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Maurer-Cartan moduli and theorems of Riemann-Hilbert typeFeb 07 2018We study Maurer-Cartan moduli spaces of dg algebras and associated dg categories and show that, while not quasi-isomorphism invariants, they are invariants of strong homotopy type, a natural notion that has not been studied before. We prove, in several ... More
$\infty$-topoi and Natural Phenomena: GenerationFeb 07 2018We show that the Segal topos of derived stacks over simplicial commutative $k$-algebras, which can be used to model natural phenomena, has a subobject classifier, something we regard as being a source from which dynamics is generated. This is done by ... More
Analogues of centralizer subalgebras for fiat 2-categories and their 2-representationsFeb 06 2018The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive $2$-representations with a fixed apex $\mathcal{J}$ of a fiat $2$-category $\cC$ and the set of equivalence classes of faithful simple transitive ... More
Crossed extensions and equivalences of topological 2-groupoidsFeb 06 2018We provide concrete models for generalized morphisms and Morita equivalences of topological 2-groupoids by introducing the notions of crossings and crossed extensions of groupoid crossed modules. A systematic study of these objects is elaborated and an ... More
Reconstruction of tensor categories from their structure invariantsFeb 03 2018In this paper, we study tensor (or monoidal) categories of finite rank over an algebraically closed field $\mathbb F$. Given a tensor category $\mathcal{C}$, we have two structure invariants of $\mathcal{C}$: the Green ring (or the representation ring) ... More
Incidence bicomodules, Möbius inversion, and a Rota formula for infinity adjunctionsJan 23 2018In the same way decomposition spaces, also known as unital 2-Segal spaces, have incidence (co)algebras, and certain relative decomposition spaces have incidence (co)modules, we identify the structures that have incidence bi(co)modules: they are certain ... More
Smooth Version of Johnson's Problem Concerning Derivations of Group AlgebrasJan 10 2018A description of the algebra of outer derivations of a group algebra of a finitely presented discrete group is given in terms of the Cayley complex of the groupoid of the adjoint action of the group. This task is a smooth version of Johnson's problem ... More
Modified trace is a symmetrised integralDec 31 2017A modified trace for a finite k-linear pivotal category is a family of linear forms on endomorphism spaces of projective objects which has cyclicity and so-called partial trace properties. We show that a non-degenerate modified trace defines a compatible ... More
A combinatorial model for the free loop fibrationDec 06 2017We introduce the abstract notion of a closed necklical set in order to describe a functorial combinatorial model of the free loop fibration $\Omega Y\rightarrow \Lambda Y\rightarrow Y$ over the geometric realization $Y=|X|$ of a path connected simplicial ... More
Grothendieck's homotopy theory, polynomial monads and delooping of spaces of long knotsDec 04 2017We extend some classical results - such as Quillen's Theorem A, the Grothendieck construction, and the characterisation of homotopically cofinal functors - from the homotopy theory of small categories to polynomial monads and their algebras. As an application ... More
Braiding for categorical algebras and crossed modules of algebras I: Associative and Lie algebrasNov 23 2017In this paper we study the categories of braided categorical associative algebras and braided crossed modules of associative algebras and we relate these structures with the categories of braided categorical Lie algebras and braided crossed modules of ... More
Non-unital polygraphs form a presheaf categoriesNov 02 2017We prove, as claimed by A.Carboni and P.T.Johnstone, that the category of non-unital polygraphs, i.e. polygraphs where the source and target of each generator are not identity arrows, is a presheaf category. More generally we develop a new criterion for ... More
When is the heart of a t-structure a Grothendieck category?Aug 24 2017Let $\mathcal D$ be a triangulated category endowed with a $t$-structure $\mathfrak t=(\mathcal U,\Sigma \mathcal V)$ and denote by $\mathcal H:=\mathcal U\cap \Sigma\mathcal V$ its heart. In this paper we study the following well-known problem: Under ... More
Derivations of Group AlgebrasAug 16 2017In the paper, a method of describing the outer derivations of the group algebra of a finitely presentable group is given. The description of derivations is given in terms of characters of the groupoid of the adjoint action of the group.
