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Monomial $G$-posets and their Lefschetz invariantsMar 20 2019Let $G$ be a finite group, and $C$ be an abelian group. We introduce the notions of $C$-monomial $G$-sets and $C$-monomial $G$-posets, and state some of their categorical properties. This gives in particular a new description of the $C$-monomial Burnside ... More

Hereditary species as monoidal decomposition spaces, comodule bialgebras, and operadic categoriesMar 19 2019We show that Schmitt's hereditary species induce monoidal decomposition spaces, and exhibit Schmitt's bialgebra construction as an instance of the general bialgebra construction on a monoidal decomposition space. We show furthermore that this bialgebra ... More

Algebras of the extended probabilistic powerdomain monadMar 18 2019We investigate the Eilenberg-Moore algebras of the extended probabilistic powerdomain monad $\mathcal V_w$ over the category $\mathbf{TOP}_0$ of $T_0$ topological spaces and continuous maps. We prove that every $\mathcal V_w$-algebra in our setting is ... More

Algebras of the extended probabilistic powerdomain monadMar 18 2019Mar 19 2019We investigate the Eilenberg-Moore algebras of the extended probabilistic powerdomain monad $\mathcal V_w$ over the category $\mathbf{TOP}_0$ of $T_0$ topological spaces and continuous maps. We prove that every $\mathcal V_w$-algebra in our setting is ... More

Relative $B$-groupsMar 17 2019This paper extends the notion of $B$-group to a relative context. For a finite group $K$ and a field $\mathbb{F}$ of characteristic 0, the lattice of ideals of the Green biset functor $\mathbb{F}B_K$ obtained by shifting the Burnside functor $\mathbb{F}B$ ... More

Foundations of Algebraic Theories and Higher Dimensional CategoriesMar 17 2019Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of universal algebra, ... More

Enriched Sets and Higher CategoriesMar 16 2019We introduce the notion of an enriched set, as an abstraction of enriched categories, and a category of enriched sets. The set of enriched sets is itself described as a set enriched over the category of enriched sets. We introduce a method for the construction ... More

Syntactic approaches to opetopesMar 14 2019Opetopes are algebraic descriptions of shapes corresponding to compositions in higher dimensions. As such, they offer an approach to higher-dimensional algebraic structures, and in particular, to the definition of weak $\omega$-categories, which was the ... More

The tricategory of formal composites and its strictificationMar 14 2019The results of this thesis allows one to replace calculations in tricategories with equivalent calculations in Gray categories (aka semistrict tricategories). In particular the rewriting calculus for Gray categories as used for example by the online proof ... More

Gabriel-Roiter measure, representation dimension and rejective chainsMar 13 2019The Gabriel-Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel-Roiter measure can be extended to abelian length categories ... More

Functorial PBW theorems for post-Lie algebrasMar 11 2019Using the categorical approach to Poincar\'e-Birkhoff-Witt type theorems from our previous work with Tamaroff, we prove three such theorems: for universal enveloping Rota-Baxter algebras of tridendriform algebras, for universal enveloping Rota--Baxter ... More

Representability theorems, up to homotopyMar 10 2019We prove two representability theorems, up to homotopy, for presheaves taking values in a closed symmetric combinatorial model category \cat V. The first theorem resembles the Freyd representability theorem, the second theorem is closer to the Brown representability ... More

Triangular Matrix Categories II: Recollements and functorially finite subcategoriesMar 10 2019In this paper we continue the study of triangular matrix categories $\mathbf{\Lambda}=\left[ \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}\right]$ initiated in [21]. First, given an additive category $\mathcal{C}$ and an ideal ... More

Triangular Matrix Categories I: Dualizing Varieties and generalized one-point extensionMar 10 2019Following Mitchell's philosophy, in this paper we define the analogous of the triangular matrix algebra to the context of rings with several objects. Given two additive categories $\mathcal{U}$ and $\mathcal{T}$ and $M\in \mathsf{Mod}(\mathcal{U}\otimes ... More

