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2-Segal objects and algebras in spansMay 16 2019We define a category parameterizing Calabi-Yau algebra objects in an infinity category of spans. Using this category, we prove that there are equivalences of infinity categories relating, firstly: 2-Segal simplicial objects in C to algebra objects in ... More

The Constituents of Sets, Numbers, and Other Mathematical Objects, Part TwoMay 15 2019The arithmetic of natural numbers has a natural and simple encoding within sets, and the simplest set whose structure is not that of any natural number extends this set-theoretic representation to positive and negative integers. The operation that implements ... More

Towards a constructive simplicial model of Univalent FoundationsMay 15 2019We provide a partial solution to the problem of defining a constructive version of Voevodsky's simplicial model of univalent foundations. For this, we prove constructive counterparts of the necessary results of simplicial homotopy theory, building on ... More

On the profinite homotopy type of log schemesMay 15 2019We complete the program, initiated in [6], to compare the many different possible definitions of the underlying homotopy type of a log scheme. We show that, up to profinite completion, they all yield the same result, and thus arrive at an unambiguous ... More

On the Universal Property of Derived ManifoldsMay 15 2019It is well known that any model for derived manifolds must form a higher category. In this paper, we propose a universal property for this higher category, classifying it up to equivalence. Namely, the $\infty$-category $\mathbf{DMfd}$ of derived manifolds ... More

A constructive account of the Kan-Quillen model structure and of Kan's Ex$^{\infty}$ functorMay 15 2019We give a fully constructive proof that there is a proper cartesian $\omega$-combinatorial model structure on the category of simplicial sets, whose generating cofibrations and trivial cofibrations are the usual boundary inclusion and horn inclusion. ... More

$\mathbb{P}^n$-functorsMay 14 2019We propose a new theory of (non-split) P^n-functors. These are F: A -> B for which the adjunction monad RF is a repeated extension of Id_A by powers of an autoequivalence H and three conditions are satisfied: the monad condition, the adjoints condition, ... More

Enriched Lawvere Theories for Operational SemanticsMay 14 2019Enriched Lawvere theories are a generalization of Lawvere theories that allow us to describe the operational semantics of formal systems. For example, a graph-enriched Lawvere theory describes structures that have a graph of operations of each arity, ... More

On the homotopy hypothesis in dimension 3May 14 2019We show that if the canonical left semi-model structure on the category of Grothendieck $n$-groupoids exists, then it satisfies the homotopy hypothesis, i.e. the associated $(\infty,1)$-category is equivalent to that of homotopy $n$-types, thus generalizing ... More

Embedding Deligne's category $\mathrm{Rep}(S_t)$ in the Heisenberg categoryMay 14 2019We define a faithful linear monoidal functor from the partition category, and hence from Deligne's category $\mathrm{Rep}(S_t)$, to the Heisenberg category. We show that the induced map on Grothendieck rings is injective and corresponds to the Kronecker ... More

The center of monoidal bicategories in 3+1D Dijkgraaf-Witten TheoryMay 12 2019In this work, for a finite group $G$ and a 4-cocycle $\omega \in Z^4(G,\mathbf{k}^\times)$, we compute explicitly the center of the monoidal bicategory $\operatorname{2Vec}_G^{\omega}$ of $\omega$-twisted $G$-graded 1-categories of finite dimensional ... More

Conformal nets V: dualizabilityMay 09 2019We prove that finite-index conformal nets are fully dualizable objects in the 3-category of conformal nets. Therefore, assuming the cobordism hypothesis applies, there exists a local framed topological field theory whose value on the point is any finite-index ... More

Higher Segal spaces and Lax $\mathbb{A}_\infty$-algebrasMay 08 2019The notion of a higher Segal space was introduced by Dyckerhoff and Kapranov as a general framework for studying higher associativity inherent in a wide range of mathematical objects. In the present work we formalize the connection between this notion ... More

The free globularily generated double category as a free objectMay 06 2019We provide a formal interpretation of the free globularily generated double category construction as a free construction in the category of globularily generated double categories. We regard this as a categorical formalization of the consideration of ... More

Characterizing the invariances of learning algorithms using category theoryMay 06 2019Many learning algorithms have invariances: when their training data is transformed in certain ways, the function they learn transforms in a predictable manner. Here we formalize this notion using concepts from the mathematical field of category theory. ... More