Quasi-coherent sheaves in differential geometryJul 04 2017Jul 28 2017It is proved that the category of simplicial complete bornological spaces over $\mathbb R$ carries a combinatorial monoidal model structure satisfying the monoid axiom. For any commutative monoid in this category the category of modules is also a monoidal ... More
The localic Istropy group of a toposJun 15 2017It has been shown by J.Funk, P.Hofstra and B.Steinberg that any Grothendieck topos T is endowed with a canonical group object, called its isotropy group, which acts functorially on every object of T. We show that this group is in fact the group of points ... More
Frobenius structures over Hilbert C*-modulesApr 19 2017Jun 05 2017We study the monoidal dagger category of Hilbert C*-modules over a commutative C*-algebra from the perspective of categorical quantum mechanics. The dual objects are the finitely presented projective Hilbert C*-modules. Special dagger Frobenius structures ... More
Some remarks on protolocalizations and protoadditive reflectionsFeb 28 2017Oct 29 2017We investigate additional properties of protolocalizations, introduced and studied by F. Borceux, M. M. Clementino, M. Gran, and L. Sousa, and of protoadditive reflections, introduced and studied by T. Everaert and M. Gran. Among other things we show ... More
Three natural subgroups of the Brauer-Picard group of a Hopf algebra with applicationsFeb 16 2017In this article we construct three explicit natural subgroups of the Brauer-Picard group of the category of representations of a finite-dimensional Hopf algebra. In examples the Brauer Picard group decomposes into an ordered product of these subgroups, ... More
Lax orthogonal factorisations in ordered structuresFeb 08 2017We give an account of lax orthogonal factorisation systems on order-enriched categories. Among them, we define and characterise the KZ-reflective ones, in a way that mirrors the characterisation of reflective orthogonal factorisation systems. We use simple ... More
Lax orthogonal factorisations in monad-quantale-enriched categoriesJan 19 2017Sep 26 2017We show that, for a quantale $V$ and a $\mathsf{Set}$-monad $\mathbb{T}$ laxly extended to $V$-$\mathsf{Rel}$, the presheaf monad on the category of $(\mathbb{T},V)$-categories is simple, giving rise to a lax orthogonal factorisation system (lofs) whose ... More
Symmetry of the Definition of Degeneration in Triangulated CategoriesDec 17 2016Dec 22 2016Module structures of an algebra on a fixed finite dimensional vector space form an algebraic variety. Isomorphism classes correspond to orbits of the action of an algebraic group on this variety and a module is a degeneration of another if it belongs ... More
Cubicial rigidification, the cobar construction, and the based loop spaceDec 14 2016Dec 15 2016We prove the following generalization of a classical result of Adams: for any pointed and connected topological space $(X,b)$, that is not necessarily simply connected, the cobar construction of the differential graded (dg) coalgebra of normalized singular ... More
A Generalization of the Curry-Howard CorrespondenceDec 08 2016We present a variant of the calculus of deductive systems developed in (Lambek 1972, 1974), and give a generalization of the Curry-Howard-Lambek theorem giving an equivalence between the category of typed lambda-calculi and the category of cartesian closed ... More
String Diagrams For Double Categories and EquipmentsDec 08 2016In this paper, we extend the string diagrams commonly used to work with monoidal categories and bicategories to double categories and proarrow equipments. We extend the proofs of Joyal and Street to show that any valid topological deformation of these ... More
A category of hybrid systemsDec 06 2016We propose a definition of the category of hybrid systems in which executions are special types of morphisms. Consequently morphisms of hybrid systems send executions to executions. We plan to use this result to define and study networks of hybrid systems. ... More
