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Via the adjunction - ∗ 1 ⊣ V (1 , -) : Span (V) → V - Mat and a cartesian monad T on an extensive category V with finite limits, we construct an adjunction - ∗ 1 ⊣ V (1 , -) : Cat (T , V) → (T ¯ , V) - Cat between categories of generalized enriched multicategories and generalized internal multicategories, provided the monad T satisfies a suitable property, which holds for several examples. We verify, moreover, that the left adjoint is fully faithful, and preserves pullbacks, provided that the copower functor - ∗ 1 : Set → V is fully faithful. We also apply this result to study descent theory of generalized enriched multicategorical structures. These results are built upon the study of base-change for generalized multicategories, which, in turn, was carried out in the context of categories of horizontal lax algebras arising out of a monad in a suitable 2-category of pseudodouble categories. [ABSTRACT FROM AUTHOR]

Copyright of Cahiers de Topologie et Geometrie Differentielle Categoriques is the property of Andree C. EHRESMANN and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The theory of presentations of enriched monads was developed by Kelly, Power, and Lack, following classic work of Lawvere, and has been generalized to apply to subcategories of arities in recent work of Bourke–Garner and the authors. We argue that, while theoretically elegant and structurally fundamental, such presentations of enriched monads can be inconvenient to construct directly in practice, as they do not directly match the definitional procedures used in constructing many categories of enriched algebraic structures via operations and equations. Retaining the above approach to presentations as a key technical underpinning, we establish a flexible formalism for directly describing enriched algebraic structure borne by an object of a V -category C in terms of parametrized J -ary operations and diagrammatic equations for a suitable subcategory of arities J ↪ C . On this basis we introduce the notions of diagrammatic J -presentation and J -ary variety, and we show that the category of J -ary varieties is dually equivalent to the category of J -ary V -monads. We establish several examples of diagrammatic J -presentations and J -ary varieties relevant in both mathematics and theoretical computer science, and we define the sum and tensor product of diagrammatic J -presentations. We show that both J -relative monads and J -pretheories give rise to diagrammatic J -presentations that directly describe their algebras. Using diagrammatic J -presentations as a method of proof, we generalize the pretheories-monads adjunction of Bourke and Garner beyond the locally presentable setting. Lastly, we generalize Birkhoff’s Galois connection between classes of algebras and sets of equations to the above setting. [ABSTRACT FROM AUTHOR]

We study Serre structures on categories enriched in pivotal monoidal categories, and apply this to study Serre structures on two types of graded k-linear categories: categories with group actions and categories with graded hom spaces. We check that Serre structures are preserved by taking orbit categories and skew group categories, and describe the relationship with graded Frobenius algebras. Using a formal version of Auslander-Reiten translations, we show that the derived category of a d-representation finite algebra is fractionally Calabi-Yau if and only if its preprojective algebra has a graded Nakayama automorphism of finite order. This connects various results in the literature and gives new examples of fractional Calabi-Yau algebras. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the completeness of L-quasi-uniform convergence spaces and L-quasi-uniform spaces. Firstly, we describe L-quasi-uniform convergence spaces as enriched categories. Then we give two kinds of completeness of L-quasi-uniform convergence spaces and show that Lawvere completeness implies Cauchy completeness. Finally, we use the Cauchy completeness of L-quasi-uniform convergence spaces to define the Cauchy completeness of L-quasiuniform spaces, and show that Cauchy completeness is equivalent to Lawvere completeness in L-quasi-uniform spaces. [ABSTRACT FROM AUTHOR]

We characterize injective objects, injective hulls and essential embeddings in the category of symmetric categories enriched in a small, integral and involutive quantaloid. In particular, injective partial metric spaces are precisely formulated. [ABSTRACT FROM AUTHOR]

We study ideals in, and continuity of, quantaloid-enriched categories (Qcategories for short) as a 'many-valued and many-typed' generalization of domain theory. Abstractly, for any (saturated) class Φ of presheaves, we define and study the Φ-continuity of Q-categories. Concretely, we compute three examples of such saturated classes of presheaves - the class of flat ideals, the class of irreducible ideals and the class of conical ideals - which are proper generalizations of ideals in domain theory. [ABSTRACT FROM AUTHOR]

We show that induced representations for a pair of diffeological Lie groups exist, in the form of an indexed colimit in the category of diffeological spaces. [ABSTRACT FROM AUTHOR]

