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Representable ( 𝕋 , V ) -categories
Abstract
Working in the framework of -categories, for a symmetric monoidal closed category V and a (not necessarily cartesian) monad , we present a common account to the study of ordered compact Hausdorff spaces and stably compact spaces on one side and monoidal categories and representable multicategories on the other one. In this setting we introduce the notion of dual for -categories.
... As an application of this technique, we construct a free Talgebra functor and the underlying T-monoid functor, which are analogues of the free monoidal category functor and the underlying multicategory functor going between the categories of monoidal categories and multicategories. We then generalize the results of[5]. The second theme of the paper is the lax formal theory of monads within a tricategory. ...
... In the case A = Mat(V ), our T-algebras are exactly the T-algebras considered in[5]. In particular: When T 0 is the free monoid monad, and V is an arbitrary monoidal category, a T-algebra is a strict monoidal V-category. ...
... In the case A = Mat(V ), the free T-algebra functor is the functor from the category of (T, V )-categories to the category of T-algebras constructed in[5]. In particular, when T 0 is taken to be the free monoid monad, then the free T-algebra functor becomes the free monoidal V-category functor on the category of V-multicategories. ...
Generalized operads, also called generalized multicategories and T-monoids,
are defined as monads within a Kleisli bicategory. With or without emphasizing
their monoidal nature, generalized operads have been considered by numerous
authors in different contexts, with examples including symmetric
multicategories, topological spaces, globular operads and Lawvere theories. In
this paper we study functoriality of the Kleisli construction, and
correspondingly that of generalized operads. Motivated by this problem we
develop a lax version of the formal theory of monads, and study its connection
to bicategorical structures.
In this communication we generalize some recent results of Rump to categories enriched in a commutative quantale V. Using these results, we show that every quantale-enriched multicategory admits an injective hull. Finally, we expose a connection between the Isbell adjunction and the construction of injective hulls for topological spaces made by Banaschewski in 1973.
In this paper we carry the construction of equilogical spaces into an arbitrary category X topological over Set, introducing the category X-Equ of equilogical objects. Similar to what is done for the category Top of topological spaces and continuous functions, we study some features of X-Equ as (co)completeness and regular (co-)well-poweredness, as well as the fact that, under some conditions, it is a quasitopos. We achieve these various properties of the category X-Equ by representing it as a category of partial equilogical objects, as a reflective subcategory of the exact completion Xex, and as the regular completion Xreg. We finish with examples in the particular cases, amongst others, of ordered, metric, and approach spaces, which can all be described using the (T,V)-Cat setting.
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.
Preface Acknowledgements Foreword Introduction 1. A primer on ordered sets and lattices 2. Order theory of domains 3. The Scott topology 4. The Lawson Topology 5. Morphisms and functors 6. Spectral theory of continuous lattices 7. Compact posets and semilattices 8. Topological algebra and lattice theory: Applications Bibliography List of symbols List of categories Index.
We put forward a revised deflnition of stably compact spaces which allows us to show their equivalence with Nachbin's compact ordered spaces in an entirely elementary fashion. We then exhibit some constructions for stably compact spaces which ap- parently have not appeared in the literature before. These constructions allow us to show that the set of (sub-)probability valuations can be equipped with a topology which turns this set into another stably compact space. The topology chosen is not random; it is the weakest topology which makes integration of lower semicontinuous functions a continuous operation.
The purpose of this paper is to develop the basic theory of stably compact spaces (viz. compact, locally compact, coherent sober spaces) and introduce in an accessible manner and with a minimum of prerequisites some significant new lines of investigation and application arising from recent research, which has arisen primarily in the theoretical computer science community. Three primary themes have developed:
(i) the property of stable compactness is preserved under a large variety of constructions involving powerdomains, hyperspaces and function spaces;(ii) the underlying de Groot duality of stably compact spaces, which finds varied expression, is reflected by duality theorems involving the just mentioned constructions; and(iii) the notion of inner and outer pavings is a useful and natural tool for such studies of stably compact spaces.
Generalized operads, also called generalized multicategories and T-monoids,
are defined as monads within a Kleisli bicategory. With or without emphasizing
their monoidal nature, generalized operads have been considered by numerous
authors in different contexts, with examples including symmetric
multicategories, topological spaces, globular operads and Lawvere theories. In
this paper we study functoriality of the Kleisli construction, and
correspondingly that of generalized operads. Motivated by this problem we
develop a lax version of the formal theory of monads, and study its connection
to bicategorical structures.
Monoidal Topology describes an active research area that, after various past proposals on how to axiomatize 'spaces' in terms of convergence, began to emerge at the beginning of the millennium. It combines Barr's relational presentation of topological spaces in terms of ultrafilter convergence with Lawvere's interpretation of metric spaces as small categories enriched over the extended real half-line. Hence, equipped with a quantale V (replacing the reals) and a monad T (replacing the ultrafilter monad) laxly extended from set maps to V-valued relations, the book develops a categorical theory of (T, V)-algebras that is inspired simultaneously by its metric and topological roots. The book highlights in particular the distinguished role of equationally defined structures within the given lax-algebraic context and presents numerous new results ranging from topology and approach theory to domain theory. All the necessary pre-requisites in order and category theory are presented in the book.