Polynomials as spansMar 10 2019The paper defines polynomials in a bicategory $\mathscr{M}$. Polynomials in bicategories $\mathrm{Spn}\mathscr{C}$ of spans in a finitely complete category $\mathscr{C}$ agree with polynomials in $\mathscr{C}$ as defined by Nicola Gambino and Joachim ... More

Infinity Operads and Monoidal Categories with Group EquivarianceMar 09 2019This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant structure. In the ... More

Model-independent comparison between factorization algebras and algebraic quantum field theory on Lorentzian manifoldsMar 08 2019This paper investigates the relationship between algebraic quantum field theories and factorization algebras on globally hyperbolic Lorentzian manifolds. Functorial constructions that map between these two types of theories in both directions are developed ... More

Nilpotent Types and Fracture Squares in Homotopy Type TheoryMar 08 2019We develop the basic theory of nilpotent types and their localizations away from sets of numbers in Homotopy Type Theory. For this, general results about the classifying spaces of fibrations with fiber an Eilenberg-Mac Lane space are proven. We also construct ... More

Higher Lawvere theoriesMar 07 2019We survey Lawvere theories at the level of infinity categories, as an alternative framework for higher algebra (rather than infinity operads). From a pedagogical perspective, they make many key definitions and constructions less technical. They are also ... More

Adelic models of tensor-triangulated categoriesMar 07 2019We show that a well behaved Noetherian, finite dimensional, stable, monoidal model category is equivalent to a model built from categories of modules over completed rings in an adelic fashion. For abelian groups this is based on the Hasse square, for ... More

Adelic cohomologyMar 07 2019The characteristic feature of the adeles is that they involve localizations of products (or equivalently restricted products of localizations). The point of this paper is to introduce an adelic style cohomological invariant of a partially ordered set ... More

Local cohomology in Grothendieck categoriesMar 06 2019Let $\mathcal{A}$ be a locally noetherian Grothendieck category. In this paper we define and study the section functor on $\mathcal{A}$ with respect to an open subset of ASpec$\mathcal{A}$. Next we define and study local cohomology theory in $\mathcal{A}$ ... More

On terminal coalgebras derived from initial algebrasMar 06 2019A number of important set functors have countable initial algebras, but terminal coalgebras are uncountable or even non-existent. We prove that the countable cardinality is an anomaly: every set functor with a nonempty initial algebra of a regular cardinality ... More

Cohomologies of a Lie algebra with a derivation and applicationsMar 06 2019The main object of study of this paper is the notion of a LieDer pair, i.e. a Lie algebra with a derivation. We introduce the concept of a representation of a LieDer pair and study the corresponding cohomologies. We show that a LieDer pair is rigid if ... More

Monoidal Adjunctions - Linearity and DualityMar 05 2019We explain two related constructions on the data of two monoidal symmetric closed categories $\mathscr{A}$ and $\mathscr{E}$ and monoidal functors $F: \mathscr{E}\to \mathscr{A}$ and $G: \mathscr{A}\to \mathscr{E}$. In a first part, we recall and partly ... More

Lenses and LearnersMar 05 2019Lenses are a well-established structure for modelling bidirectional transformations, such as the interactions between a database and a view of it. Lenses may be symmetric or asymmetric, and may be composed, forming the morphisms of a monoidal category. ... More

Topos quantum theory with short posetsMar 05 2019Topos quantum mechanics, developed by Isham et. al., creates a topos of presheaves over the poset V(N) of abelian von Neumann subalgebras of the von Neumann algebra N of bounded operators associated to a physical system, and established several results, ... More

Tensor product of correspondence functorsMar 05 2019As part of the study of correspondence functors, the present paper investigates their tensor product and proves some of its main properties. In particular, the correspondence functor associated to a finite lattice has the structure of a commutative algebra ... More

