Latest in math.ct

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Covers and direct limits: A contramodule-based approachJul 12 2019We present applications of contramodule techniques to the Enochs conjecture about covers and direct limits, both in the categorical tilting context and beyond. In the $n$-tilting-cotilting correspondence situation, if $\mathsf A$ is a Grothendieck abelian ... More
The constructive Kan-Quillen model structure: two new proofsJul 11 2019We present two new proofs of Simon Henry's result that the category of simplicial sets admits a constructive counterpart of the classical Kan-Quillen model structure. Our proofs are entirely self-contained and avoid complex combinatorial arguments on ... More
Deriving Dagger CompactnessJul 11 2019Dagger compact structure is a common assumption in the study of physical process theories, but lacks a clear interpretation. Here we derive dagger compactness from more operational axioms on a category. We first characterise the structure in terms of ... More
Simple-minded reductions of triangulated categoriesJul 11 2019We will introduce a new reduction process of triangulated category, which is analogue to the silting reduction and Calabi-Yau reduction. For a triangulated category $\cal T$ with a pre-simple-minded collection (=pre-SMC) $\cal R$, we construct a new triangulated ... More
Artin glueings of frames as semidirect productsJul 11 2019Artin glueings provide a way to reconstruct a frame from a closed sublocale and its open complement. We show that Artin glueings can be described as split extensions satisfying a Schreier-type condition in the category frames with finite-meet preserving ... More
Matlis category equivalences for a ring epimorphismJul 11 2019Under mild assumptions, we construct the two Matlis additive category equivalences for an associative ring epimorphism $u\colon R\to U$. Assuming that the ring epimorphism is homological of flat/projective dimension $1$, we discuss the abelian categories ... More
The universal property of infinite direct sums in C$^*$-categories and W$^*$-categoriesJul 10 2019When formulating universal properties for objects in a dagger category, one usually expects a universal property to characterize the universal object up to unique unitary isomorphism. We observe that this is automatically the case in the important special ... More
A functorial approach to monomorphism categories for species IJul 10 2019For any generalised species over a locally bounded quiver we investigate abstract versions of the monomorphism category as studied by Ringel and Schmidmeier. We prove that analogues of the kernel and cokernel functor send almost split sequences over the ... More
Word operads and admissible orderingsJul 09 2019We use Giraudo's construction of combinatorial operads from monoids to offer a conceptual explanation of the origins of Hoffbeck's path sequences of shuffle trees, and use it to define new monomial orders of shuffle trees. One such order is utilised to ... More
Homotopy-coherent algebra via Segal conditionsJul 09 2019Many homotopy-coherent algebraic structures can be described by Segal-type limit conditions determined by an "algebraic pattern", by which we mean an $\infty$-category equipped with a factorization system and a collection of "elementary" objects; examples ... More
Generalized bornological coarse spaces and coarse motivic spectraJul 09 2019We generalize the notion of a bornology by omitting the condition that a one-point-subset is bounded and obtain a complete and co-complete generalization of the category of bornological coarse spaces. Then we imitate the construction of motivic coarse ... More
$L'$-localization in an $\infty$-toposJul 08 2019We prove that, given any reflective subfibration $L_\bullet$ on an $\infty$-topos $\mathcal{E}$, there exists a reflective subfibration $L'_\bullet$ on $\mathcal{E}$ whose local maps are the $L$-separated maps, that is, the maps whose diagonals are $L$-local. ... More
Localization theory in an $\infty$-toposJul 08 2019We develop the theory of reflective subfibrations on an $\infty$-topos $\mathcal{E}$. A reflective subfibration $L_\bullet$ on $\mathcal{E}$ is a pullback-compatible assignment of a reflective subcategory $\mathcal{D}_X\subseteq \mathcal{E}{/X}$, for ... More
From weight structures to (orthogonal) $t$-structures and backJul 08 2019A $t$-structure $t=(C_{t\le 0},C_{t\ge 0})$ on a triangulated category $C$ is right adjacent to a weight structure $w=(C_{w\le 0}, C_{w\ge 0})$ if $C_{t\ge 0}=C_{w\ge 0}$; then $t$ can be uniquely recovered from $w$ and vice versa. We prove that if $C$ ... More
Pseudo-dualizing complexes of bicomodules and pairs of t-structuresJul 07 2019This paper is a coalgebra version of arXiv:1703.04266 and a sequel to arXiv:1607.03066. We present the definition of a pseudo-dualizing complex of bicomodules over a pair of coassociative coalgebras $\mathcal C$ and $\mathcal D$. For any such complex ... More
Cyclic structures and broken cyclesJul 07 2019We introduce a new way to encode semicyclic structures using a stack of broken cycles. (We also prove an analogue for paracyclic structures.) This was motivated not only by higher algebra but also by Fukaya-categorical considerations. We also openly speculate ... More
The equivalence of the categories of Giry-algebras and super convex spacesJul 07 2019Using the property that the category of super convex spaces has a codense subcategory, we prove that the category of super convex spaces is equivalent to the category of Giry-algebras.