Notions and techniques of enriched category theory can be used to study topological structures, like metric spaces, topological spaces and approach spaces, in the context of topological theories. Recently in [D. Hofmann, Injective spaces via adjunction, arXiv:math.CT/0804.0326] the construction of a Yoneda embedding allowed to identify injectivity of spaces as cocompleteness and to show monadicity of the category of injective spaces and left adjoints over Set. In this paper we generalise these results, studying cocompleteness with respect to a given class of distributors. We show in particular that the description of several semantic domains presented in [M. Escardó and B. Flagg, Semantic domains, injective spaces and monads, Electronic Notes in Theoretical Computer Science 20 (1999)] can be translated into the V-enrichcd setting. 2009-02-15.
A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of ‘category with finite products’. To capture such distinctions, we consider on a 2-category those 2-monads for which algebra structure is essentially unique if it exists, giving a precise mathematical definition of ‘essentially unique’ and investigating its consequences. We call such 2-monads property-like. We further consider the more restricted class of fully property-like 2-monads, consisting of those property-like 2-monads for which all 2-cells between (even lax) algebra morphisms are algebra 2-cells. The consideration of lax morphisms leads us to a new characterization of those monads, studied by Kock and Zoeberlein, for which ‘structure is adjoint to unit’, and which we now call lax-idempotent 2-monads: both these and their colax-idempotent duals are fully property-like. We end by showing that (at least for finitary 2-monads) the classes of property-likes, fully property-likes, and lax-idempotents are each coreflective among all 2-monads.
The standard contravariant adjunction between TOP (the category of topological spaces) and LAT (the category of distributive lattices) induces a triple Λ on LAT and a triple Σ on TOP. We show that the category LATΛ of Λ-algebras is just the category of frames, and describe the category TOPΣ of Σ-algebras as a subcategory of TOP.
Recent work of several authors shows that many categories of interest to topologists can be represented as categories of lax algebras. In this paper we introduce the concept of a topological theory as a syntactical tool to deal with lax algebras, and show the usefulness of our approach by applying it to the study of function spaces.
Let CmptPoSp denote the category of compact pospaces with continuous monotone maps and let PoSet denote the category of partially ordered sets and monotone maps. In this paper we show that the forgetful functor G: CmptPoSp → PoSet is monadic; that is, G has a left-adjoint and CmptPoSp is isomorphic to the category of algebras PoSetB for the monad B on PoSet induced by the adjunction. This result, which is an asymmetric version of Manes' theorem, shows that the notion of compact pospace is algebraic in a precise sense and provides a useful tool for investigating the category CmptPoSp. As a corollary we obtain the theorem of Simmons and Wyler which says that CmptPoSp is also algebraic over the category of topological spaces and continuous maps. This makes explicit the connection between the Salbany and the prime Wallman compactifications. We also give an explicit construction—as the prime spectrum of the lattice of upper sets—of the Stone-Čech-Nachbin order compactification for a discrete ordered space.
We give an explicit description of the free completion EM ( K ) of a 2-category K under the Eilenberg–Moore construction, and show that this has the same underlying category as the 2-category Mnd ( K ) of monads in K . We then demonstrate that much of the formal theory of monads can be deduced using only the universal property of this completion, provided that one is willing to work with EM ( K ) as the 2-category of monads rather than Mnd ( K ) . We also introduce the wreaths in K ; these are the objects of EM ( EM ( K )) , and are to be thought of as generalized distributive laws. We study these wreaths, and give examples to show how they arise in a variety of contexts.
The analogy between dist (a, b)+dist (b, c)≥dist (a, c) and hom (A, B) ⊗ hom (B, C)→hom (A, C) is rigorously developed to display many general results about metric spaces as consequences of a «generalized pure logic»
whose «truth-values» are taken in an arbitrary closed category.
In questo articolo viene rigorosamente sviluppata l'analogia fra dist (a, b)+dist (b, c)≥dist (a, c) e hom (A, B) ⊗ hom (B, C)→ hom (A, C), giungendo a numerosi risultati generali sugli spazi metrici, come conseguenza di una «logica pura generalizzata» i cui
«valori di verità» sono scelti in una arbitraria categoria chiusa.
This article shows that the distributive laws of Beck in the bicategory of sets and matrices, wherein monads are categories, determine strict factorization systems on their composite monads. Conversely, it is shown that strict factorization systems on categories give rise to distributive laws. Moreover, these processes are shown to be mutually inverse in a precise sense. Strict factorization systems are shown to be the strict algebras for the 2-monad on the 2-category of categories. Further, an extension of the distributive law concept provides a correspondence with the classical factorization systems.