A recurrent formula of $A_{\infty}$-quasi inverses of dg-natural transformations between dg-lifts of derived functorsMar 05 2019A dg-natural transformation between dg-functors is called an objectwise homotopy equivalence if its induced morphism on each object admits a homotopy inverse. In general an objectwise homotopy equivalence does not have a dg-inverse but has an $A_{\infty}$ ... More

Injectives types in univalent mathematicsMar 04 2019We investigate the injective types and the algebraically injective types in univalent mathematics, both in the absence and in the presence of propositional resizing. Injectivity is defined by the surjectivity of the restriction map along any embedding, ... More

Bicategories in Univalent FoundationsMar 04 2019We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories. To construct examples of those, we develop the notion of ... More

Differentiable Causal Computations via Delayed TraceMar 04 2019We investigate causal computations taking sequences of inputs to sequences of outputs where the $n$th output depends on the first $n$ inputs only. We model these in category theory via a construction taking a Cartesian category $C$ to another category ... More

Locally small spaces with an applicationMar 03 2019We develop the theory of locally small spaces in a new simple language and apply this simplification to re-build the theory of locally definable spaces over structures with topologies.

More finite sets coming from non-commutative countingMar 01 2019Mar 04 2019This note contains a written form of a talk given by the first author at the conference on Mirror Symmetry and Related Topics, Miami, January 28-February 2, 2019. Details and related remarks are added. In our previous papers we introduced categorical ... More

Unifying notions of pasting diagramsMar 01 2019In this work, we relate the three main formalisms for the notion of pasting diagram in strict $\omega$-categories: Street's parity complexes, Johnson's pasting schemes and Steiner's augmented directed complexes. We first show that parity complexes and ... More

Abandoning Monomorphisms: Partial Maps, Fractions and FactorizationsFeb 28 2019For a composition-closed and pullback-stable class S of morphisms in a category C containing all isomorphisms, we form the category Span(C, S) of S-spans (s, f) in C with first "leg" s lying in S, and give an alternative construction of its quotient category ... More

$n$-Cotorsion pairsFeb 28 2019Motivated by some properties satisfied by Gorenstein projective and Gorenstein injective modules over an Iwanaga-Gorenstein ring, we present the concept of left and right $n$-cotorsion pairs in an abelian category $\mathcal{C}$. Two classes $\mathcal{A}$ ... More

On deformations of diagrams of commutative algebrasFeb 27 2019In this paper we study classical deformations of diagrams of commutative algebras over a field of characteristic 0. In particular we determine several homotopy classes of DG-Lie algebras, each one of them controlling this above deformation problem: the ... More

A cobordism category attached to Khovanov-Rozansky link homologies based on operadsFeb 27 2019We consider colored operads and their actions on categories. As a special example we construct a cobordism category with a colored operad action arising from oriented planar arc diagrams. This is used to construct an invariant of oriented tangle diagrams ... More

A cobordism category attached to Khovanov-Rozansky link homologies based on operadsFeb 27 2019Mar 15 2019We consider colored operads and their actions on categories. As a special example we construct a cobordism category with a colored operad action arising from oriented planar arc diagrams. This is used to construct an invariant of oriented tangle diagrams ... More

Tameness, powerful images, and large cardinalsFeb 26 2019We provide comprehensive, level-by-level characterizations of large cardinals, in the range from weakly compact to strongly compact, by closure properties of powerful images of accessible functors. In the process, we show that these properties are also ... More

Induction, Coinduction, and Fixed Points in PL Type TheoryFeb 26 2019Recently we presented a concise survey of the formulation of the induction and coinduction principles, and some concepts related to them, in programming languages type theory and four other mathematical disciplines. The presentation in type theory involved ... More

Simple and projective correspondence functorsFeb 26 2019A correspondence functor is a functor from the category of finite sets and correspondences to the category of $k$-modules, where $k$ is a commutative ring. We determine exactly which simple correspondence functors are projective. Moreover, we analyze ... More