An Introduction to Higher Categorical AlgebraJul 05 2019This article is a survey of algebra in the $\infty$-categorical context, as developed by Lurie in "Higher Algebra", and is a chapter in the "Handbook of Homotopy Theory". We begin by introducing symmetric monoidal stable $\infty$-categories, such as the ... More
Span composition using fake pullbacksJul 05 2019The construction of a category of spans can be made in some categories $\CC$ which do not have pullbacks in the traditional sense. The PROP for monoids is a good example of such a $\CC$. The 2012 book concerning homological algebra by Marco Grandis gives ... More
Functorial Approach to Graph and Hypergraph TheoryJul 04 2019We provide a new approach to categorical graph and hypergraph theory by using categorical syntax and semantics. For each monoid $M$ and action on a set $X$, there is an associated presheaf topos of $(X,M)$-graphs where each object can be interpreted as ... More
Enriched Regular TheoriesJul 04 2019Regular and exact categories were first introduced by Michael Barr in 1971; since then, the theory has developed and found many applications in algebra, geometry, and logic. In particular, a small regular category determines a certain theory, in the sense ... More
A pullback diagram in the coarse categoryJul 04 2019This paper studies the limit of a pullback diagram in the coarse category.
A homotopy coherent cellular nerve for bicategoriesJul 03 2019The subject of this paper is a nerve construction for bicategories introduced by Leinster, which defines a fully faithful functor from the category of bicategories and normal pseudofunctors to the category of presheaves over Joyal's category $\Theta_2$. ... More
Left properness of multipointed d-spacesJul 02 2019We prove that the q-model structure and the m-model structure of multipointed $d$-spaces are left proper. We also use the techniques developed in this paper to consider the limit case in which all lengths of execution paths are equal to $0$. We then obtain, ... More
Biased permutative equivariant categoriesJul 01 2019For a finite group G, we introduce the complete suboperad $Q_G$ of the categorical G-Barratt-Eccles operad $P_G$. We prove that $P_G$ is not finitely generated, but $Q_G$ is finitely generated and is a genuine $E_\infty$ G-operad (i.e., it is $N_\infty$ ... More
A Compositional Framework for Scientific Model AugmentationJul 01 2019Scientists construct and analyze computational models to understand the world. That understanding comes from efforts to augment, combine, and compare models of related phenomena. We propose SemanticModels.jl, a system that leverages techniques from static ... More
On Two notions of Gerbe over StackJun 30 2019Let $\mathcal{G}$ be a Lie groupoid. The category $B\mathcal{G}$ of principal $\mathcal{G}$-bundles define a geometric stack. On the other hand, given a geometric stack $\mathcal{D}$, there exists a Lie groupoid $\mathcal{H}$ such that $B\mathcal{H}$ ... More
Erratum and Addendum: The factorization of the Giry monadJun 30 2019The category of super convex spaces, a proper subcategory of convex spaces, possesses the property that it has a codense subcategory. This codense subcategory allows for an elementary proof that the Giry monad factorizes through the category of super ... More
Coextension of scalars in operad theoryJun 28 2019The functor between operadic algebras given by restriction along an operad map generally has a left adjoint. We give a necessary and sufficient condition for the restriction functor to admit a right adjoint. The condition is a factorization axiom which ... More
Comparison of spaces associated to DGLA via higher holonomyJun 27 2019Fof a nilpotent differential graded Lie algebra whose components vanish in degrees below -1 we construct an explicit equivalence between the nerve of the Deligne 2-groupoid and the simplicial set of differential forms with values in the Lie algebra introduced ... More
2-representations of Soergel bimodulesJun 27 2019In this paper we study the graded 2-representation theory of Soergel bimodules for a finite Coxeter group. We establish a precise connection between the graded 2-representation theory of this non-semisimple 2-category and the 2-representation theory of ... More