Fluxes, bundle gerbes and 2-Hilbert spacesDec 06 2016We elaborate on the construction of a prequantum 2-Hilbert space from a bundle gerbe over a 2-plectic manifold, providing the first steps in a program of higher geometric quantisation of closed strings in flux compactifications and of M5-branes in C-fields. ... More
Derived Recollements and Generalised AR FormulasDec 06 2016The Defect Recollement, Evaluation Recollement, Restriction Recollement, and Auslander-Gruson-Jensen-Recollement are shown to be instances of a general construction using derived functors and methods from stable module theory. The right derived functors ... More
Derived Recollements and Generalised AR FormulasDec 06 2016Dec 07 2016The Defect Recollement, Evaluation Recollement, Restriction Recollement, and Auslander-Gruson-Jensen-Recollement are shown to be instances of a general construction using derived functors and methods from stable module theory. The right derived functors ... More
Globular: an online proof assistant for higher-dimensional rewritingDec 04 2016This article introduces Globular, an online proof assistant for the formalization and verification of proofs in higher-dimensional category theory. The tool produces graphical visualizations of higher-dimensional proofs, assists in their construction ... More
Mass growth of objects and categorical entropyDec 03 2016In the pioneer work by Dimitrov-Haiden-Katzarkov-Kontsevich, they introduced various categorical analogies from classical theory of dynamical systems. In particular, they defined the entropy of an endofunctor on a triangulated category with a split generator. ... More
Cospan construction of the graph category of Borisov and ManinNov 30 2016It is shown how the graph category of Borisov and Manin can be constructed from (a variant of) the graph category of Joyal and Kock, essentially by reversing the generic morphisms. More precisely, the morphisms in the Borisov-Manin category are exhibited ... More
Guarded Cubical Type TheoryNov 28 2016This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, ... More
Relative $2$-Segal spacesNov 28 2016We introduce a relative version of the $2$-Segal simplicial spaces defined by Dyckerhoff and Kapranov and G\'alvez-Carrillo, Kock and Tonks. Examples of relative $2$-Segal spaces include the categorified unoriented cyclic nerve, real pseudo-holomorphic ... More
On some categorical-algebraic conditions in S-protomodular categoriesNov 28 2016In the context of semi-abelian categories, several additional conditions have been considered in order to obtain a closer group-like behavior. Among them are locally algebraic cartesian closedness and algebraic coherence. The recent notion of S-protomodular ... More
A geometric approach to Hall algebras I: Higher AssociativityNov 28 2016We construct a geometric system from which the Hall algebra can be recovered. This system inherently satisfies higher associativity conditions and thus leads to a categorification of the Hall algebra. We then suggest how to use this approach to construct ... More
Hall monoidal categories and categorical modulesNov 24 2016We construct so called Hall monoidal categories (and Hall modules thereover) and exhibit them as a categorification of classical Hall and Hecke algebras (and certain modules thereover). The input of the (functorial!) construction are simplicial groupoids ... More
Torsion pairs in silting theoryNov 24 2016In the setting of compactly generated triangulated categories, we show that the heart of a (co)silting t-structure is a Grothendieck category if and only if the (co)silting object satisfies a purity assumption. Moreover, in the cosilting case the previous ... More
Spans of cospansNov 23 2016We introduce the notion of a span of cospans and define, for them, horizonal and vertical composition. These compositions satisfy the interchange law if working in a topos $\mathbf{C}$ and if the span legs are monic. A bicategory is then constructed from ... More