Gabriel-Ulmer duality for topoi and its relation with site presentationsFeb 25 2019Given a regular cardinal $\kappa$, we define a $\kappa$-prototopos as a $\kappa$-small limit theory whose category of models is a Grothendieck topoi. A characterization of $\aleph_0$-prototopoi can be found in [A. Carboni, M. C. Pedicchio, and J. Rosick\'y, ... More

Étale inverse semigroupoids - the fundamentalsFeb 25 2019In this article we will study semigroupoids, and more specifically inverse semigroupoids. These are a common generalization to both inverse semigroups and groupoids, and provide a natural language on which several types of dynamical structures may be ... More

$\mathcal G$-systemsFeb 25 2019Mar 03 2019A $\mathcal G$-system is a collection of $\mathbb Z$-bases of $\mathbb Z^n$ with some extra axiomatic conditions. There are two kinds of actions "mutations" and "twists" naturally acting on a $\mathcal G$-system, which provide the combinatorial structure ... More

$\mathcal G$-systemsFeb 25 2019A $\mathcal G$-system is a collection of $\mathbb Z$-bases of $\mathbb Z^n$ with some extra axiomatic conditions. There are two kinds of actions "mutations" and "twists" naturally acting on a $\mathcal G$-system, which provide the combinatorial structure ... More

(Lack of) Model Structures on the Category of GraphsFeb 25 2019In the present article, we study model structures on the category of graphs with $\times$-homotopy equivalences as the weak equivalences, namely, $(\mathcal{G},\times)$. We show that the analog of Strom-Hurewicz model structure in the category of graphs ... More

The operad that corepresents enrichmentFeb 24 2019I show that the theories of enrichment in a monoidal infinity-category defined by Hinich and by Gepner-Haugseng agree, and that the identification is unique. Among other things, this makes the Yoneda lemma available in the former model.

Inner horns for 2-quasi-categoriesFeb 23 2019The purpose of this paper is to provide a characterisation of fibrations into 2-quasi-categories using inner horn inclusions. There are two slightly different definitions of horn inclusion for $\Theta_2$-sets; one is the usual subcomplex generated by ... More

Examples of (non)-braided crossed productFeb 21 2019In [A. Mejia, M. Mombelli, Crossed extensions of the corepresentation category of finite supergroup algebras, Int. J. Math.], eight tensor categories were introduced which are extensions of the category Comod(H) of comodules over a supergroup algebra ... More

Nearness PosetsFeb 21 2019We extend nearness frames to posets representing bases and even subbases of $T_1$ spaces. This allows us to put a classic duality due to Wallman, between compact $T_1$ spaces and abstract simplicial complexes, into a general nearness framework. Within ... More

The Lawvere condition and a classification theorem for Mal'tsev categoriesFeb 20 2019A classification theorem for three different sorts of Mal'tsev categories is proven. The theorem provides a classification for Mal'tsev category, naturally Malt'sev category, and weakly Mal'tsev category in terms of classifying classes of spans. The class ... More

Characterizations of majority categoriesFeb 19 2019In universal algebra, it is well known that varieties admitting a majority term admit several Mal'tsev-type characterizations. The main aim of this paper is to establish categorical counterparts of some of these characterizations for regular categories. ... More

The center of a Green biset functorFeb 17 2019For a Green biset functor $A$, we define the commutant and the center of $A$ and we study some of their properties and their relationship. This leads in particular to the main application of these constructions: the possibility of splitting the category ... More

On Group-Like MagmoidsFeb 16 2019A magmoid is a non-empty set with a partial binary operation; group-like magmoids generalize group-like magmas such as semigroups, monoids and groups. In this article, we first consider the many ways in which the notions of associative multiplication, ... More

A Fubini rule for $\infty$-coendsFeb 16 2019We prove a Fubini rule for $\infty$-co/ends of $\infty$-functors $F : \mathcal C^\text{op}\times\mathcal C\to \mathcal D$. This allows to lay down "integration rules", similar to those in classical co/end calculus, also in the setting of $\infty$-categories. ... More

A standard theorem on adjunctions in two variablesFeb 16 2019We record an explicit proof of the theorem that lifts a two-variable adjunction to the arrow categories of its domains.

