Latest in math.ct

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Pro-equivalences of diagramsSep 18 2019This paper presents a model structure for natural transformations of diagrams of simplicial presheaves of a fixed shape, in which the weak equivalences are defined by analogy with pro-equivalences between pro-objects.
Path categories and quasi-categoriesSep 18 2019This paper gives an introduction to the homotopy theory of quasi-categories. Weak equivalences between quasi-categories are characterized as maps which induce equivalences on a naturally defined system of groupoids. These groupoids effectively replace ... More
Homotopy Types of Abstract Elementary ClassesSep 17 2019We prove that for any homotopy type $X$, there is an elementary class $\calC$, with joint embedding, almagamation and no maximal models such that the classifying space realizes the homotopy type $X$. We provide a few explicit examples.
Representable diagrammatic sets as a model of weak higher categoriesSep 17 2019Developing an idea of Kapranov and Voevodsky, we introduce a model of weak omega-categories based on directed complexes, combinatorial presentations of pasting diagrams. We propose this as a convenient framework for higher-dimensional rewriting. We define ... More
Enriched categories and tropical mathematicsSep 17 2019We point out a connection of enriched category theory over a quantale and tropical mathematics. Quantales or complete idempotent semirings, as well as matrices with coefficients in them, are fundamental objects in both fields. We first survey standard ... More
A Linear Exponential Comonad in s-finite Transition Kernels and Probabilistic Coherent SpacesSep 17 2019This paper concerns a stochastic construction of probabilistic coherent spaces by employing novel ingredients (i) linear exponential comonads arising properly in the measure-theory (ii) continuous orthogonality between measures and measurable functions. ... More
Locally (co)Cartesian fibrations as realisation fibrations and the classifying space of cospansSep 16 2019We show that the conditions in Steimle's 'additivity theorem for cobordism categories' can be weakened to only require \emph{locally} (co)Cartesian fibrations, making it applicable to a larger class of functors. As an application we compute the difference ... More
Weak comp algebras and cup products in secondary Hochschild cohomology of entwining structuresSep 12 2019We define the secondary Hochschild complex for an entwining structure over a commutative $k$-algebra $B$. We show that this complex carries the structure of a weak comp algebra. We obtain cup product structures and Hodge type decomposition for the secondary ... More
D-Ultrafilters and their MonadsSep 11 2019For a number of locally finitely presentable categories K we describe the codensity monad of the full embedding of all finitely presentable objects into K. We introduce the concept of D-ultrafilter on an object, where D is a "nice" cogenerator of K. We ... More
Accessible set endofunctors are universalSep 11 2019It is shown that every concretizable category can be fully embedded into the category of accessible set functors and natural transformations.
Noncommutative tensor triangular geometrySep 10 2019We develop a general noncommutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (M$\Delta$Cs). Insight from noncommutative ring theory is used to obtain a framework for prime, semiprime, ... More
Covers in the Canonical Grothendieck TopologySep 08 2019We explore the canonical Grothendieck topology in some specific circumstances. First we use a description of the canonical topology to get a variant of Giraud's Theorem. Then we explore the canonical Grothendieck topology on the categories of sets and ... More
Mapping finite state machines to zk-SNARKS Using Category TheorySep 06 2019We provide a categorical procedure to turn graphs corresponding to state spaces of finite state machines into boolean circuits, leveraging on the fact that boolean circuits can be easily turned into zk-SNARKS. Our circuits verify that a given sequence ... More
Finitely Presentable Algebras For Finitary MonadsSep 05 2019For finitary regular monads T on locally finitely presentable categories we characterize the finitely presentable objects in the category of T-algebras in the style known from general algebra: they are precisely the algebras presentable by finitely many ... More
Homology Groups and Categorical DiagonalizationSep 05 2019A sequence of homology groups as an object in a monoidal homotopy category is discussed. For a morphism which has a fixed object (a sequence of chain complexes) as a domain in a monoidal homotopy category, we consider a mapping cone of the morphism with ... More
Connected monads weakly preserve productsSep 05 2019If $F$ is a (not necessarily associative) monad on $Set$, then the natural transformation $F(A\times B)\to F(A)\times F(B)$ is surjective if and only if $F(\boldsymbol{1})=\boldsymbol{1}$. Specializing $F$ to $F_{\mathcal{V}}$, the free algebra functor ... More
On the axiomatisability of the dual of compact ordered spacesSep 04 2019We provide a direct and elementary proof of the fact that the category of Nachbin's compact ordered spaces is dually equivalent to an Aleph_1-ary variety of algebras. Further, we show that Aleph_1 is a sharp bound: compact ordered spaces are not dually ... More