Some remarks on blueprints and ${\mathbb F}_1$-schemesMay 03 2019Over the past two decades several different approaches to defining a geometry over ${\mathbb F}_1$ have been proposed. In this paper, relying on To\"en and Vaqui\'e's formalism (2009), we investigate the category ${\mathsf{Sch}}_{\mathsf B}$ of schemes ... More

Wide subcategories and lattices of torsion classesMay 03 2019In this paper, we study the relationship between wide subcategories and torsion classes of an abelian length category $\mathcal{A}$ from the point of view of lattice theory. Motivated by $\tau$-tilting reduction of Jasso, we mainly focus on intervals ... More

Hopf-Frobenius algebras and a new Drinfeld doubleMay 02 2019The ZX-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of $\dag$-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of this structure, ... More

Computational Petri Nets: Adjunctions Considered HarmfulApr 29 2019We review some of the endeavors in trying to connect Petri nets with free symmetric monoidal categories. We give a list of requirement such connections should respect if they are meant to be useful for practical/implementation purposes. We show how previous ... More

Computational Petri Nets: Adjunctions Considered HarmfulApr 29 2019May 08 2019We review some of the endeavors in trying to connect Petri nets with free symmetric monoidal categories. We give a list of requirement such connections should respect if they are meant to be useful for practical/implementation purposes. We show how previous ... More

On morphisms killing weights and Hurewicz-type theoremsApr 29 2019We study "canonical weight decompositions" slightly generalizing that defined by J. Wildeshaus. For an triangulated category $C$, any integer $n$, and a weight structure $w$ on $C$ a triangle $LM\to M\to RM\to LM[1]$, where $LM$ is of weights at most ... More

Polynomial functors and two-parameter quantum symmetric pairsApr 29 2019Apr 30 2019We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups $GL_n$, the two-parameter polynomial functors give a new ... More

Polynomial functors and two-parameter quantum symmetric pairsApr 29 2019We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups $GL_n$, the two-parameter polynomial functors give a new ... More

Polynomial functors and two-parameter quantum symmetric pairsApr 29 2019May 07 2019We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups $GL_n$, the two-parameter polynomial functors give a new ... More

The Yoneda Ext bifunctor and arbitrary products and coproducts in abelian categoriesApr 27 2019There are well known identities that involve the Ext bifunctor, coproducts, and products in Ab4 and Ab4* abelian categories with enough projectives and enough injectives. Namely, for every such category $\mathcal{A}$, the isomorphisms $\operatorname{Ext}^n ... More

The Category of Node-and-Choice Forms, with Subcategories for Choice-Sequence Forms and Choice-Set FormsApr 27 2019The literature specifies extensive-form games in many styles, and eventually I hope to formally translate games across those styles. Toward that end, this paper defines $\mathbf{NCF}$, the category of node-and-choice forms. The category's objects are ... More

Ambiguous representations of semilattices and imperfect informationApr 26 2019Crisp and lattice-valued ambiguous representations of one continuous semilattice in another one are introduced and operation of taking pseudo-inverse of the above relations is defined. It is shown that continuous semilattices and their ambiguous representations, ... More

Shifted Coisotropic CorrespondencesApr 25 2019We define (iterated) coisotropic correspondences between derived Poisson stacks, and construct symmetric monoidal higher categories of derived Poisson stacks where the $i$-morphisms are given by $i$-fold coisotropic correspondences. Assuming an expected ... More

Accessible categories, set theory, and model theory: an invitationApr 25 2019We give a self-contained introduction to accessible categories and how they shed light on both model- and set-theoretic questions. We survey for example recent developments on the study of presentability ranks, a notion of cardinality localized to a given ... More

The Way of the DaggerApr 24 2019A dagger category is a category equipped with a functorial way of reversing morphisms, i.e. a contravariant involutive identity-on-objects endofunctor. Dagger categories with additional structure have been studied under different names e.g. in categorical ... More

Finite torsors on projective schemes defined over a discrete valuation ringApr 24 2019Given a Henselian and Japanese discrete valuation ring $A$ and a flat and projective $A$-scheme $X$, we follow the approach of Biswas-dos Santos to introduce a full subcategory of coherent modules on $X$ which is then shown to be Tannakian. We then prove ... More

Pyknotic objects, I. Basic notionsApr 22 2019Pyknotic objects in are sheaves on the site of compacta. These provide a convenient way to do algebra and homotopy theory with additional topological information present. This appears, for example, when trying to contemplate the derived category of a ... More