Burnside rings for Real $2$-representation theory: The linear theoryJun 26 2019Jul 02 2019This paper is a fundamental study of the Real $2$-representation theory of $2$-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a $2$-equivariant Morita bicategory, where a novel construction of induction ... More
Burnside rings for Real $2$-representation theory: The linear theoryJun 26 2019This paper is a fundamental study of the Real $2$-representation theory of $2$-groups. It also contains many new results in the usual (non-Real) case. Our framework relies on a $2$-equivariant Morita bicategory, where a novel construction of induction ... More
The Schwarz-Voronov embedding of $\mathbb{Z}_2^n$-manifoldsJun 24 2019Informally, $\mathbb{Z}_2^n$-manifolds are `manifolds' with $\mathbb{Z}_2^n$-graded coordinates and a sign rule determined by the standard scalar product of their $\mathbb{Z}_2^n$-degrees. Such manifolds can be understood in a sheaf-theoretic framework, ... More
On the uniqueness of unitary structure for unitarizable fusion categoriesJun 24 2019We prove that every unitarizable fusion category admits a unique unitary structure. More generally, we show that the forgetful 2-functor from the 2-groupoid of unitary fusion categories, unitary monoidal equivalences and unitary monoidal natural isomorphisms ... More
Reflecting Algebraically Compact FunctorsJun 23 2019A compact T-algebra is an initial T-algebra whose inverse is a final T-coalgebra. Functors with this property are said to be algebraically compact. This is a very strong property used in programming semantics which allows one to interpret recursive datatypes ... More
Mixed Linear and Non-linear Recursive TypesJun 22 2019We describe a type system with mixed linear and non-linear recursive types called LNL-FPC (the linear/non-linear fixpoint calculus). The type system supports linear typing which enhances the safety properties of programs, but also supports non-linear ... More
Description of unitary representations of the group of infinite $p$-adic integer matricesJun 22 2019We classify irreducible unitary representations of the group of all infinite matrices over a $p$-adic field ($p\ne 2$) with integer elements equipped with a natural topology. Any irreducible representation passes through a group $GL$ of infinite matrices ... More
A co-reflection of cubical sets into simplicial sets with applications to model structuresJun 21 2019We show that the category of simplicial sets is a co-reflective subcategory of the category of cubical sets with connections, with the inclusion given by a version of the straightening functor. We show that using the co-reflector, one can transfer any ... More
Preservation theorems for strong first-order logicsJun 21 2019We prove preservation theorems for $\mathcal{L}_{\omega_1, G}$, the countable fragment of Vaught's closed game logic. These are direct generalizations of the theorems of \L{}o\'s-Tarski (resp. Lyndon) on sentences of $\mathcal{L}_{\omega_1, \omega}$ preserved ... More
A topos-theoretic proof of Shelah's eventual categoricity conjecture for abstract elementary classesJun 21 2019Assuming $GCH$ and that there is a measurable cardinal, we give a topos-theoretic proof of Shelah's eventual categoricity conjecture for abstract elementary classes (AEC's). We also show that the large cardinal assumption can be spared assuming instead ... More
A topos-theoretic proof of Shelah's eventual categoricity conjecture for abstract elementary classesJun 21 2019Jul 03 2019Assuming $GCH$ and that there is a measurable cardinal, we give a topos-theoretic proof of Shelah's eventual categoricity conjecture for abstract elementary classes (AEC's). We also show that the large cardinal assumption can be spared assuming instead ... More
Endomorphism operads of functorsJun 21 2019We consider the endomorphism operad of a functor, which is roughly the object of natural transformations from (monoidal) powers of that functor to itself. There are many examples from geometry, topology, and algebra where this object has already been ... More
Endomorphism operads of functorsJun 21 2019Jul 03 2019We consider the endomorphism operad of a functor, which is roughly the object of natural transformations from (monoidal) powers of that functor to itself. There are many examples from geometry, topology, and algebra where this object has already been ... More