On the axiomatisability of the dual of compact ordered spacesSep 04 2019Sep 13 2019We provide a direct and elementary proof of the fact that the category of Nachbin's compact ordered spaces is dually equivalent to an Aleph_1-ary variety of algebras. Further, we show that Aleph_1 is a sharp bound: compact ordered spaces are not dually ... More
Tensor products of finitely presented functorsAug 31 2019We study right exact tensor products on the category of finitely presented functors. As our main technical tool, we use a multilinear version of the universal property of so-called Freyd categories. Furthermore, we compare our constructions with the Day ... More
Regular and relational categories: Revisiting 'Cartesian bicategories I'Aug 30 2019Regular logic is the fragment of first order logic generated by $=$, $\top$, $\wedge$, and $\exists$. A key feature of this logic is that it is the minimal fragment required to express composition of binary relations; another is that it is the internal ... More
Runge-Kutta and NetworksAug 29 2019We categorify the RK family of numerical integration methods (explicit and implicit). Namely we prove that if a pair of ODEs are related by an affine map then the corresponding discrete time dynamical systems are also related by the map. We show that ... More
Cellular Monads from Positive GSOS SpecificationsAug 29 2019We give a leisurely introduction to our abstract framework for operational semantics based on cellular monads on transition categories. Furthermore, we relate it for the first time to an existing format, by showing that all Positive GSOS specifications ... More
Locally type $\text{FP}_n$ and $n$-coherent categoriesAug 28 2019We study finiteness conditions in Grothendieck categories by introducing the concepts of objects of type $\text{FP}_n$ and studying their closure properties with respect to short exact sequences. This allows us to propose a notion of locally type $\text{FP}_n$ ... More
Foundations of brick diagramsAug 28 2019We discuss the foundations of 2-dimensional graphical languages, with a view towards their computer implementation in a 'compiler' for monoidal categories. In particular, we discuss the close relationship between string diagrams, pasting diagrams, linear ... More
A universal characterization of standard Borel spacesAug 28 2019We prove that the category $\mathsf{SBor}$ of standard Borel spaces is the (bi-)initial object in the 2-category of countably complete Boolean (countably) extensive categories. This means that $\mathsf{SBor}$ is the universal category admitting some familiar ... More
Networks of hybrid open systemsAug 27 2019We generalize the results of "Networks of open systems" by the first author to the setting of hybrid systems. In particular we introduce the notions of hybrid open systems, their networks and maps between networks. A network of systems is a blueprint ... More
Hilbert spaces and ${C}^\ast$-algebras are not finitely concreteAug 27 2019Sep 04 2019We show that no faithful functor from the category of Hilbert spaces with linear isometries into the category of sets preserves directed colimits. Thus Hilbert spaces cannot form an abstract elementary class, even up to change of language. We deduce an ... More
Hilbert spaces and ${C}^\ast$-algebras are not finitely concreteAug 27 2019We show that no faithful functor from the category of Hilbert spaces with linear isometries into the category of sets preserves directed colimits. Thus Hilbert spaces cannot form an abstract elementary class, even up to change of language. We deduce an ... More
Hilbert spaces and ${C}^\ast$-algebras are not finitely concreteAug 27 2019Sep 16 2019We show that no faithful functor from the category of Hilbert spaces with linear isometries into the category of sets preserves directed colimits. Thus Hilbert spaces cannot form an abstract elementary class, even up to change of language. We deduce an ... More
Model Theory of Proalgebraic GroupsAug 27 2019We lay the foundations for a model theoretic study of proalgebraic groups. Our axiomatization is based on the tannakian philosophy. Through a tensor analog of skeletal categories we are able to consider neutral tannakian categories with a fibre functor ... More
Deformation cohomology of Schur-Weyl categories. Free symmetric categoriesAug 24 2019The deformation cohomology of a tensor category controls deformations of its monoidal structure. Here we describe the deformation cohomology of tensor categories generated by one object (the so-called Schur-Weyl categories). Using this description we ... More
Minimal index and dimension for inclusions of von Neumann algebras with finite-dimensional centersAug 24 2019The notion of index for inclusions of von Neumann algebras goes back to a seminal work of Jones on subfactors of type ${I\!I}_1$. In the absence of a trace, one can still define the index of a conditional expectation associated to a subfactor and look ... More
First-order homotopical logicAug 23 2019We introduce a homotopy-theoretic interpretation of intuitionistic first-order logic based on ideas from Homotopy Type Theory. We provide a categorical formulation of this interpretation using the framework of Grothendieck fibrations. We then use this ... More
On the dependent product in toposesAug 22 2019We give an explicit construction of the dependent product in an elementary topos, and a site-theoretic description for it in the case of a Grothendieck topos.