Categories in control: applied PROPsNov 23 2016Control theory uses `signal-flow diagrams' to describe processes where real-valued functions of time are added, multiplied by scalars, differentiated and integrated, duplicated and deleted. These diagrams can be seen as string diagrams for the PROP FinRel_k, ... More
A topos associated with a colored categoryNov 22 2016We show that a functor category whose domain is a colored category is a topos.The topos structure enables us to introduce cohomology of colored categories including quasi-schemoids. If the given colored category arises from an association scheme, then ... More
A topos associated with a colored categoryNov 22 2016Nov 23 2016We show that a functor category whose domain is a colored category is a topos.The topos structure enables us to introduce cohomology of colored categories including quasi-schemoids. If the given colored category arises from an association scheme, then ... More
Hopf polyads, Hopf categories and Hopf group monoids viewed as Hopf monadsNov 16 2016We associate, in a functorial way, a monoidal bicategory $\mathsf{Span}| \mathcal V$ to any monoidal bicategory $\mathcal V$. Two examples of this construction are of particular interest: Hopf polyads (due to Brugui\`eres) can be seen as Hopf monads in ... More
Operator algebras in rigid C*-tensor categoriesNov 14 2016In this article, we define operator algebras internal to a rigid C*-tensor category $\mathcal{C}$. A C*/W*-algebra object in $\mathcal{C}$ is an algebra object $\mathbf{A}$ in $\operatorname{ind}$-$\mathcal{C}$ whose category of free modules ${\sf FreeMod}_{\mathcal{C}}(\mathbf{A})$ ... More
Operator algebras in rigid C*-tensor categoriesNov 14 2016Dec 01 2016In this article, we define operator algebras internal to a rigid C*-tensor category $\mathcal{C}$. A C*/W*-algebra object in $\mathcal{C}$ is an algebra object $\mathbf{A}$ in $\operatorname{ind}$-$\mathcal{C}$ whose category of free modules ${\sf FreeMod}_{\mathcal{C}}(\mathbf{A})$ ... More
Linear Logic Properly DisplayedNov 13 2016We introduce proper display calculi for intuitionistic, bi-intuitionistic and classical linear logics with exponentials, which are sound, complete, conservative, and enjoy cut-elimination and subformula property. Based on the same design, we introduce ... More
The Hopf monoid of coloring problemsNov 13 2016We study coloring problems, which are induced subposets P of a Boolean lattice, paired with an order ideal I from the poset of intervals, ordered by inclusion. We study a quasisymmetric function associated to coloring problems, called the chromatic quasisymmetric ... More
Subgroups of Quantum GroupsNov 12 2016We investigate the notion of a subgroup of a quantum group. We suggest a general definition, which takes into account the work that has been done for quantum homogeneous spaces. We further restrict our attention to reductive subgroups, where some faithful ... More
2-Calabi-Yau categories with a directed cluster-tilting subcategoryNov 11 2016As a generalization of acyclic 2-Calabi-Yau categories, we consider 2-Calabi-Yau categories with a directed cluster-tilting subcategory; we study their cluster-tilting subcategories and the cluster combinatorics that they encode. We show that such categories ... More
Higher cyclic operadsNov 08 2016We introduce a convenient definition for higher cyclic operads, which is based on unrooted trees and Segal conditions. More specifically, we introduce a category $\Xi$ of trees, which carries a tight relationship to the Moerdijk-Weiss category of rooted ... More
Unified Functorial Signal Representation II: Category action, Base Hierarchy, Base structured GeometryNov 08 2016In this paper we study the base structured category $\mathcal{X} \rtimes_{F} \mathbf{C}$, developing the perspective of a functor as a multi-object category action. Using an elementary example of a permutation action on a finite set, we introduce the ... More
Topoi of parametrized objectsNov 07 2016We give necessary and sufficient conditions on a presentable infinity-category C so that families of objects of C form an infinity-topos. In particular, we prove a conjecture of Joyal that this is the case whenever C is stable.