Copyright of Cahiers de Topologie et Geometrie Differentielle Categoriques is the property of Andree C. EHRESMANN and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Commuting pairs of algebraic structures on a set have been studied by several authors and may be described equivalently as algebras for the tensor product of Lawvere theories, or more basically as certain bifunctors that here we call bifold algebras. The much less studied notion of commutant for Lawvere theories was first introduced by Wraith and generalizes the notion of centralizer clone in universal algebra. Working in the general setting of enriched algebraic theories for a system of arities, we study the interaction of the concepts of bifold algebra and commutant. We show that the notion of commutant arises via a universal construction in a two-sided fibration of bifold algebras over various theories. On this basis, we study special classes of bifold algebras that are related to commutants, introducing the notions of commutant bifold algebra and balanced bifold algebra. We establish several adjunctions and equivalences among these categories of bifold algebras and related categories of algebras over various theories, including commutative, contracommutative, saturated, and balanced algebras. We also survey and develop examples of commutant bifold algebras, including examples that employ Pontryagin duality and a theorem of Ehrenfeucht and Łoś on reflexive abelian groups. Along the way, we develop a functorial treatment of fundamental aspects of bifold algebras and commutants, including tensor products of theories and the equivalence of bifold algebras and commuting pairs of algebras. Because we work relative to a (possibly large) system of arities in a closed category V , our main results are applicable to arbitrary V -monads on a finitely complete V , the enriched theories of Borceux and Day, the enriched Lawvere theories of Power relative to a regular cardinal, and other notions of algebraic theory. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of Grothendieck enriched categories for categories enriched over a sufficiently nice Grothendieck monoidal category V , generalizing the classical notion of Grothendieck categories. Then we establish the Gabriel-Popescu type theorem for Grothendieck enriched categories. We also prove that the property of being Grothendieck enriched categories is preserved under the change of the base monoidal categories by a monoidal right adjoint functor. In particular, if we take as V the monoidal category of complexes of abelian groups, we obtain the notion of Grothendieck dg categories. As an application of the main results, we see that the dg category of complexes of quasi-coherent sheaves on a quasi-compact and quasi-separated scheme is an example of Grothendieck dg categories. [ABSTRACT FROM AUTHOR]

For a finite-index II 1 subfactor N ⊂ M , we prove the existence of a universal Hopf ∗-algebra (or, a discrete quantum group in the analytic language) acting on M in a trace-preserving fashion and fixing N pointwise. We call this Hopf ∗-algebra the quantum Galois group for the subfactor and compute it in some examples of interest, notably for arbitrary irreducible finite-index depth-two subfactors. Along the way, we prove the existence of universal acting Hopf algebras for more general structures (tensors in enriched categories), in the spirit of recent work by Agore, Gordienko and Vercruysse. [ABSTRACT FROM AUTHOR]

We develop a general framework for studying signatures, presentations, and algebraic colimits of enriched monads for a subcategory of arities, even when the base of enrichment V is not locally presentable. When V satisfies the weaker requirement of local boundedness, the resulting framework is sufficiently general to apply to the Φ-accessible monads of Lack and Rosický and the J-ary monads of the first author, while even without local boundedness our framework captures in full generality the presentations of strongly finitary monads of Lack and Kelly as well as Wolff's presentations of V-categories by generators and relations. Given any small subcategory of arities j : J → C in an enriched category C, satisfying certain assumptions, we prove results on the existence of free J-ary monads, the monadicity of J-ary monads over J-signatures, and the existence of algebraic colimits of J-ary monads. We study a notion of presentation for J-ary monads and show that every such presentation presents a J-ary monad. Certain of our results generalize earlier results of Kelly, Power, and Lack for finitary enriched monads in the locally finitely presentable setting, as well as analogous results of Kelly and Lack for strongly finitary monads on cartesian closed categories. Our main results hold for a wide class of subcategories of arities in locally bounded enriched categories. [ABSTRACT FROM AUTHOR]

We introduce the notion of M-locally generated category for a factorization system (E, M) and study its properties. We offer a Gabriel-Ulmer duality for these categories, introducing the notion of nest. We develop this theory also from an enriched point of view. We apply this technology to Banach spaces showing that it is equivalent to the category of models of the nest of finite-dimensional Banach spaces. [ABSTRACT FROM AUTHOR]

The familiar adjunction between ordered sets and completely distributive lattices can be extended to generalised metric spaces, that is, categories enriched over a quantale (a lattice of "truth values"), via an appropriate distributive law between the "down-set" monad and the "up-set" monad on the category of quantale-enriched categories. If the underlying lattice of the quantale is completely distributive, and if powers distribute over non-empty joins in the quantale, then this distributive law can be concretely formulated in terms of operations, equations and choice functions, similar to the familiar distributive law of lattices. [ABSTRACT FROM AUTHOR]