Idempotents and one-sided units II. Lattice invariants and a semigroup of functors on the category of monoidsAug 22 2019For a monoid $M$, we denote by $\mathbb G(M)$ the group of units, $\mathbb E(M)$ the submonoid generated by the idempotents, and $\mathbb G_L(M)$ and $\mathbb G_R(M)$ the submonoids consisting of all left or right units. Writing $\mathcal M$ for the (monoidal) ... More
Good Fibrations through the Modal PrismAug 21 2019Homotopy type theory is a formal language for doing abstract homotopy theory -- the study of identifications. But in unmodified homotopy type theory, there is no way to say that these identifications come from identifying the path-connected points of ... More
Iterated traces in bicategories and Lefschetz theoremsAug 20 2019While not obvious from its initial motivation in linear algebra, there are many context where iterated traces can be defined. In this paper we prove a very general theorem about iterated 2-categorical traces. We show that many Lefschetz-type theorems ... More
A synthetic approach to Markov kernels, conditional independence, and theorems on sufficient statisticsAug 19 2019Sep 01 2019We develop Markov categories as a framework for synthetic probability and statistics, following work of Golubtsov as well as Cho and Jacobs. This means that we treat the following concepts in purely abstract categorical terms: conditioning and disintegration; ... More
A synthetic approach to Markov kernels, conditional independence, and theorems on sufficient statisticsAug 19 2019We develop Markov categories as a framework for synthetic probability and statistics, following work of Golubtsov as well as Cho and Jacobs. This means that we treat the following concepts in purely abstract categorical terms: conditioning and disintegration; ... More
A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statisticsAug 19 2019Sep 11 2019We develop Markov categories as a framework for synthetic probability and statistics, following work of Golubtsov as well as Cho and Jacobs. This means that we treat the following concepts in purely abstract categorical terms: conditioning and disintegration; ... More
Covariant & Contravariant Homotopy TheoriesAug 19 2019Given a locally presentable category together with a suitable functorial cylinder object, we construct model structures which are sensitive to the `direction' of the cylinder. We show that the Covariant and Contravariant model structures on simplicial ... More
Higher Equipments, Double Colimits and Homotopy ColimitsAug 16 2019This document is centered around a main idea: simplicial categories, by which we mean simplicial objects in the category of categories, can be treated as a two-fold categorical structure and their double category theory is homotopically meaningful. The ... More
The Injective Spectrum of a Right Noetherian Ring II: Sheaves and Torsion TheoriesAug 16 2019This is the second of two papers on the injective spectrum of a right noetherian ring. In the prequel, we considered the injective spectrum as a topological space associated to a ring (or, more generally, a Grothendieck category), which generalises the ... More
The Injective Spectrum of a Right Noetherian Ring I: Injective Spectra and Krull DimensionAug 16 2019The injective spectrum is a topological space associated to a ring $R$, which agrees with the Zariski spectrum when $R$ is commutative noetherian. We consider injective spectra of right noetherian rings (and locally noetherian Grothendieck categories) ... More
The Jordan-Hölder property and Grothendieck monoids of exact categoriesAug 15 2019We investigate the Jordan-H\"older property (JHP) in exact categories. First we introduce a new invariant of exact categories, the Grothendieck monoids, and show that (JHP) holds if and only if the Grothendieck monoid is free. Moreover, we give a criterion ... More
The homotopy theory of coherently commutative monoidal quasi-categoriesAug 14 2019The main objective of this paper is to construct a symmetric monoidal closed model category of coherently commutative monoidal quasi-categories.