Frobenius objects in the category of relationsJun 20 2019We give a characterization, in terms of simplicial sets, of Frobenius objects in the category of relations. This result generalizes a result of Heunen, Contreras, and Cattaneo showing that special dagger Frobenius objects in the category of relations ... More
Denseness conditions, morphisms and equivalences of toposesJun 20 2019We establish a general theorem providing necessary and sufficient explicit conditions for a morphism of sites to induce an equivalence of toposes. This results from a detailed analysis of arrows in Grothendieck toposes and denseness conditions, which ... More
The Mathematical Specification of the Statebox LanguageJun 18 2019This document defines the mathematical backbone of the Statebox programming language. In the simplest way possible, Statebox can be seen as a clever way to tie together different theoretical structures to maximize their benefits and limit their downsides. ... More
Galois descent criteriaJun 14 2019This paper gives an introduction to homotopy descent, and its applications in algebraic $K$-theory computations for fields. On the \'etale site of a field, a fibrant model of a simplicial presheaf can be constructed from naive Galois cohomological objects ... More
Cosimplicial spaces and cocyclesJun 14 2019Standard results from non-abelian cohomology theory specialize to a theory of torsors and stacks for cosimplicial groupoids. The space of global sections of the stack completion of a cosimplicial groupoid $G$ is weakly equivalent to the Bousfield-Kan ... More
A complete language for faceted dataflow programsJun 13 2019We present a complete categorical axiomatization of a wide class of dataflow programs. This gives a three-dimensional diagrammatic language for workflows, more expressive than the directed acyclic graphs generally used for this purpose. This calls for ... More
An extensional $λ$-model with $\infty$-grupoid structureJun 13 2019From a topological space, a set with $\infty$-grupoid structure is built and this construction is applied to the case of ordered sets equipped with the Scott topology. The main purpose is to project the $\lambda$-model $D_\infty$ of Dana Scott to an extensional ... More
On the denotational semantics of Linear Logic with least and greatest fixed points of formulasJun 13 2019We develop a denotational semantics of Linear Logic with least and greatest fixed points in coherence spaces (where both fixed points are interpreted in the same way) and in coherence spaces with totality (where they have different interpretations). These ... More
Rewriting Structured Cospans: A Syntax For Open SystemsJun 13 2019The concept of a system has proliferated through natural and social sciences. While myriad theories of systems exist, there is no mathematical general theory of systems. In this thesis, we take a first step towards formulating such a theory. Our focus ... More
Sectional algebras of semigroupoid bundlesJun 13 2019In this article we use semigroupoids to describe a notion of algebraic bundles, mostly motivated by Fell ($C^*$-algebraic) bundles, and the sectional algebras associated to them. As the main motivational example, Steinberg algebras may be regarded as ... More
Virtual classes of parabolic $\mathrm{SL}_2(\mathbb{C})$-character varietiesJun 12 2019In this paper, we compute the virtual classes in the Grothendieck ring of algebraic varieties of $\mathrm{SL}_2(\mathbb{C})$-character varieties over compact orientable surfaces with parabolic points of semi-simple type. When the parabolic punctures are ... More
Using Category Theory in Modeling Generics in OOP (Outline)Jun 12 2019Modeling generics in object-oriented programming languages such as Java and C# is a challenge. Recently we proposed a new order-theoretic approach to modeling generics. Given the strong relation between order theory and category theory, in this extended ... More
Compact inverse categoriesJun 10 2019The Ehresmann-Schein-Nambooripad theorem gives a structure theorem for inverse monoids: they are inductive groupoids. A particularly nice case due to Jarek is that commutative inverse monoids become semilattices of abelian groups. It has also been categorified ... More
Rewriting modulo isotopies in pivotal linear $(2,2)$-categoriesJun 10 2019In this paper, we study rewriting modulo a set of algebraic axioms in categories enriched in linear categories, called linear~$(2,2)$-categories. We introduce the structure of linear~$(3,2)$-polygraph modulo as a presentation of a linear~$(2,2)$-category ... More