We study effective descent V -functors for cartesian monoidal categories V with finite limits. This study is carried out via the properties enjoyed by the 2-functor V ↦ Fam (V) , results about effective descent of bilimits of categories, and the fact that the enrichment 2-functor preserves certain bilimits. Since these results rely on an understanding of (effective) descent morphisms in Fam (V) , we carefully study these morphisms in free coproduct completions. Finally, we provide refined conditions when V is a regular category. [ABSTRACT FROM AUTHOR]

We develop universal algebra over an enriched category and relate it to finitary enriched monads over. Using it, we deduce recent results about ordered universal algebra where inequations are used instead of equations. Then we apply it to metric universal algebra where quantitative equations are used instead of equations. This contributes to understanding of finitary monads on the category of metric spaces. [ABSTRACT FROM AUTHOR]

Using generalized enriched categories, in this paper we show that Rosický's proof of cartesian closedness of the exact completion of the category of topological spaces can be extended to a wide range of topological categories over Set , like metric spaces, approach spaces, ultrametric spaces, probabilistic metric spaces, and bitopological spaces. In order to do so we prove a sufficient criterion for exponentiability of (T , V) -categories and show that, under suitable conditions, every injective (T , V) -category is exponentiable in (T , V) - Cat. [ABSTRACT FROM AUTHOR]

In contrast to the fact that every completely distributive lattice is necessarily continuous in the sense of Scott, it is shown that complete distributivity of a category enriched over the closed category obtained by endowing the unit interval with a continuous t-norm does not imply its continuity in general. Necessary and sufficient conditions for the implication are presented. [ABSTRACT FROM AUTHOR]

The quantale of distance distributions is of fundamental importance for understanding probabilistic metric spaces as enriched categories. Motivated by the categorical interpretation of partial metric spaces, we are led to investigate the quantaloid of diagonals between distance distributions, which is expected to establish the categorical foundation of probabilistic partial metric spaces. Observing that the quantale of distance distributions w.r.t. an arbitrary continuous t-norm is non-divisible, we precisely characterize diagonals between distance distributions, and prove that one-step functions are the only distance distributions on which the set of diagonals coincides with the generated down set. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of stable future, past and total component systems on a directed space with no loops. Then, we associate the stable component category to a stable (future, past or total) component system. Stable component categories are enriched in some monoidal category, eg. the homotopy category of spaces, and carry information about the spaces of directed paths between particular points. It is shown that the geometric realizations of finite pre-cubical sets with no loops admit unique minimal stable (future/past/total) component systems. These constructions provide a new family of invariants for directed spaces. [ABSTRACT FROM AUTHOR]

Abstract We define and study opfibrations of V -enriched categories when V is an extensive monoidal category whose unit is terminal and connected. This includes sets, simplicial sets, categories, or any locally cartesian closed category with disjoint coproducts and connected unit. We show that for an ordinary category B , there is an equivalence of 2-categories between V -enriched opfibrations over the free V -category on B , and pseudofunctors from B to the 2-category of V -categories. This generalizes the classical (Set -enriched) Grothendieck correspondence. [ABSTRACT FROM AUTHOR]

It is well known that the category of Gray-categories does not admit a monoidal biclosed structure that models weak higher-dimensional transformations. In this paper, the first of a series on the topic, we describe several skew monoidal closed structures on the category of Gray-categories, one of which captures higher lax transformations, and another which models higher pseudo-transformations. [ABSTRACT FROM AUTHOR]

In this work, we describe an adjunction between the comma category of Set-based monads under the V -powerset monad and the category of associative lax extensions of Set-based monads to the category of V -relations. In the process, we give a general construction of the Kleisli extension of a monad to the category of V-relations.

Descent theory for linear categories is developed. Given a linear category as an extension of a diagonal category, we introduce descent data, and the category of descent data is isomorphic to the category of representations of the diagonal category, if some flatness assumptions are satisfied. Then Hopf–Galois descent theory for linear Hopf categories, the Hopf algebra version of a linear category, is developed. This leads to the notion of Hopf–Galois category extension. We have a dual theory, where actions by dual linear Hopf categories on linear categories are considered. Hopf–Galois category extensions over groupoid algebras correspond to strongly graded linear categories. [ABSTRACT FROM AUTHOR]