Ideal categories of rings and ring of conesAug 14 2019In this paper we describe the ideal category of a ring R as preadditive proper category. Further it is also shown that the cones in this category is a ring with appropriate addition and multiplication.
Morita Invariance of Equivariant Lusternik-Schnirelmann Category and Invariant Topological ComplexityAug 14 2019We use the homotopy invariance of equivariant principal bundles to prove that the equivariant $\A$-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann category of ... More
Morita Invariance of Equivariant Lusternik-Schnirelmann Category and Invariant Topological ComplexityAug 14 2019Aug 15 2019We use the homotopy invariance of equivariant principal bundles to prove that the equivariant $\A$-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann category of ... More
Morita Invariance of Equivariant Lusternik-Schnirelmann Category and Invariant Topological ComplexityAug 14 2019Aug 20 2019We use the homotopy invariance of equivariant principal bundles to prove that the equivariant ${\mathcal A}$-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann ... More
Morita Invariance of Equivariant Lusternik-Schnirelmann Category and Invariant Topological ComplexityAug 14 2019Aug 16 2019We use the homotopy invariance of equivariant principal bundles to prove that the equivariant ${\mathcal A}$-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann ... More
An informal introduction to dg categoriesAug 13 2019In this informal introduction to dg categories, the slogan is that dg categories are more rudimentary than triangulated categories. We recall some details on the dg quotient category introduced by Bernhard Keller and Vladimir Drinfeld.
Hausdorff coalgebrasAug 12 2019As composites of constant, (co)product, identity, and powerset functors, Kripke polynomial functors form a relevant class of $\mathsf{Set}$-functors in the theory of coalgebras. The main goal of this paper is to expand the theory of limits in categories ... More
Compactly generated spaces and quasi-spaces in topologyAug 12 2019The notions of compactness and Hausdorff separation for generalized enriched categories allow us, as classically done for the category $\mathsf{Top}$ of topological spaces and continuous functions, to study $\textit{compactly generated spaces}$ and $\textit{quasi-spaces}$ ... More
Compatible actions in semi-abelian categoriesAug 12 2019The concept of a pair of compatible actions was introduced in the case of groups by Brown and Loday and in the case of Lie algebras by Ellis. In this article we extend it to the context of semi-abelian categories (that satisfy the Smith-is-Huq condition). ... More
Methods of constructive category theoryAug 12 2019We give an introduction to constructive category theory by answering two guiding computational questions. The first question is: how do we compute the set of all natural transformations between two finitely presented functors like $\mathrm{Ext}$ and $\mathrm{Tor}$ ... More
Pretorsion theories in general categoriesAug 09 2019We present a setting for the study of torsion theories in general categories. The idea is to associate, with any pair $(\mathcal T, \mathcal F)$ of full replete subcategories in a category $\mathcal C$, the corresponding full subcategory $\mathcal Z = ... More
Representing Polish groupoids via metric structuresAug 08 2019We prove that every open $\sigma$-locally Polish groupoid $G$ is Borel equivalent to the groupoid of models on the Urysohn sphere $\mathbb{U}$ of an $\mathcal{L}_{\omega_1\omega}$-sentence in continuous logic. In particular, the orbit equivalence relations ... More
Realizing the braided Temperley-Lieb-Jones C*-tensor categories as Hilbert C*-modulesAug 07 2019We associate to each Temperley-Lieb-Jones C*-tensor category $\mathcal{T}\!\mathcal{L}\mathcal{J}(\delta)$ with parameter $\delta$ in the discrete range $\{2\cos(\pi/(k+2))\,:\,k=1,2,\ldots\}\cup\{2\}$ a certain C*-algebra $\mathcal{B}$ of compact operators. ... More
Supplying bells and whistles in symmetric monoidal categoriesAug 07 2019It is common to encounter symmetric monoidal categories $\mathcal{C}$ for which every object is equipped with an algebraic structure, in a way that is compatible with the monoidal product and unit in $\mathcal{C}$. We define this formally and say that ... More