Prime Ideals and Topos Points of MonoidsNov 07 2016Let $M$ be a commutative monoid such that $M/M^*$ is finitely generated, where $M^*$ is the group of invertible elements of $M$. We will show that for such monoids, the points of the topos of $M$-sets are in a one-to-one relation with the prime ideals ... More
On the graded dual numbers, arcs, and non-crossing partitions of the integersNov 07 2016We give a combinatorial model for the bounded derived category of graded modules over the dual numbers in terms of arcs on the integer line with a point at infinity. Using this model we describe the lattice of thick subcategories of the bounded derived ... More
A generalization of finite-dimensional Gorenstein algebrasNov 05 2016We introduce and study abelian categories equipped with a comonad and a Nakayama functor relative to the comonad. These categories generalize important features of the module category of a finite-dimensional algebra from the viewpoint of Gorenstein homological ... More
Some invariance properties of cyclic cohomology with coefficientsNov 04 2016In this paper, we further explore the conceptual approach to cyclic cohomology with coefficients. In particular we give a derived version of the definition with better invariance properties. We show that the new definition agrees with the old under certain ... More
On torsion pairs, (well generated) weight structures, adjacent $t$-structures, and related (co)homological functorsNov 02 2016The paper contains a rich collection of results related to weight structures and (more generally) to torsion pairs. For any weight structure $w$ we study (co)homological pure functors; these "ignore all weights except weight zero" and have already found ... More
On torsion pairs, (well generated) weight structures, adjacent $t$-structures, and related (co)homological functorsNov 02 2016Dec 05 2016The paper contains a collection of results related to weight structures, $t$-structures, and (more generally) to torsion pairs. For any weight structure $w$ we study (co)homological pure functors; these "ignore all weights except weight zero" and have ... More
Inverting operations in operadsNov 02 2016We construct a localization for operads with respect to one-ary operations based on the Dwyer-Kan hammock localization. For an operad O and a sub-monoid of one-ary operations W we associate an operad LO and a canonical map O to LO which takes elements ... More
Relational PK-Nets for Transformational Music AnalysisNov 02 2016In the field of transformational music theory, which emphasizes the possible transformations between musical objects, Klumpenhouwer networks (K-Nets) constitute a useful framework with connections in both group theory and graph theory. Recent attempts ... More
Symmetry and Complete Regularity: Kopperman's duality {\it à la quantale}Nov 02 2016Nearly three decades from his celebrated result, we study a modern refinement and strengthening of Kopperman's full metrisabilty of all topological spaces. Within this new theory of \emph{V-spaces}, developed by Flagg and Weiss, we investigate several ... More
Phantom Ideals and Cotorsion Pairs in Extriangulated CategoriesNov 02 2016In this paper, we introduce and study relative phantom morphisms in extriangulated categories defined by Nakaoka and Palu. Then using their properties, we show that if $(\C,\E,\s)$ is an extriangulated category with enough injective objects and projective ... More
Classification of Roberts actions of strongly amenable C^* tensor categories on the injective factor of type III_1Nov 02 2016In this paper, we generalize Izumi's result on uniqueness of realization of finite C$^*$-tensor categories in the endomorphism category of the injective factor of type II_1 for finitely generated strongly amenable C$^*$-tensor categories by applying Popa's ... More
Persistence Diagrams as Diagrams: A Categorification of the Stability TheoremOct 31 2016Persistent homology, a central tool of topological data analysis, provides invariants of data called barcodes (also known as persistence diagrams). A barcode is simply a multiset of real intervals. Recent work of Edelsbrunner, Jablonski, and Mrozek suggests ... More
A Grothendieck-Witt space for stable infinity categories with dualityOct 31 2016We construct a Grothendieck-Witt space for any stable infinity category with duality. If we apply our construction to perfect complexes over a commutative ring in which 2 is invertible we recover the classical Grothendieck-Witt space. Our Grothendieck-Witt ... More
Infinity categories with duality and hermitian multiplicative infinite loop space machinesOct 31 2016We show that any symmetric monoidal infinity category with duality gives rise to a direct sum hermitian K-theory spectrum. This assignment is lax symmetric monoidal, thereby producing E-infinity ring spectra from bimonoidal infinity categories with duality. ... More
Quasi-Coxeter categories and quantum groupsOct 31 2016Nov 02 2016We define the notion of braided quasi-Coxeter category, which is informally a tensor category carrying commuting actions of a generalised braid group B_W and Artin's braid groups B_n on the tensor powers of its objects. The data which defines the action ... More
Derived Categories -- a TextbookOct 30 2016This is the first part of a book on derived categories. The purpose of the book is to provide solid foundations for the theory of derived categories, and to present several applications of this theory to algebra and algebraic geometry. The emphasis is ... More
A classification of nullity classes in abelian categoriesOct 29 2016Nov 01 2016We give a classification of nullity classes (or torsion classes) in an abelian category by forming a spectrum of equivalence classes of premonoform objects. This is parallel to Kanda's classification of Serre subcategories.
A classification of nullity classes in abelian categoriesOct 29 2016Nov 22 2016We give a classification of nullity classes (or torsion classes) in an abelian category by forming a spectrum of equivalence classes of premonoform objects. This is parallel to Kanda's classification of Serre subcategories.