Partial silting objects and smashing subcategoriesFeb 15 2019We study smashing subcategories of a triangulated category with coproducts via silting theory. Our main result states that for derived categories of dg modules over a non-positive differential graded ring, every compactly generated localising subcategory ... More

On Finitary Functors and Finitely Presentable AlgebrasFeb 15 2019A simple criterion for a functor to be finitary is presented: we call $F$ finitely bounded if for all objects $X$ every finitely generated subobject of $FX$ factorizes through the $F$-image of a finitely generated subobject of $X$. This is equivalent ... More

Nerves of 2-categories and 2-categorification of $(\infty,2)$-categoriesFeb 14 2019We show that the homotopy theory of strict 2-categories embeds in that of $(\infty,2)$-categories in the form of 2-precomplicial sets. More precisely, we construct a nerve-categorification adjunction that is a Quillen pair between Lack's model structure ... More

Pretorsion theories, stable category and preordered setsFeb 14 2019We show that in the category of preordered sets, there is a natural notion of pretorsion theory, in which the partially ordered sets are the torsion-free objects and the sets endowed with an equivalence relation are the torsion objects. Correspondingly, ... More

Correspondence functors and latticesFeb 13 2019A correspondence functor is a functor from the category of finite sets and correspondences to the category of k-modules, where k is a commu-tative ring. A main tool for this study is the construction of a correspondence functor associated to any finite ... More

Correspondence functors and finiteness conditionsFeb 13 2019We investigate the representation theory of finite sets. The correspondence functors are the functors from the category of finite sets and correspondences to the category of k-modules, where k is a commutative ring. They have various specific properties ... More

Morita Bicategories of Algebras and Duality InvolutionsFeb 13 2019The notion of a weak duality involution on a bicategory was recently introduced by Shulman in [arXiv:1606.05058]. We construct a weak duality involution on the fully dualisable part of $\text{Alg}$, the Morita bicategory of finite-dimensional k-algebras. ... More

The algebra of Boolean matrices, correspondence functors, and simplicityFeb 13 2019We determine the dimension of every simple module for the algebra of the monoid of all relations on a finite set (i.e. Boolean matrices). This is in fact the same question as the determination of the dimension of every evaluation of a simple correspondence ... More

Integral and differential structure on the free $C^{\infty}$-ring modalityFeb 12 2019This paper develops an example of an integral category whose integral transformation operates on smooth 1-forms. Further, we revisit the differential structure of this category, and we investigate derivations, coderelictions, and Rota-Baxter algebras ... More

High-level methods for homotopy construction in associative $n$-categoriesFeb 11 2019A combinatorial theory of associative $n$-categories has recently been proposed, with strictly associative and unital composition in all dimensions, and the weak structure arising as a combinatorial notion of homotopy with a natural geometrical interpretation. ... More

Monoidal structures on the categories of quadratic dataFeb 11 2019The notion of 2--monoidal category used here was introduced by B.~Vallette in 2007 for applications in the operadic context. The starting point for this article was a remark by Yu. Manin that in the category of quadratic algebras (that is, "quantum linear ... More

Monoidal structures on the categories of quadratic dataFeb 11 2019Feb 28 2019The notion of 2--monoidal category used here was introduced by B.~Vallette in 2007 for applications in the operadic context. The starting point for this article was a remark by Yu. Manin that in the category of quadratic algebras (that is, "quantum linear ... More

Derivator Six-Functor-Formalisms - Construction IIFeb 10 2019Starting from very simple and obviously necessary axioms on a (derivator enhanced) four-functor-formalism, we construct derivator six-functor-formalisms using compactifications. This works, for example, for various contexts over topological spaces and ... More

On d-Categories and d-OperadsFeb 09 2019We extend the theory of d-categories, by providing an explicit description of the right mapping spaces of the d-homotopy category of an $\infty$-category. Using this description, we deduce an invariant $\infty$-categorical characterization of the d-homotopy ... More