Matrix formulation for non-Abelian familiesAug 07 2019We generalize the $K$ matrix formulation to non-trivial non-Abelian families of 2+1D topological orders. Given a topological order $\mathcal C$, any topological order in the same non-Abelian family as $\mathcal C$ can be efficiently described by $\boldsymbol{a}=(a_I)$ ... More
Generalized Lens Categories via functors $\mathcal{C}^{\rm op}\to\mathsf{Cat}$Aug 06 2019Aug 07 2019Lenses have a rich history and have recently received a great deal of attention from applied category theorists. We generalize the notion of lens by defining a category $\mathsf{Lens}_F$ for any category $\mathcal{C}$ and functor $F\colon \mathcal{C}^{\rm ... More
Generalized Lens Categories via functors $\mathcal{C}^{\rm op}\to\mathsf{Cat}$Aug 06 2019Lenses have a rich history and have recently received a great deal of attention from applied category theorists. We generalize the notion of lens by defining a category Lens_F for any category $\mathcal{C}$ and functor $F: \mathcal{C}^{\rm op}\to\mathsf{Cat}$, ... More
A Larson-Sweedler Theorem for Hopf V-CategoriesAug 06 2019The aim of this paper is to extend the classical Larson-Sweedler theorem, namely that a $k$-bialgebra has a non-singular integral (and in particular is Frobenius) if and only if it is a finite dimensional Hopf algebra, to the `many-object' setting of ... More
Algebraic realization of noncommutative near-group fusion categoriesAug 05 2019Noncommutative near-group fusion categories were completely classified in the previous work of the first named author by using an operator algebraic method (and hence under the assumption of unitarity), and they were shown to be group theoretical though ... More
Equivalence of the categories of group triples and of hypergroups over the groupAug 04 2019The main result of this paper is that the categories of (right) hypergroups over the group and of triples, consisting of a group, its subgroup and a (right) transversal to this subgroup, are equivalent.
Bicategories of fractions revisited: towards small homs and canonical 2-cellsAug 03 2019This paper introduces a set of conditions on a class of arrows in a bicategory which is weaker than the one given in the literature but still allows a bicalculus of fractions. These conditions allow us to invert a smaller collection of arrows so that ... More
2-Biproducts in 2-CategoriesAug 03 2019An algebraic definition for weak 2-biproducts in 2-categories is introduced. It is shown that in a locally semiadditive distributive 2-category (a 2-category whose 2-morphisms horizontally and vertically distribute over monoid additions and whose Hom-categories ... More
Completely distributive enriched categories are not always continuousAug 03 2019In contrast to the fact that every completely distributive lattice is necessarily continuous in the sense of Dana Scott, it is shown that complete distributivity of a category enriched over the closed category obtained by endowing the unit interval with ... More
Locally finitely presented and coherent heartsAug 01 2019Given a torsion pair $\mathbf{t}=(\mathcal{T},\mathcal{F})$ in a Grothendieck category $\mathcal{G}$, we study when the heart $\mathcal{H}_{\mathbf t}$ of the associated Happel-Reiten-Smalo $t$-structure in the derived category ${\mathbf D}(\mathcal{G})$ ... More
Gorenstein homological dimensions for extriangulated categoriesAug 01 2019Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. The authors introduced and studied $\xi$-$\mathcal{G}$projective and $\xi$-$\mathcal{G}$injective in \cite{HZZ}. In this paper, ... More
On the relation between n-cotorsion pairs and (n+1)-cluster tilting subcategoriesJul 31 2019A notion of $n$-cotorsion pairs in an extriangulated category with enough projectives and enough injectives is defined in this article. We show that there exists a one-to-one correspondence between $n$-cotorsion pairs and $(n+1)$-cluster tilting subcategories. ... More
Diagonal $p$-permutation functorsJul 30 2019Let $k$ be an algebraically closed field of positive characteristic $p$, and $\mathbb{F}$ be an algebraically closed field of characteristic 0. We consider the $\mathbb{F}$-linear category $\mathbb{F} pp_k^\Delta$ of finite groups, in which the set of ... More