A classification of nullity classes in the derived category of a ringOct 29 2016For a commutative Noetherian ring $R$ with finite Krull dimension, we study the nullity classes in $D^c_{fg}(R)$, the full triangulated subcategory $D^c_{fg}(R)$ of the derived category $D(R)$ consisting of objects which can be represented by cofibrant ... More
On the notion of flat 2-functorsOct 28 2016In this paper we develop the 2-dimensional theory of flat functors. We define a $\mathcal{C}at$-valued 2-functor $P$ to be flat when its left bi-Kan extension $P^*$ along the Yoneda 2-functor $h$ is left exact. By left bi-Kan extension we understand the ... More
Connections in Tangent CategoriesOct 27 2016Connections are an important tool of differential geometry. This paper investigates their definition and structure in the abstract setting of tangent categories. At this level of abstraction we derive several classically important results about connections, ... More
The compactness locus of a geometric functor and the formal construction of the Adams isomorphismOct 26 2016We introduce the compactness locus of a geometric functor between rigidly-compactly generated tensor-triangulated categories, and describe it for several examples arising in equivariant homotopy theory and algebraic geometry. It is a subset of the tensor-triangular ... More
A note on triangulated monads and categories of module spectraOct 26 2016Consider a monad on an idempotent complete triangulated category with the property that its Eilenberg-Moore category of modules is also triangulated, in the unique compatible way. We show that any other triangulated adjunction realizing this monad is ... More
Alexander-Beck modules detect the unknotOct 26 2016We introduce the Alexander-Beck module of a knot as a canonical refinement of the classical Alexander module, and we prove that this new invariant is an unknot-detector.
From weak cofibration categories to model categoriesOct 25 2016In [BaSc2] the authors introduced a much weaker homotopical structure than a model category, called a "weak cofibration category". We further showed that a small weak cofibration category induces in a natural way a model category structure on its ind-category, ... More
Bootstrapping structural properties, via accessible imagesOct 25 2016We present several new model-theoretic applications of the fact that, under a mild large cardinal assumption, the powerful image of any accessible functor is accessible. In particular, we generalize to the context of accessible categories the results ... More
Bootstrapping structural properties, via accessible imagesOct 25 2016Nov 01 2016We present several new model-theoretic applications of the fact that, under a mild large cardinal assumption, the powerful image of any accessible functor is accessible. In particular, we generalize to the context of accessible categories the results ... More
Categorical ComplexityOct 25 2016We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several examples of ... More
Higher Categories and Topological Quantum Field TheoriesOct 24 2016We construct a Turaev-Viro type invariant of smooth closed oriented $4$-manifolds out of a $G$-crossed braided spherical fusion category ($G$-BSFC) for $G$ a finite group. The construction can be extended to obtain a $(3+1)$-dimensional topological quantum ... More
Classifying exact categories via Wakamatsu tiltingOct 24 2016Using the Morita-type embedding, we show that any exact category with enough projectives has a realization as a (pre)resolving subcategory of a module category. When the exact category has enough injectives, the image of the embedding can be described ... More
Classifying exact categories via Wakamatsu tiltingOct 24 2016Dec 01 2016Using the Morita-type embedding, we show that any exact category with enough projectives has a realization as a (pre)resolving subcategory of a module category. When the exact category has enough injectives, the image of the embedding can be described ... More
Modelling and Computing Homotopy Types: IOct 24 2016The aim of this article is to explain a philosophy for applying higher dimensional Seifert-van Kampen Theorems, and how the use of groupoids and strict higher groupoids resolves some foundational anomalies in algebraic topology at the border between homology ... More
Modelling and Computing Homotopy Types: IOct 24 2016Nov 01 2016The aim of this article is to explain a philosophy for applying higher dimensional Seifert-van Kampen Theorems, and how the use of groupoids and strict higher groupoids resolves some foundational anomalies in algebraic topology at the border between homology ... More