Controlled objects as a symmetric monoidal functorFeb 08 2019The goal of this paper is to associate functorially to every symmetric monoidal additive category $\mathbf{A}$ with a strict $G$-action a lax symmetric monoidal functor $\mathbf{V}_{\mathbf{A}}^{G}:G\mathbf{BornCoarse}\to \mathbf{Add}_{\infty}$ from the ... More

Majority categoriesFeb 08 2019We introduce the notion of a majority category --- the categorical counterpart of varieties of universal algebras admitting a majority term. This notion can be thought to capture properties of the category of lattices, in a way that parallels how Mal'tsev ... More

Deformations of Kupershmidt operators on Leibniz algebras and Leibniz bialgebrasFeb 08 2019In this paper, we study (proto-, quasi-)twilled Leibniz algebras and the associated L-infty-algebras and differential graded Lie algebras. As applications, first we study the twilled Leibniz algebra corresponding to the semidirect product of a Leibniz ... More

Operads without cogebrasFeb 07 2019We give an example of a non-trivial linear operad that only admits trivial cogebras and give sufficient conditions ensuring that the cofree cogebra functor be faithful.

Model categories of quiver representationsFeb 06 2019Gillespie's Theorem gives a systematic way to construct model category structures on $\mathscr{C}( \mathscr{M} )$, the category of chain complexes over an abelian category $\mathscr{M}$. We can view $\mathscr{C}( \mathscr{M} )$ as the category of representations ... More

Semantic Factorization and DescentFeb 04 2019Feb 05 2019Let $\mathbb{A}$ be a $2$-category with suitable opcomma objects and pushouts. We give a direct proof that, provided that the codensity monad of a morphism $p$ exists and is preserved by a suitable morphism, the factorization given by the lax descent ... More

Weighted limits in an $(\infty,1)$-categoryFeb 02 2019We introduce the notion of weighted limit in an arbitrary quasi-category, suitably generalizing ordinary limits in a quasi-category, and classical weighted limits in an ordinary category. This is accomplished by generalizing Joyal's approach: we identify ... More

On the Objects and Morphisms of Category of Soft setsFeb 01 2019Soft set theory can deal uncertainties in nature by parametrization process. In this paper, we explore the objects and morphisms of category of soft sets, Sset(U) in detail. Also, gives characterizations of monomorphisms and epimorphisms in Sset(U).

From non-commutative diagrams to anti-elementary classesFeb 01 2019Anti-elementarity is a strong way of ensuring that a class of structures , in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form $\mathscr{L}_{\infty\lambda}$. We prove that many ... More

Completeness of infinitary heterogeneous logicJan 31 2019Given a regular cardinal $\kappa$ such that $\kappa^{<\kappa}=\kappa$ (e.g., if the Generalized Continuum Hypothesis holds), we develop a proof system for classical infinitary logic that includes heterogeneous quantification (i.e., infinite alternate ... More

Categorical representations and KLR algebrasJan 30 2019We prove that the KLR algebra associated with the cyclic quiver of length $e$ is a subquotient of the KLR algebra associated with the cyclic quiver of length $e+1$. We also give a geometric interpretation of this fact. This result has an important application ... More

A Coalgebraic View on ReachabilityJan 30 2019Coalgebras for an endofunctor provide a category-theoretic framework for modeling a wide range of state-based systems of various types. We provide an iterative construction of the reachable part of a given pointed coalgebra that is inspired and resembles ... More

A Coalgebraic View on ReachabilityJan 30 2019Mar 11 2019Coalgebras for an endofunctor provide a category-theoretic framework for modeling a wide range of state-based systems of various types. We provide an iterative construction of the reachable part of a given pointed coalgebra that is inspired by and resembles ... More

The Gray monoidal product of double categoriesJan 30 2019The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category $\mathbb A$, the corresponding internal hom functor $|[ \mathbb A,-]|$ sends a double category $\mathbb B$ to the double ... More