Quantale semantics of Lambek calculus with subexponential modalitiesJul 30 2019In this paper, we consider the polymodal version of Lambek calculus with subexponential modalities initially introduced by Kanovich, Kuznetsov, Nigam, and Scedrov and its quantale semantics. In our approach, subexponential modalities have an interpretation ... More
Quantale semantics of Lambek calculus with subexponential modalitiesJul 30 2019Aug 08 2019In this paper, we consider the polymodal version of Lambek calculus with subexponential modalities initially introduced by Kanovich, Kuznetsov, Nigam, and Scedrov and its quantale semantics. In our approach, subexponential modalities have an interpretation ... More
Walls and asymptotics for Bridgeland stability conditions on 3-foldsJul 29 2019We consider Bridgeland stability conditions for three-folds conjectured by Bayer-Macr\`i-Toda in the case of Picard rank one. We study the differential geometry of numerical walls, characterizing when they are bounded, discussing possible intersections, ... More
Projective discrete modules over profinite groupsJul 29 2019We show that the category of discrete modules over an infinite profinite group has no non-zero projective objects and does not satisfy Ab4*. We also prove the same types of results in a generalized setting using a ring with linear topology.
Invariants of 4-manifolds from Khovanov-Rozansky link homologyJul 29 2019We use Khovanov-Rozansky gl(N) link homology to define pivotal 4-categories, which give rise to invariants of oriented smooth 4-manifolds. The technical heart of this construction is a proof of the sweep-around property, which makes these link homologies ... More
Invariants of 4-manifolds from Khovanov-Rozansky link homologyJul 29 2019Aug 14 2019We use Khovanov-Rozansky gl(N) link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals. The technical heart of this construction is a proof of the sweep-around property, ... More
An Open Petri Net Implementation of Gene Regulatory NetworksJul 25 2019Gene regulatory network (GRN) plays a central role in system biology and genomics. It provides a promising way to model and study complex biological processes. Several computational methods have been developed for the construction and analysis of GRN. ... More
The contraction category of graphsJul 25 2019We study the category whose objects are graphs of fixed genus and whose morphisms are contractions. We show that the corresponding contravariant module categories are Noetherian and we study two families of modules over these categories. The first takes ... More
Serre dimension and stability conditionsJul 25 2019We study relations between the Serre dimension defined as the growth of entropy of the Serre functor and the global dimension of Bridgeland stability conditions due to Ikeda-Qiu. A fundamental inequality between the Serre dimension and the infimum of ... More
Inner automorphisms of groupoidsJul 24 2019Bergman has given the following abstract characterisation of the inner automorphisms of a group $G$: they are exactly those automorphisms of $G$ which can be extended functorially along any homomorphism $G \rightarrow H$ to an automorphism of $H$. This ... More
Sheaf theoretic characterization of topological etale groupoidsJul 24 2019A sheaf can be defined in two ways. As the etale space, or as the contravariant functor. In this paper, by analogy, we characterize the topological etale groupoids by a sheaf theoritic way. For that purpose, we introduce a pseudogroup sheaf as generalization ... More
Torsion pairs and quasi-abelian categoriesJul 23 2019We define torsion pairs for quasi-abelian categories and give several characterisations. We show that many of the torsion theoretic concepts translate from abelian categories to quasi-abelian categories. As an application, we generalise the recently defined ... More
Relative rigid objects in extriangulated categoriesJul 23 2019In this paper, we study a close relationship between relative cluster tilting theory in extriangulated categories and tau-tilting theory in module categories. Our main results show that relative rigid objects are in bijection with $\tau$-rigid pairs, ... More
The word problem for double categoriesJul 23 2019We solve the word problem for double categories by translating it to the word problem for 2-categories. This yields a quadratic algorithm deciding the equality of diagrams in a free double category. The translation is of interest in its own right since ... More