Pyknotic objects, I. Basic notionsApr 22 2019Apr 30 2019Pyknotic objects are (hyper)sheaves on the site of compacta. These provide a convenient way to do algebra and homotopy theory with additional topological information present. This appears, for example, when trying to contemplate the derived category of ... More

Gorenstein dimension of abelian categoriesApr 22 2019Let C be triangulated category and X a cluster tilting subcategory of C. Koenig and Zhu showed that the quotient category C/X is Gorenstein of Gorenstein dimension at most one. The notion of an extriangulated category was introduced by Nakaoka and Palu ... More

Introduction to Gestural Similarity in Music. An Application of Category Theory to the OrchestraApr 22 2019Mathematics, and more generally computational sciences, intervene in several aspects of music. Mathematics describes the acoustics of the sounds giving formal tools to physics, and the matter of music itself in terms of compositional structures and strategies. ... More

Fuzzy set and presheavesApr 21 2019This note presents a presheaf theoretic approach to the construction of fuzzy sets, which builds on Barr's description of fuzzy sets as sheaves of monomorphisms on a locale. A presheaf-theoretic method is used to show that the category of fuzzy sets is ... More

Quantum channels as a categorical completionApr 21 2019We propose a categorical foundation for the connection between pure and mixed states in quantum information and quantum computation. The foundation is based on distributive monoidal categories. First, we prove that the category of all quantum channels ... More

Quantum channels as a categorical completionApr 21 2019Apr 24 2019We propose a categorical foundation for the connection between pure and mixed states in quantum information and quantum computation. The foundation is based on distributive monoidal categories. First, we prove that the category of all quantum channels ... More

Compositionality of Rewriting Rules with ConditionsApr 19 2019We extend the notion of compositional associative rewriting as recently studied in the rule algebra framework literature to the setting of rewriting rules with conditions. Our methodology is category-theoretical in nature, where the definition of rule ... More

Generalized Petri NetsApr 19 2019We give a definition of $\mathsf{Q}$-$\mathsf{Net}$; a generalization of Petri nets based on a Lawvere theory $\mathsf{Q}$ for which many existing variants of Petri nets are a special case. This definition is functorial with respect to change in Lawvere ... More

A complete classification of pivotal fusion categories $\otimes$-generated by an object of dimension $\frac{1 + \sqrt{5}}{2}$Apr 18 2019In this paper we give a complete classification of pivotal fusion categories $\otimes$-generated by an object of dimension $\frac{1 + \sqrt{5}}{2}$. We show that all such categories arise as certain wreath products of either the Fibonacci category, or ... More

A unified framework for notions of algebraic theoryApr 18 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

A Categorical Approach to L-ConvexityApr 17 2019We investigate an enriched-categorical approach to a field of discrete mathematics. The main result is a duality theorem between a class of enriched categories (called $\overline{\mathbb{Z}}$- or $\overline{\mathbb{R}}$-categories) and that of what we ... More

A 2-Categorical Study of Graded and Indexed MonadsApr 17 2019In the study of computational effects, it is important to consider the notion of computational effects with parameters. The need of such a notion arises when, for example, statically estimating the range of effects caused by a program, or studying the ... More

All $(\infty,1)$-toposes have strict univalent universesApr 15 2019We prove the conjecture that any Grothendieck $(\infty,1)$-topos can be presented by a Quillen model category that interprets homotopy type theory with strict univalent universes. Thus, homotopy type theory can be used as a formal language for reasoning ... More

All $(\infty,1)$-toposes have strict univalent universesApr 15 2019Apr 26 2019We prove the conjecture that any Grothendieck $(\infty,1)$-topos can be presented by a Quillen model category that interprets homotopy type theory with strict univalent universes. Thus, homotopy type theory can be used as a formal language for reasoning ... More

Endomorphisms of functors $\vdash$ representationsApr 15 2019The functor that takes a ring to its category of modules has an adjoint if one remembers the forgetful functor to abelian groups: the endomorphism ring of linear natural transformations. This uses the self-enrichment of the category of abelian groups. ... More

On the naturalness of Mal'tsev categoriesApr 14 2019Mal'tsev categories turned out to be a central concept in categorical algebra. On one hand, the simplicity and the beauty of the notion is revealed through a lot of characterizations of different flavour. Depending on the context, one can define Mal'tsev ... More

Injective Semimodules - RevisitedApr 14 2019Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no non-zero injective semimodules (e.g. the semiring of non-negative integers). In this paper, we study ... More