Wreaths, mixed wreaths and twisted coactionsOct 24 2016Distributive laws between two monads in a 2-category $\CK$, as defined by Jon Beck in the case $\CK=\mathrm{Cat}$, were pointed out by the author to be monads in a 2-category $\mathrm{Mnd}\CK$ of monads. Steve Lack and the author defined wreaths to be ... More
Cocompletion of restriction categoriesOct 23 2016Restriction categories were introduced as a way of generalising the notion of partial map categories. In this paper, we define cocomplete restriction category, and give the free cocompletion of a small restriction category as a suitably defined category ... More
Categorifying rationalizationOct 23 2016We solve a problem proposed by Khovanov by constructing, for any set of primes $S$, a triangulated category (in fact a stable $\infty$-category) whose Grothendieck group is $S^{-1}\mathbf{Z}$. More generally, for any exact $\infty$-category $E$, we construct ... More
Data structures for quasistrict higher categoriesOct 21 2016We present new data structures for quasistrict higher categories, in which associativity and unit laws hold strictly. Our approach has low axiomatic complexity compared to traditional algebraic definitions of higher categories, and we use it to give a ... More
Complicial sets, an overtureOct 21 2016The aim of these notes is to introduce the intuition motivating the notion of a "complicial set", a simplicial set with certain marked "thin" simplices that witness a composition relation between the simplices on their boundary. By varying the marking ... More
tCG Torsion PairsOct 20 2016We investigate conditions for when the $t$-structure of Happel-Reiten-Smal\o \ associated to a torsion pair is a compactly generated $t$-structure. The concept of a {$t$CG} torsion pair is introduced and for any ring $R$, we prove that $\mathbf{t}=(\mathcal{T},\mathcal{F})$ ... More
$μ$-Bicomplete Categories and Parity GamesOct 20 2016For an arbitrary category, we consider the least class of functors con- taining the projections and closed under finite products, finite coproducts, parameterized initial algebras and parameterized final coalgebras, i.e. the class of functors that are ... More
Continuous metric spaces and injective approach spacesOct 20 2016The notion of Scott distance between points and subsets in a metric space, a $[0,\infty]$-enriched version of Scott topology on ordered sets, is introduced, making a metric space into an approach space in the sense of R. Lowen. It is proved that the specialization ... More
Operadic categories and their skew monoidal categories of collectionsOct 20 2016I describe a generalization of the notion of operadic category due to Batanin and Markl. For each such operadic category I describe a skew monoidal category of collections, such that a monoid in this skew monoidal category is precisely an operad over ... More
Unified Functorial Signal Representation I: From Grothendieck Fibration to general Base structured CategoriesOct 19 2016The Grothendieck construction in category theory is the categorical generalization of the usual semidirect product which produces a fibred category from a given (contravariant) pseudo-functor into the category of small categories $\mathbf{Cat}$. In this ... More
The smooth Hom-stack of an orbifoldOct 19 2016For a compact manifold M and a differentiable stack \cX presented by a Lie groupoid X, we show the Hom-stack Hom(M,\cX) is presented by a Fr\'echet--Lie groupoid Map(M,X) and so is an infinite-dimensional differentiable stack. We further show that if ... More
Cubical sets and the topological toposOct 17 2016Coquand's cubical set model for homotopy type theory provides the basis for a computational interpretation of the univalence axiom and some higher inductive types, as implemented in the cubical proof assistant. This paper contributes to the understanding ... More
Computing the Minimal Model for the Quantum Symmetric AlgebraOct 17 2016We prove that the quantum symmetric algebra is morita equivalent to the horizontal strip category defined by Sam and Snowden in arXiv:1206.2233
Globularily generated double categories II: The free globularily generated double category constructionOct 17 2016We introduce the free globularily generated double category construction. We aim to establish an analogy between the way double categories and globularily generated double categories relate to bicategories. We introduce the notion of decorated bicategory ... More
Cluster subalgebras and cotorsion pairs in Frobenius extriangulated categoriesOct 17 2016Nakaoka and Palu introduced the notion of extriangulated categories by extracting the similarities between exact categories and triangulated categories. In this paper, we study cotorsion pairs in a Frobenius extriangulated category $\C$. Especially, for ... More