Antipodes, preantipodes and Frobenius functorsJun 08 2019We prove that a quasi-bialgebra admits a preantipode if and only if the associated free quasi-Hopf bimodule functor is Frobenius, if and only if the relative (opmonoidal) monad is a Hopf monad. The same results hold in particular for a bialgebra, tightening ... More
Some Ideas on Categories and SheavesJun 08 2019We firstly introduce some key concepts in category theory, such as quotient category, completion of limits, $\mathrm{Mor}$ category, and so on; then give the concept of topology algebras and sheaves, and discuss how to restore the structue of sheaves ... More
Notes on Jordan-Hölder property for exact categoriesJun 07 2019We prove a generalised version of Jordan-H\"older theorem for pre-abelian exact categories, which allow us to study the relative length function.
Three real Artin-Tate motivesJun 07 2019We analyze the spectrum of the tensor-triangulated category of Artin-Tate motives over the base field R of real numbers, with integral coefficients. Away from 2, we obtain the same spectrum as for complex Tate motives, previously studied by the second-named ... More
A topological groupoid representing the topos of presheaves on a monoidJun 06 2019Butz and Moerdijk famously showed that every (Grothendieck) topos with enough points is equivalent to the category of sheaves on some topological groupoid. We give an alternative, more algebraic construction in the special case of a topos of presheaves ... More
Split extensions and semidirect products of unitary magmasJun 05 2019We develop a theory of split extensions of unitary magmas, which includes defining such extensions and describing them via suitably defined semidirect product, yielding an equivalence between the categories of split extensions and of (suitably defined) ... More
Monoidal characterisation of groupoids and connectorsJun 05 2019We study internal structures in regular categories using monoidal methods. Groupoids in a regular Goursat category can equivalently be described as special dagger Frobenius monoids in its monoidal category of relations. Similarly, connectors can equivalently ... More
Formulating basic notions of finite group theory via the lifting propertyJun 05 2019We reformulate several basic notions of notions in finite group theory in terms of iterations of the lifting property (orthogonality) with respect to particular morphisms. Our examples include the notions being nilpotent, solvable, perfect, torsion-free; ... More
A dévissage theorem of non-connective $K$-theoryJun 04 2019The purpose of this article is to show a version of d\'evissage theorem of non-connective $K$-theory. Our theorem contains Quillen's d\'evissage theorem, Waldhausen's cell filtration theorem and theorem of heart as special cases. In this sense, we give ... More
Bialgebraic Semantics for String DiagramsJun 04 2019Jul 02 2019Turi and Plotkin's bialgebraic semantics is an abstract approach to specifying the operational semantics of a system, by means of a distributive law between its syntax (encoded as a monad) and its dynamics (an endofunctor). This setup is instrumental ... More
Bialgebraic Semantics for String DiagramsJun 04 2019Turi and Plotkin's bialgebraic semantics is an abstract approach to specifying the operational semantics of a system, by means of a distributive law between its syntax (encoded as a monad) and its dynamics (an endofunctor). This setup is instrumental ... More
Type-theoretic algebraic weak factorisation systemsJun 04 2019Motivated by Homotopy Type Theory, we introduce type-theoretic algebraic weak factorisation systems and show how they give rise to models of Martin-L\"of type theory. This is done by showing that the comprehension category associated to a type-theoretic ... More
The F-Symbols for the H3 Fusion CategoryJun 04 2019We present a solution for the F-symbols of the H3 fusion category, which is Morita equivalent to the even parts of the Haagerup subfactor. This solution has been computed by solving the pentagon equations and using several properties of trivalent categories. ... More
Modular operads and the nerve theoremJun 04 2019We describe a category of undirected graphs which comes equipped with a faithful functor into the category of (colored) modular operads. The associated singular functor from modular operads to presheaves is fully faithful, and its essential image can ... More