Involution algebroids: a generalisation of Lie algebroids for tangent categoriesApr 13 2019We define involution algebroids which generalise Lie algebroids to the abstract setting of tangent categories. As a part of this generalisation the Jacobi identity which appears in classical Lie theory is replaced by an identity similar to the Yang-Baxter ... More

Involution algebroids: a generalisation of Lie algebroids for tangent categoriesApr 13 2019Apr 16 2019We define involution algebroids which generalise Lie algebroids to the abstract setting of tangent categories. As a part of this generalisation the Jacobi identity which appears in classical Lie theory is replaced by an identity similar to the Yang-Baxter ... More

Involution algebroids: a generalisation of Lie algebroids for tangent categoriesApr 13 2019May 10 2019We define involution algebroids which generalise Lie algebroids to the abstract setting of tangent categories. As a part of this generalisation the Jacobi identity which appears in classical Lie theory is replaced by an identity similar to the Yang-Baxter ... More

Information Based Method for Approximate Solving Stochastic Control ProblemsApr 12 2019An information based method for solving stochastic control problems with partial observation has been proposed. First, the information-theoretic lower bounds of the cost function has been analysed. It has been shown, under rather weak assumptions, that ... More

Uniform Interpolation and Compact CongruencesApr 12 2019Uniform interpolation properties are defined for equational consequence in a variety of algebras and related to properties of compact congruences on first the free and then the finitely presented algebras of the variety. It is also shown, following related ... More

Sheaves and DualityApr 11 2019It has long been known in universal algebra that any distributive sublattice of congruences of an algebra which consists entirely of commuting congruences yields a sheaf representation of the algebra. In this paper we provide a generalisation of this ... More

Weak factorization systems and stable independenceApr 11 2019We exhibit a bridge between the theory of weak factorization systems, a categorical concept used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the cofibrantly generated ... More

Weak factorization systems and stable independenceApr 11 2019Apr 23 2019We exhibit a bridge between the theory of weak factorization systems, a categorical concept used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the cofibrantly generated ... More

Quantale-valued dissimilarityApr 11 2019In the same spirit of the theory of apartness relations of Scott, a positive theory of dissimilarity valued in an involutive quantale $\mathsf{Q}$ is established without the aid of negation, which dualizes the theory of $\mathsf{Q}$-valued sets in the ... More

A counterexample in quasi-category theoryApr 10 2019We give an example of a morphism of simplicial sets which is a monomorphism, bijective on 0-simplices, and a weak categorical equivalence, but which is not inner anodyne. This answers an open question of Joyal. Furthermore, we use this morphism to refute ... More

A counterexample in quasi-category theoryApr 10 2019Apr 19 2019We give an example of a morphism of simplicial sets which is a monomorphism, bijective on 0-simplices, and a weak categorical equivalence, but which is not inner anodyne. This answers an open question of Joyal. Furthermore, we use this morphism to refute ... More

A note on abelian quotient categoriesApr 09 2019Let C be a triangulated category with a Serre functor S and X a non-zero contravariantly finite rigid subcategory of C. Then X is cluster tilting if and only if the quotient category C/X is abelian and S(X)=X[2]. As an application, this result generalizes ... More

The Constituents of Sets, Numbers, and Other Mathematical Objects, Part OneApr 09 2019The sets used to construct other mathematical objects are pure sets, which means that all of their elements are sets, which are themselves pure. One set may therefore be within another, not as an element, but as an element of an element, or even deeper, ... More

Categorified Chern character and cyclic cohomologyApr 08 2019We examine Hopf cyclic cohomology in the same context as the analysis of the geometry of loop spaces $LX$ in derived algebraic geometry and the resulting close relationship between $S^1$-equivariant quasi-coherent sheaves on $LX$ and $D_X$-modules. Furthermore, ... More

Six model categories for directed homotopyApr 08 2019We construct a q-model structure, a h-model structure and a m-model structure on multipointed $d$-spaces and on flows. The two q-model structures are combinatorial and coincide with the combinatorial model structures already known on these categories. ... More

A General Framework for the Semantics of Type TheoryApr 08 2019We propose an abstract notion of a type theory to unify the semantics of various type theories including Martin-L\"{o}f type theory, two-level type theory and cubical type theory. We establish basic results in the semantics of type theory: every type ... More