We define strict and weak duality involutions on 2-categories, and prove a coherence theorem that every bicategory with a weak duality involution is biequivalent to a 2-category with a strict duality involution. For this purpose we introduce \2-categories with contravariance", a sort of enhanced 2-category with a basic notion of \contravariant morphism", which can be regarded either as generalized multicategories or as enriched categories. This enables a universal characterization of duality involutions using absolute weighted colimits, leading to a conceptual proof of the coherence theorem. [ABSTRACT FROM AUTHOR]

We give a general categorical construction that yields several monads of measures and distributions as special cases, alongside several monads of filters. The construction takes place within a categorical setting for generalized functional analysis, called a functional-analytic context , formulated in terms of a given monad or algebraic theory T enriched in a closed category V . By employing the notion of commutant for enriched algebraic theories and monads, we define the functional distribution monad associated to a given functional-analytic context. We establish certain general classes of examples of functional-analytic contexts in cartesian closed categories V , wherein T is the theory of R -modules or R -affine spaces for a given ring or rig R in V , or the theory of R-convex spaces for a given preordered ring R in V . We prove theorems characterizing the functional distribution monads in these contexts, and on this basis we establish several specific examples of functional distribution monads. [ABSTRACT FROM AUTHOR]

Autonomous differential equations induced by continuous vector fields usually appear in non-smooth mechanics and other scientific contexts. For these type of equations, given an initial condition, one has existence theorems but, in general, the uniqueness of the solution can not be ensured. For continuous vector fields, the equation solutions do not generally present a continuous flow structure; one particular but interesting case, occurs when under some initial conditions one can ensure existence of solutions and uniqueness in forward time obtaining in this case continuous semi-flows. The discretization and return Poincaré techniques induce the corresponding discrete flows and semi-flows and some inverse methods as the suspension can construct a flow from a discrete flow or semi-flow. The objective of this work is to give categorical models for the diverse phase spaces of continuous and discrete semi-flows and flows and for the relations between these different phase spaces. We also introduce some new constructions such as the prolongation of continuous and discrete semi-flows and the telescopic functors. We consider small Top-categories (weakly enriched over the category Top of topological spaces) and we take as categorical models of the solutions of these differential equations some categories of continuous functors from a small Top-category to the category of topological spaces. Moreover, the processes of discretizations, suspensions, prolongations, et cetera are described in terms of adjoint functors. The main contributions of this paper are the construction of a tensor product associated to a functor between small Top-categories and the interpretation of prolongations, suspensions and telescopes as particular cases of this general tensor product. In general, the paper is focused on the establishment of links between category theory and dynamical systems more than on the study of differential equations using some categorical terminology. [ABSTRACT FROM AUTHOR]

In this article we study function spaces (rise) and descent (fall) in quantale-enriched categories, paying particular attention to enrichment in the non-negative reals, the quantale of distribution functions and the unit interval equipped with a continuous t-norm. [ABSTRACT FROM AUTHOR]

We prove an adjoint functor theorem in the setting of categories enriched in a monoidal model category V admitting certain limits. When V is equipped with the trivial model structure this recaptures the enriched version of Freyd's adjoint functor theorem. For non-trivial model structures, we obtain new adjoint functor theorems of a homotopical flavour — in particular, when V is the category of simplicial sets we obtain a homotopical adjoint functor theorem appropriate to the ∞-cosmoi of Riehl and Verity. We also investigate accessibility in the enriched setting, in particular obtaining homotopical cocompleteness results for accessible ∞-cosmoi. [ABSTRACT FROM AUTHOR]

We provide a uniform definition of higher order Toda brackets in a general setting, covering the known cases of long Toda brackets for topological spaces and Massey products for differential graded algebras, among others. [ABSTRACT FROM AUTHOR]

We introduce Hopf categories enriched over braided monoidal categories. The notion is linked to several recently developed notions in Hopf algebra theory, such as Hopf group (co)algebras, weak Hopf algebras and duoidal categories. We generalize the fundamental theorem for Hopf modules and some of its applications to Hopf categories. [ABSTRACT FROM AUTHOR]

In fairly elementary terms this paper presents, and expands upon, a recent result by Garner by which the notion of topologicity of a concrete functor is subsumed under the concept of total cocompleteness of enriched category theory. Motivated by some key results of the 1970s, the paper develops all needed ingredients from the theory of quantaloids in order to place essential results of categorical topology into the context of quantaloid-enriched category theory, a field that previously drew its motivation and applications from other domains, such as quantum logic and sheaf theory. [ABSTRACT FROM AUTHOR]