A graphical category for higher modular operadsJun 04 2019We present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy. This homotopy theory takes the form of a Quillen model structure on the collection of simplicial presheaves for a certain ... More
An algebraic representation of globular setsJun 03 2019We describe a fully faithful embedding of the category of (reflexive) globular sets into the category of counital cosymmetric coalgebras. This embedding is a lift of the usual functor of chains and the extra structure consists of a derived form of cup ... More
Automorphisms of categories of schemesJun 03 2019Given two schemes $S$ and $S'$, we prove that every equivalence between $\mathbf{Sch}_S$ and $\mathbf{Sch}_{S'}$ comes from a unique isomorphism between $S$ and $S'$. This eliminates all Noetherian and finite type hypotheses from a result of Mochizuki ... More
Descent Data and Absolute Kan ExtensionsJun 03 2019Jun 06 2019The fundamental construction underlying descent theory, the lax descent category, comes with a functor that forgets the \textit{descent data}. We prove that, in any $2$-category with lax descent objects, the forgetful morphisms create all absolute Kan ... More
Descent Data and Absolute Kan ExtensionsJun 03 2019The fundamental construction underlying descent theory, the lax descent category, comes with a functor that forgets the \textit{descent data}. We prove that, in any $2$-category with lax descent objects, the forgetful morphisms create all absolute Kan ... More
Type-theoretic weak factorization systemsJun 01 2019This article presents three characterizations of the weak factorization systems on finitely complete categories that interpret intensional dependent type theory with Sigma-, Pi-, and Id-types. The first characterization is that the weak factorization ... More
Equivariant Grothendieck-Riemann-Roch theorem via formal deformation theoryJun 01 2019We use the formalism of traces in higher categories to prove a common generalization of the holomorphic Atiyah-Bott fixed point formula and the Grothendieck-Riemann-Roch theorem. The proof is quite different from the original one proposed by Grothendieck ... More
Matrix factorizations for self-orthogonal categories of modulesMay 31 2019For a commutative ring $S$ and self-orthogonal subcategory $\mathsf{C}$ of $\mathsf{Mod}(S)$, we consider matrix factorizations whose modules belong to $\mathsf{C}$. Let $f\in S$ be a regular element. If $f$ is $M$-regular for every $M\in \mathsf{C}$, ... More
Probabilistic mappings and Bayesian nonparametricsMay 27 2019Jun 25 2019In this paper we develop a functorial language of probabilistic mappings and apply it to some basic problems in Bayesian nonparametrics. First we extend and unify the Kleisli category of probabilistic mappings proposed by Lawvere and Giry with the category ... More
Probabilistic mappings and Bayesian nonparametricsMay 27 2019In this paper we develop a functorial language of probabilistic mappings and apply it to basic problems in Bayesian nonparametrics. First we extend and unify the Kleisli category of probabilistic mappings proposed by Lawvere and Giry with the category ... More
Symmetric approximation sequences, derived equivalences and quotient algebras of locally $Φ$-Beilinson-Green algebrasMay 27 2019In this paper, we introduce a class of locally $\Phi$-Beilinson-Green algebras which is a way of obtaining algebras from C-constructions, where $\Phi$ is an infinite admissible set of the integral numbers, and show that symmetric approximation sequences ... More
On the equvialence of colimits and 2-colimitsMay 27 2019We compare the colimit and 2-colimit of strict 2-functors in the 2-category of groupoids, over a certain type of posets. These posets are of special importance, as they correspond to coverings of a topological space. The main result of this paper gives ... More
Homotopies in Grothendieck fibrationsMay 25 2019We define a natural 2-categorical structure on the base category of a large class of Grothendieck fibrations. Given any model category $\mathbf{C}$, we construct a fibration whose fibers are the homotopy categories of the slice categories $\mathbf{C}/A$, ... More