We introduce the notion of representable multicategory, which stands in the same relation to that of monoidal category as fibration does to contravariant pseudofunctor (into at). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe represent ability in elementary terms via universal arrows. We also give a doctrinal characterisation of representability based on a fundamental monadic adjunction between the 2-category of multicategories and that of strict monoidal categories. The first main result is the coherence theorem for representable multicategories, asserting their equivalence to strict ones, which we establish via a new technique based on the above doctrinal characterisation. The other main result is a 2-equivalence between the 2-category of representable multicategories and that of monoidal categories and strong monoidal functors. This correspondence extends smoothly to one between bicategories and a several object version of representable multicategories.
The paper discusses interactions between order and topology on a given set which do not presuppose any separation conditions for either of the two structures, but which lead to the existing notions established by Nachbin in more special situations. We pursue this discussion at the much more general level of lax algebras, so that our categories do not concern just ordered topological spaces, but also sets with two interacting orders, approach spaces with an additional metric, etc.
The work of the present author and his coauthors over the past years gives evidence that it may be useful to regard each topological space as a kind of enriched category, by interpreting the convergence relation x→x between ultrafilters and points of a topological space X as arrows in X. Naturally, this point of view opens the door to the use of concepts and ideas from enriched Category Theory for the investigation of topological spaces. Topological theories introduced by the author provide a convenient general setting for appropriately transferring these concepts and ideas to the world of topological spaces and some other geometric objects such as approach spaces. Using tools like adjunction and the Yoneda lemma, we show that the cocomplete spaces are precisely the injective spaces, and they are algebras for a suitable monad on . This way we obtain enriched versions of known results about injective topological spaces and continuous lattices.
We analyse the 2-dimensional categorical algebra underlying the process of completing categories, or posets. The algebra explains why and how completeness of a category is describable in monad theoretic terms, and why the limit formation for freely completed categories admits a further adjoint.
The main source of inspiration for the present paper is the work of R. Rosebrugh and R.J. Wood on constructive complete distributive lattices where the authors employ elegantly the concepts of adjunction and module in their study of ordered sets. Both notions (suitably adapted) are available in topology too, which permits us to investigate topological, metric and other kinds of spaces in a similar spirit. Therefore, relative to a choice of modules, we consider spaces which admit all colimits with weight in , as well as (suitably defined) -distributive and -algebraic spaces. We show that the category of -distributive spaces and -colimit preserving maps is dually equivalent to the idempotent splitting completion of a category of spaces and convergence relations between them. We explain the connection of these results to the traditional duality of spaces with frames, and conclude further duality theorems. Finally, we study properties and structures of the resulting categories, in particular monoidal (closed) structures.
For a symmetric monoidal-closed category V and a suitable monad T on the category of sets, we introduce the notion of reflexive and transitive (T , V)-algebra and show that various old and new structures are instances of such algebras. Lawvere's presentation of a metric space as a V-category is included in our setting, via the Betti-Carboni-Street-Walters interpretation of a V-category as a monad in the bicategory of V-matrices, and so are Barr's presentation of topological spaces as lax algebras, Lowen's approach spaces, and Lambek's multicategories, which enjoy renewed interest in the study of n-categories. As a further example, we introduce a new structure called ultracategory which simultaneously generalizes the notions of topological space and of category.
Given a 2-category K admitting a calculus of bimodules, and a 2-monad T on it compatible with such calculus, we construct a 2-category L with a 2-monad S on it such that: S has the adjoint-pseudo-algebra property. The 2-categories of pseudo-algebras of S and T are equivalent. Thus, coherent structures (pseudo-T-algebras) are transformed into universally characterised ones (adjoint-pseudo-S-algebras). The 2-category L consists of lax algebras for the pseudo-monad induced by T on the bicategory of bimodules of K. We give an intrinsic characterisation of pseudo- S-algebras in terms of representability . Two major consequences of the above transformation are the classications of lax and strong morphisms, with the attendant coherence result for pseudo-algebras. We apply the theory in the context of internal categories and examine monoidal and monoidal globular categories (including their monoid classiers) as well as pseudo-functors into Cat. Contents 1
It is known since 1973 that Lawvere's notion of (Cauchy-)complete enriched category is meaningful for metric spaces: it captures exactly Cauchy-complete metric spaces. In this paper we introduce the corresponding notion of Lawvere completeness for -categories and show that it has an interesting meaning for topological spaces and quasi-uniform spaces: for the former ones means weak sobriety while for the latter means Cauchy completeness. Further, we show that has a canonical -category structure which plays a key role: it is Lawvere-complete under reasonable conditions on the setting; permits us to define a Yoneda embedding in the realm of -categories.
Continuous lattices and domains, volume 93 of Encyclopedia of Mathematics and its Applications
- G Gierz
- K H Hofmann
- K Keimel
- J Lawson
- M W Mislove
- D Scott
Topologia e Ordem. Univ. of Chicago Press
- L Nachbin
Semantic domains, injective spaces and monads. In: Mathematical foundations of programming semantics
- M H Escardó
- R Flagg
Časopis pro pěstování matematiky a fysiky
- M H Stone
- MH Stone