Network Models from Petri Nets with CatalystsApr 07 2019Petri networks and network models are two frameworks for the compositional design of systems of interacting entities. Here we show how to combine them using the concept of a "catalyst": an entity that is neither destroyed nor created by any process it ... More

The Mathematics of Text StructureApr 06 2019In previous work we gave a mathematical foundation, referred to as DisCoCat, for how words interact in a sentence in order to produce the meaning of that sentence. To do so, we exploited the perfect structural match of grammar and categories of meaning ... More

Torsion pairs in categories of modules over a preadditive categoryApr 05 2019We present a natural extension to functor categories over small preadditive categories of the classical results of Gabriel and Jans classifying, respectively, hereditary torsion pairs and TTF triples in terms of Gabriel topologies and idempotent ideals. ... More

Auslander-Reiten triangles and Grothendieck groups of triangulated categoriesApr 04 2019We prove that if the Auslander-Reiten triangles generate the relations for the Grothendieck group of a Hom-finite Krull-Schmidt triangulated category with a cogenerator, then the category has only finitely many isomorphism classes of indecomposable objects ... More

Conjugacy classes and centralizers for pivotal fusion categoriesApr 03 2019A criterion for M\"uger centralizer of a fusion subcategory of a braided non-degenerate fusion category is given. Along the way we extend some identities on the space of class functions of a fusion category introduced by Shimizu in \cite{scalg}. We also ... More

On coherent topoi & coherent $1$-localic $\infty$-topoiApr 03 2019In this note we prove the following useful fact that seems to be missing from the literature: the $\infty$-category of coherent ordinary topoi (in the sense of SGA4) is equivalent to the $\infty$-category of coherent $1$-localic $\infty$-topoi (in Lurie's ... More

Inversion, Iteration, and the Art of Dual WieldingApr 02 2019The humble $\dagger$ ("dagger") is used to denote two different operations in category theory: Taking the adjoint of a morphism (in dagger categories) and finding the least fixed point of a functional (in categories enriched in domains). While these two ... More

Pushouts and e-Projective SemimodulesApr 02 2019Projective modules play an important role in the study of the category of modules over rings and in the characterization of various classes of rings. Several characterizations of projective objects which are equivalent for modules over rings are not necessarily ... More

Graph cohomologies and rational homotopy type of configuration spacesApr 02 2019We compare the cohomology complex defined by Baranovsky and Sazdanovi\'{c}, that is the $E_{1}$ page of a spectral sequence converging to the homology of the configuration space depending on a graph, with the rational model for the configuration space ... More

Graph cohomologies and rational homotopy type of configuration spacesApr 02 2019May 10 2019We compare the cohomology complex defined by Baranovsky and Sazdanovi\'{c}, that is the $E_{1}$ page of a spectral sequence converging to the homology of the configuration space depending on a graph, with the rational model for the configuration space ... More

The weight complex functor is symmetric monoidalApr 02 2019Bondarko's (strong) weight complex functor is a triangulated functor from Voevodsky's triangulated category of motives to the homotopy category of chain complexes of classical Chow motives. Its construction is valid for any dg enhanced triangulated category ... More

Affine structures on Lie groupoidsApr 02 2019Affine structures on a Lie groupoid, including affine $k$-vector fields, $k$-forms and $(p,q)$-tensors are studied. We show that the space of affine structures is a 2-vector space over the space of multiplicative structures. Moreover, the space of affine ... More

Temporal Landscapes: A Graphical Temporal Logic for ReasoningApr 01 2019We present an elementary introduction to a new logic for reasoning about behaviors that occur over time. This logic is based on temporal type theory. The syntax of the logic is similar to the usual first-order logic; what differs is the notion of truth ... More

Homotopical categories: from model categories to $(\infty,1)$-categoriesApr 01 2019Apr 18 2019This chapter, written for a forthcoming volume on spectra to appear in the MSRI Publications Series with Cambridge University Press, surveys the history of homotopical categories, from Gabriel and Zisman's categories of fractions to Quillen's model categories, ... More

Homotopical categories: from model categories to $(\infty,1)$-categoriesApr 01 2019This chapter, written for a forthcoming volume on spectra to appear in the MSRI Publications Series with Cambridge University Press, surveys the history of homotopical categories, from Gabriel and Zisman's categories of fractions to Quillen's model categories, ... More

Higher symmetries in abstract stable homotopy theoriesApr 01 2019This survey offers an overview of an on-going project on uniform symmetries in abstract stable homotopy theories. This project has calculational, foundational, and representation-theoretic aspects, and key features of this emerging field on abstract representation ... More