Toposes of Discrete Monoid ActionsMay 24 2019Properties of toposes of right $M$-sets are studied, and these toposes are characterised up to equivalence by their canonical points. The solution to the corresponding Morita equivalence problem is presented in the form of an equivalence between a 2-category ... More
Dependent products and 1-inaccessible universesMay 24 2019The purpose of this writing is to show that, if we use the definition of elementary $\infty$-topos that has been proposed by Mike Shulman, then the fact that every geometric $\infty$-topos satisfies the required axioms, more specifically the last one ... More
$\mathcal{M}$-coextensivity and the strict refinement propertyMay 24 2019The notion of an $\mathcal{M}$-coextensive object is introduced in an arbitrary category $\mathbb{C}$, where $\mathcal{M}$ is a distinguished class of morphisms from $\mathbb{C}$. This notion allows for a categorical treatment of the strict refinement ... More
Koszul duality in exact categoriesMay 24 2019In this paper we establish Koszul duality type results in the setting of chain complexes in exact categories. In particular we prove generalisations of Vallette's cooperadic Koszul duality theorem, and operadic Koszul duality along the lines of Lurie. ... More
The Moore Complex of a Simplicial Cocommutative Hopf AlgebraMay 23 2019We introduce the Moore complex of a simplicial cocommutative Hopf algebra through Hopf kernels. The most striking result to emerge from this construction is the coherent definition of 2-crossed modules of cocommutative Hopf algebras. This unifies the ... More
Von Neumann Regular $\mathcal{C}^{\infty}-$Rings and ApplicationsMay 23 2019In this paper we present the notion of a von Neumann regular $\mathcal{C}^{\infty}-$ring, we prove some results about them and we describe some of their properties. We prove, using two different methods, that the category of von Neumann regular $\mathcal{C}^{\infty}-$rings ... More
Condensations in higher categoriesMay 23 2019We present a higher-categorical generalization of the "Karoubi envelope" construction from ordinary category theory, and prove that, like the ordinary Karoubi envelope, our higher Karoubi envelope is the closure for absolute limits. Our construction replaces ... More
Rank-based persistenceMay 22 2019Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been attempted over time, some topological in flavor, based on the vector space-valued homology functor, other combinatorial, based ... More
Ontological models for quantum theory as functorsMay 22 2019We interpret ontological models for finite-dimensional quantum theory as functors from the category of finite-dimensional C*-algebras and completely positive maps to the category of measurable spaces and Markov kernels. This uniformises several earlier ... More
Recovery and convergence rate of the Frank-Wolfe Algorithm for the m-EXACT-SPARSE ProblemMay 22 2019We study the properties of the Frank-Wolfe algorithm to solve the m-EXACT-SPARSE reconstruction problem, where a signal y must be expressed as a sparse linear combination of a predefined set of atoms, called dictionary. We prove that when the signal is ... More
Complete Positivity for Mixed Unitary CategoriesMay 21 2019In this article we generalize the $\CP^\infty$-construction of dagger monoidal categories to mixed unitary categories. Mixed unitary categories provide a setting, which generalizes (compact) dagger monoidal categories and in which one may study quantum ... More
A bound for Hall's criterion for nilpotence in semi-abelian categoriesMay 21 2019In this paper, we focus on Hall's criterion for nilpotence in semi-abelian categories, and we improve the bound of Gray's main theorem of [3,Theorem 3.4] (see Main Theorem). And this bound is best possible.
Continuous-variable nonlocality and contextualityMay 20 2019Contextuality is a non-classical behaviour that can be exhibited by quantum systems. It is increasingly studied for its relationship to quantum-over-classical advantages in informatic tasks. To date, it has largely been studied in discrete variable scenarios, ... More
Cotorsion pairs and adjoint functors in the homotopy category of $N$-complexesMay 20 2019In this paper, we first construct some complete cotorson pairs on the category $\mathbb{C}_N(\mathcal{G})$ of unbounded $N$-complexes of Grothendieck category $\mathcal{G}$, from two given cotorsion pairs in $\mathcal{G}$. Next as an application, we focus ... More