Universal Properties in Quantum TheoryJan 29 2019We argue that notions in quantum theory should have universal properties in the sense of category theory. We consider the completely positive trace preserving (CPTP) maps, the basic notion of quantum channel. Physically, quantum channels are derived from ... More

Degenerating $0$ in Triangulated CategoriesJan 28 2019In previous work, based on work of Zwara and Yoshino, we defined and studied degenerations of objects in triangulated categories analogous to degeneration of modules. In triangulated categories it is surprising that the zero object may degenerate. We ... More

Cycles over DGH-categories and pairings in categorical Hopf-cyclic cohomologyJan 28 2019Let $H$ be a Hopf algebra and let $\mathcal D_H$ be a Hopf-module category. Let $M$ be a stable anti-Yetter-Drinfeld module over $H$. We describe the Hopf-cyclic cohomology $HC^\bullet_H(\mathcal D_H,M)$ using characters of differential graded Hopf-module ... More

Three notions of dimension for triangulated categoriesJan 27 2019In this note we discuss three notions of dimension for triangulated categories: Rouquier dimension, diagonal dimension and Serre dimension. We prove some basic properties of these dimensions, compare them and discuss open problems.

Limits of the Banach spaces associated with positive operator $a$ affiliated with von Neumann algebra, which are neither purely projective nor purely inductiveJan 26 2019Feb 11 2019We consider linear normed spaces of opearators dominated by positive operator affiliated with the von Neumann algebra powered by real positive parameter. We consider and define different natural constructions of the limits spaces, based on projective ... More

Categorical semantics of metric spaces and continuous logicJan 25 2019Using the category of metric spaces as a template, we develop a metric analogue of the categorical semantics of classical/intuitionistic logic, and show that the natural notion of predicate in this "continuous semantics" is equivalent to the a priori ... More

Homotopy quotients and comodules of supercommutative Hopf algebrasJan 25 2019Jan 28 2019We study induced model structures on Frobenius categories. In particular we consider the case where $\mathcal{C}$ is the category of comodules of a supercommutative Hopf algebra $A$ over a field $k$. Given a graded Hopf algebra quotient $A \to B$ satisfying ... More

On the Monoidal Center of Deligne's Category Rep(S_t)Jan 24 2019We explicitly compute a monoidal subcategory of the monoidal center of Deligne's interpolation category Rep(S_t), for t not necessarily a natural number, and we show that this subcategory is a ribbon category. For t=n, a natural number, there exists a ... More

A Report on Subobject Classifiers and MonadsJan 23 2019A two-part report, containing (unrelated) essays on subobject classifiers and on monads.

The Freyd-Mitchell Embedding TheoremJan 23 2019Given a small abelian category $\mathcal{A}$, the Freyd-Mitchell embedding theorem states the existence of a ring $R$ and an exact full embedding $\mathcal{A} \rightarrow R$-Mod. This theorem is useful as it allows one to prove general results about abelian ... More

$W$-Types in Categories of CoalgebrasJan 19 2019We construct $W$-types in the category of coalgebras for a cartesian comonad. It generalizes the constructions of $W$-types in presheaf toposes and gluing toposes.

Beyond topological persistence: Starting from networksJan 19 2019Nowadays, data generation, representation and analysis occupy central roles in human society. Therefore, it is necessary to develop frameworks of analysis able of adapting to diverse data structures with minimal effort, much as guaranteeing robustness ... More

The Tale of Two Categories: Inductive groupoids and Cross-connectionsJan 17 2019Jan 24 2019A groupoid is a small category in which all morphisms are isomorphisms. An inductive groupoid is a specialised groupoid whose object set is a regular biordered set and the morphisms admit a partial order. A normal category is a specialised small category ... More

The singularity category of an algebra with radical square zeroJan 15 2019Feb 05 2019This paper studies the singularity category of a locally bounded $k$-linear category $\mathscr{C}$ with radical square zero. Following the work of Bautista and Liu [4], we give a complete description of $D_{sg}(\mathscr{C})$. Examples are provided to ... More