Relative Serre functor for comodule algebrasMar 31 2019Let $\mathcal{C}$ be a finite tensor category, and let $\mathcal{M}$ be an exact left $\mathcal{C}$-module category. A relative Serre functor of $\mathcal{M}$, introduced by Fuchs, Schaumann and Schweigert, is an endofunctor $\mathbb{S}$ on $\mathcal{M}$ ... More

Generating linear categories of partitionsMar 30 2019We present an algorithm for approximating linear categories of partitions (of sets). We report on concrete computer experiments based on this algorithm and how we found new examples of compact matrix quantum groups (so called ``non-easy'' quantum groups) ... More

On the "three subobjects lemma" and its higher-order generalisationsMar 30 2019We solve a problem mentioned in an article of Berger and Bourn: we prove that in the context of an algebraically coherent semi-abelian category, two natural definitions of the lower central series coincide. In a first, "standard" approach, nilpotency ... More

Derived categories of one-sided exact categories and their localizationsMar 29 2019The construction of the derived category of an exact category can be extended to one-sided exact categories in a straightforward way. We discuss some properties of the derived category of a (one-sided) exact category and its embedding into its derived ... More

A topological phase transition on the edge of the 2d $\mathbb{Z}_2$ topological orderMar 29 2019The unified mathematical theory of gapped and gapless edges of 2d topological orders was developed by two of the authors. It provides a powerful tool to study pure edge topological phase transitions on the edges of 2d topological orders (without altering ... More

A factorization homology primerMar 26 2019This chapter amalgamates some foundational developments and calculations in factorization homology.

Localizations of one-sided exact categoriesMar 26 2019One-sided exact categories were introduced by S. Bazzoni and S. Crivei by weakening the axioms of Quillen exact categories. In this paper, we consider quotients of one-sided exact categories by percolating subcategories. This generalizes the quotient ... More

A two-category of Hamiltonian manifolds, and a (1+1+1) field theoryMar 26 2019We define an extended field theory in dimensions $1+1+1$, that takes the form of a `quasi 2-functor' with values in a strict 2-category $\widehat{\mathcal{H}am}$, defined as the `completion of a partial 2-category' $\mathcal{H}am$, notions which we define. ... More

Functorial invariants of trees and their conesMar 25 2019We study the category whose objects are trees (with or without roots) and whose morphisms are contractions. We show that the corresponding contravariant module categories are Noetherian, and we study two natural families of modules over these categories. ... More

Universal AF-algebrasMar 25 2019We study the approximately finite-dimensional (AF) $C^*$-algebras that appear as inductive limits of sequences of finite-dimensional $C^*$-algebras and left-invertible embeddings. We show that there is such a separable AF-algebra $\mathcal A_\mathfrak{F}$ ... More

What is the spectral category?Mar 24 2019For a category $\mathcal{C}$ with finite limits and a class $\mathcal{S}$ of monomorphisms in $\mathcal{C}$ that is pullback stable, contains all isomorphisms, is closed under composition, and has the strong left cancellation property, we use pullback ... More

What is the spectral category?Mar 24 2019Apr 19 2019For a category $\mathcal{C}$ with finite limits and a class $\mathcal{S}$ of monomorphisms in $\mathcal{C}$ that is pullback stable, contains all isomorphisms, is closed under composition, and has the strong left cancellation property, we use pullback ... More

Monochromatic homotopy theory is asymptotically algebraicMar 24 2019In previous work, we used an $\infty$-categorical version of ultraproducts to show that, for a fixed height $n$, the symmetric monoidal $\infty$-categories of $E_{n,p}$-local spectra are asymptotically algebraic in the prime $p$. In this paper, we prove ... More

Permutads via operadic categories, and the hidden associahedronMar 21 2019The present article exploits the fact that permutads (aka shuffle algebras) are algebras over a terminal operad in a certain operadic category Per. In the first, classical part we formulate and prove a claim envisaged by Loday and Ronco that the cellular ... More

Multi-adjoint concept lattices via quantaloid-enriched categoriesMar 21 2019With quantaloids carefully constructed from multi-adjoint frames, it is shown that multi-adjoint concept lattices, multi-adjoint property-oriented concept lattices and multi-adjoint object-oriented concept lattices are derivable from Isbell adjunctions, ... More

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 22 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