The word problem for double categoriesJul 23 2019Aug 18 2019We solve the word problem for free double categories without equations between generators by translating it to the word problem for 2-categories. This yields a quadratic algorithm deciding the equality of diagrams in a free double category. The translation ... More
The word problem for double categoriesJul 23 2019Jul 28 2019We solve the word problem for free double categories without equations between generators by translating it to the word problem for 2-categories. This yields a quadratic algorithm deciding the equality of diagrams in a free double category. The translation ... More
Rewriting modulo isotopies in Khovanov-Lauda-Rouquier's categorification of quantum groupsJul 23 2019We study a presentation of Khovanov - Lauda - Rouquier's candidate $2$-categorification of a quantum group using algebraic rewriting methods. We use a computational approach based on rewriting modulo the isotopy axioms of its pivotal structure to compute ... More
Smash Products for Non-cartesian Internal PrestacksJul 23 2019The smash product construction (or the Grothendieck construction) takes a functor (or prestack) $F \colon B^{op} \to \mathbf{Cat}$ and returns a fibration $p \colon A \to B$. In this paper, we develop an analogue of the smash product for prestacks internal ... More
Functors on Posets Left Kan Extend to Cosheaves: an ErratumJul 22 2019In this note we give a self-contained proof of a fundamental statement in the study of cosheaves over a poset. Specifically, if a functor has domain a poset and co-domain a co-complete category, then the left Kan extension of that functor along the embedding ... More
On weakly negative subcategories, weight structures, and (weakly) approximable triangulated categoriesJul 22 2019We prove that certain triangulated categories are (weakly) approximable in the sense of A. Neeman. We prove that a triangulated $C$ that is compactly generated by a single object $G$ is weakly approximable if $C(G,G[i])=0$ for $i>1$ (we say that $G$ is ... More
Non-commutative disintegrations: existence and uniqueness in finite dimensionsJul 22 2019We utilize category theory to define non-commutative disintegrations, regular conditional probabilities, and optimal hypotheses for finite-dimensional C*-algebras. In the process, we introduce a notion of a.e. equivalence for positive maps and show that ... More
Galois theory and the categorical Peiffer commutatorJul 22 2019We show that the Peiffer commutator previously defined by Cigoli, Mantovani and Metere can be used to characterize central extensions of precrossed modules with respect to the subcategory of crossed modules in any semi-abelian category satisfying an additional ... More
A category for bijective combinatoricsJul 21 2019The category of matchings between finite sets extends to the category of cobordisms of signed sets. A chain of cobordisms that starts and ends with unsigned sets A and B yields a matching from A to B. This is a convenient way to package the involution ... More
Derived invariance of the numbers $h^{0,p}(X)$Jul 19 2019Let $X_1$ and $X_2$ be derived equivalent smooth projective varieties over the field of complex numbers. We prove that the numbers $h^{0,p}(X_1)$ and $h^{0,p}(X_2)$ are equal for any $p$.
Derived invariance of the numbers $h^{0,p}(X)$Jul 19 2019Jul 27 2019Let $X_1$ and $X_2$ be derived equivalent smooth projective varieties over the field of complex numbers. We prove that the numbers $h^{0,p}(X_1)$ and $h^{0,p}(X_2)$ are equal for any $p$.
A Quillen model structure on the category of Kontsevich-Soibelman weakly unital dg categoriesJul 18 2019In this paper, we study weakly unital dg categories as they were defined by Kontsevich and Soibelman [KS, Sect.4]. We construct a cofibrantly generated Quillen model structure on the category $\mathrm{Cat}_{\mathrm{dgwu}}(\Bbbk)$ of small weakly unital ... More
Finite-dimensional differential graded algebras and their geometric realizationsJul 16 2019Jul 20 2019We prove that for any finite-dimensional differential graded algebra with separable semisimple part the category of perfect modules is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme with a full separable ... More