the $\mathrm{C}^*$-3-category of 3-Hilbert spaces with a self-enrichment. We prove a Unitary Yoneda Lemma/Riesz Representation Theorem for 3-Hilbert spaces: the Yoneda embedding is an isometric equivalence. Finally, we define a unitary version of the Deligne product on 3-Hilbert spaces and prove that it satisfies an isometric version of the folding trick.

Given a Lie group $G$, a level $k$, and a Lie subgroup $H$ one can construct 2d conformal field theories by either 1.) gauging a nonanomalous $H$ symmetry of the WZW model constructed from $(G,k)$ or 2.) using an algebraic procedure known as the GKO coset construction. The two models are closely related, but not precisely the same: The gauged WZW model is identified with the corresponding GKO model coupled to a 2d topological field theory. The topological theory is characterized by a commutative Frobenius algebra derived from the endomorphisms of an algebra object in a modular tensor category constructed from $(G,H,k)$. The partition function on the torus of the two models differ by a factor of the dimension of this algebra of endomorphisms. Concrete examples are constructed and some applications to string theory and 2d Yang-Mills coupled to nonanomalous matter are briefly discussed. This paper is a summary of a longer companion paper.

We prove that the $\infty$-category of localizing motives in the sense of Blumberg–Gepner–Tabuada is not compactly generated, extending a result of Efimov.

Nakamura–Reyes showed that the category of commutative mosaics—unital reversible hypermagmas, without associativity—is complete, cocomplete, and has free objects, in contrast to commutative polygroups. Krasner's multivalued addition is designed around ultrametric balls; we take that valuation-theoretic motivation as primary and equip mosaics with valuations, as unit-reflecting unitary morphisms into the tropical polygroup $T(\Gamma)$. Value-preserving morphisms form the slice over $T(\Gamma)$, which is complete and cocomplete. The larger lax category of valued mosaics over a fixed ordered abelian group $\Gamma$ is finitely complete and has all small coproducts; it recovers the category of commutative mosaics for the trivial value group, while lax coequalizers for nontrivial $\Gamma$ remain open. Associativity is analysed via factor nesting, which implies it under totality and characterises it among total product-ultrametric mosaics such as the Krasner hyperfield and $T(\Gamma)$, yet is strictly weaker without totality. Among total valued mosaics satisfying factor nesting, the Krasner ball axiom yields associativity and upgrades the weak valuation so that sums are ultrametric balls and the superiorly canonical package follows.

This paper studies simple Lie algebras in the Verlinde category, with canonical fundamental group action, using combinatorial methods. We give an exhaustive list of such Lie algebras which either have length at most 3, or appear as a subalgebra of the restriction of the Lie algebra of a simple algebraic group along a principal morphism. The results and exceptions answer several questions and conjectures regarding Verlinde categories of algebraic groups and their Lie algebras.

We study natural topologies on spaces of subobjects of a fixed object in a suitable category, with the aim of determining when these spaces are spectral. A central step in our approach, which is also of independent interest, is the comparison between the categorical notion of a $\lambda$-generated object and the order-theoretic notion of a $\lambda$-compact element in a lattice of subobjects. We prove that these notions coincide under some natural and mild assumptions . In the finitary case, this allows us to describe the finitely generated subobjects purely in order-theoretic terms and to construct, inside each interval of a subobject lattice, a canonical algebraic core. We study this construction abstractly for complete lattices and characterize it by a universal property. We then introduce the categorical Zariski topology on spaces of subobjects, relate it to the Scott topology, and obtain spectrality criteria for the whole subobject space and for its algebraic core.