Symmetric monoidal closed categories may be related to one another not only by the functors between them but also by enrichment of one in another, and it was known to G. M. Kelly in the 1960s that there is a very close connection between these phenomena. In this first part of a two-part series on this subject, we show that the assignment to each symmetric monoidal closed category V its associated V -enriched category V extends to a 2-functor valued in an op-2-fibred 2-category of symmetric monoidal closed categories enriched over various bases. For a fixed V , we show that this induces a 2-functorial passage from symmetric monoidal closed categories over V (i.e., equipped with a morphism to V ) to symmetric monoidal closed V -categories over V . As a consequence, we find that the enriched adjunction determined a symmetric monoidal closed adjunction can be obtained by applying a 2-functor and, consequently, is an adjunction in the 2-category of symmetric monoidal closed V -categories. [ABSTRACT FROM AUTHOR]

In this paper we show how to modify cofibrations in a monoidal model category so that the tensor unit becomes cofibrant while keeping the same weak equivalences. We obtain applications to enriched categories and coloured operads in stable homotopy theory. [ABSTRACT FROM AUTHOR]

We introduce the notion of solid monoid and rigid monoid in monoidal categories and study the formal properties of these objects in this framework. We show that there is a one to one correspondence between solid monoids, smashing localizations and mapping colocalizations, and prove that rigid monoids appear as localizations of the unit of the monoidal structure. As an application, we study solid and rigid ring spectra in the stable homotopy category and characterize connective solid ring spectra as Moore spectra of subrings of the rationals. [ABSTRACT FROM AUTHOR]

The importance of accessible categories has been widely recognized; they can be described as those freely generated in some precise sense by a small set of objects and, because of that, satisfy many good properties. More specifically finitely accessible categories can be characterized as: (a) free cocompletions of small categories under filtered colimits, and (b) categories of flat presheaves on some small category. The equivalence between (a) and (b) is what makes the theory so general and fruitful. Notions of enriched accessibility have also been considered in the literature for various bases of enrichment, such as Ab , SSet , Cat and Met. The problem in this context is that the equivalence between (a) and (b) is no longer true in general. The aim of this paper is then to: 1. give sufficient conditions on V so that (a) ⇔ (b) holds; 2. give sufficient conditions on V so that (a) ⇔ (b) holds up to Cauchy completion; 3. explore some examples not covered by (1) or (2). [ABSTRACT FROM AUTHOR]

This survey paper, specifically targeted at a readership of fuzzy logicians and fuzzy set theorists, aims to provide a gentle introduction to the basic notions of quantaloid-enriched category theory. We discuss at length the definitions of quantaloid, quantaloid-enriched category, distributor and functor, always giving several examples that – hopefully – appeal to the intended readership. To indicate the strength of this general theory, we explain in considerable detail how (co)limits are dealt with, and particularly how the Yoneda embedding of a quantaloid-enriched category in its free (co)completion comes to be. Our insistence on quantaloid-enrichment (rather than quantale-enrichment) is duly explained by examples requiring a notion of “partial elements” (sheaves, partial metric spaces). A final section glosses over some further topics, providing ample references for the interested reader. [ABSTRACT FROM AUTHOR]

We study the totality of categories weakly enriched in a monoidal bicategory using a notion of enriched icon as 2-cells. We show that when the monoidal bicategory in question is symmetric then this process can be iterated. We show that starting from the symmetric monoidal bicategory Cat and performing the construction twice yields a convenient symmetric monoidal bicategory of partially strict tricategories. We show that restricting to the doubly degenerate ones immediately gives the correct bicategory of "2-tuply monoidal categories" missing from our earlier studies of the Periodic Table. We propose a generalisation to all k-tuply monoidal n-categories. [ABSTRACT FROM AUTHOR]

Given a monad T on a suitable enriched category B equipped with a proper factorization system (E,M), we define notions of T-completion, T-closure, and T-density. We show that not only the familiar notions of completion, closure, and density in normed vector spaces, but also the notions of sheafification, closure, and density with respect to a Lawvere-Tierney topology, are instances of the given abstract notions. The process of T-completion is equally the enriched idempotent monad associated to T (which we call the idempotent core of T), and we show that it exists as soon as every morphism in B factors as a T-dense morphism followed by a T-closed M-embedding. The latter hypothesis is satisfied as soon as B has certain pullbacks as well as wide intersections of M-embeddings. Hence the resulting theorem on the existence of the idempotent core of an enriched monad entails Fakir's existence result in the non-enriched case, as well as adjoint functor factorization results of Applegate-Tierney and Day. [ABSTRACT FROM AUTHOR]