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In this paper, considering L a complete residuated lattice, we present a lattice-valued category TNG (resp., TTNG) of (topological) ⊤-neighborhood groups, where the object is defined as a group equipped with a (topological) ⊤-neighborhood space such that the group operations are continuous with respect to the (topological) ⊤-neighborhood space. It is proved that: (1) The ⊤-neighborhood space associated with a ⊤-neighborhood group is topological, so the category TNG is equivalent to the category TTNG. Hence TTNG is redundant, and we need only discuss TNG. (2) The category TNG has nice characterizations, localization and uniformization. (3) The category TNG has initial structure, so it is a topological category, and each initial structure has an ordered representation. (4) The category NG of neighborhood groups can be embedded in TNG as a reflective subcategory. (5) The category TNG can be embedded in the category SLNG of stratified L-neighborhood groups as a reflective subcategory when the underlying lattice L is a meet-continuous lattice. (6) The category TNG is equivalent to the category StrLTOPG of strong L-topological groups when L is a complete MV-algebra.

It is a classic result in modal logic, often referred to as Jónsson-Tarski duality, that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality for boolean algebras. Our goal is to generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead of working with the boolean algebra of clopen subsets of a Stone space, we work with the ring of continuous real-valued functions on a compact Hausdorff space. The main novelty is to define a modal operator on such a ring utilizing a continuous relation on a compact Hausdorff space.Our starting point is the well-known Gelfand duality between the category ${\sf KHaus}$ of compact Hausdorff spaces and the category $\boldsymbol {\mathit {uba}\ell }$ of uniformly complete bounded archimedean $\ell $ -algebras. We endow a bounded archimedean $\ell $ -algebra with a modal operator, which results in the category $\boldsymbol {\mathit {mba}\ell }$ of modal bounded archimedean $\ell $ -algebras. Our main result establishes a dual adjunction between $\boldsymbol {\mathit {mba}\ell }$ and the category ${\sf KHF}$ of what we call compact Hausdorff frames; that is, Kripke frames equipped with a compact Hausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between ${\sf KHF}$ and the reflective subcategory $\boldsymbol {\mathit {muba}\ell }$ of $\boldsymbol {\mathit {mba}\ell }$ consisting of uniformly complete objects of $\boldsymbol {\mathit {mba}\ell }$ . This generalizes both Gelfand duality and Jónsson-Tarski duality.

Zhao and Ho asked in a recent paper that for each T0 space X, whether KB(X) (the set of all irreducible closed sets of X whose suprema exist) is the canonical k-bounded sobrification of X in the sense of Keimel and Lawson. In this paper, we construct a counterexample to give a negative answer. We also consider the subcategory Topκ of the category Top0 of T0 spaces, and prove that the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category Topκ.

We introduce an endofunctor H on the category bal of bounded archimedean l-algebras and show that there is a dual adjunction between the category Alg ( H ) of algebras for H and the category Coalg ( V ) of coalgebras for the Vietoris endofunctor V on the category of compact Hausdorff spaces. We prove that Gelfand duality lifts to a dual equivalence between Coalg ( V ) and the full reflective subcategory Alg u ( H ) of Alg ( H ) . We provide an alternate view of Alg u ( H ) by introducing an endofunctor H u on the full reflective subcategory of bal consisting of uniformly complete objects of bal and showing that Alg ( H u ) is isomorphic to Alg u ( H ) . On the one hand, these results generalize those of [1] , [19] for the category of coalgebras of the Vietoris endofunctor on the category of Stone spaces. On the other hand, they provide an alternate, more categorical proof of a recent result of [6] .

In this paper, a new approach to fuzzy convergence theory in the framework of stratified L-topological spaces is provided. Firstly, the concept of stratified L-prefilter convergence structures is introduced and it is shown that the resulting category is a Cartesian closed topological category. Secondly, the relations between the category of stratified L-prefilter convergence spaces and the category of stratified L-topological spaces are studied and it is proved that the latter can be embedded in the former as a reflective subcategory. Finally, the relations between the category of stratified L-prefilter convergence spaces and the category of stratified L-Min convergence spaces (fuzzy convergence spaces in the sense of Min) are investigated and it is shown that the former can be embedded in the latter as a reflective subcategory.

In this paper, we present an investigation of quantale algebras (Q-algebras for short) as lattice-valued quantales (Q-quantales for short). First, we prove that the set of all fuzzy ideals of a commutative ring with appropriate operations is a [0, 1]-quantale. Furthermore, we discuss some properties of localic nuclei on Q-algebras, and show that the category of Q-algebras with the quantale structures being frames is a full reflective subcategory of the category of Q-algebras. From this result, we can conclude that the category of L-frames is a full reflective subcategory of the category of L-quantales, where L is a frame. Finally, we build and characterize the Q-quantale completions of a Q-ordered semigroup.

Using the intrinsic lattice-valued inclusion orders of L-subsets and of stratified L-filters, the concept of a stratified L-preuniform convergence structure is proposed. This convergence structure is asymmetric and can be used to establish a framework of asymmetric lattice-valued space structures. The category of stratified L-preuniform convergence spaces is introduced and some categorical properties are presented. A reflective subcategory of stratified L-preuniform convergence spaces is found that is categorically isomorphic to strong stratified L-convergence spaces (originally called stratified L-ordered convergence spaces). Several subcategories of stratified L-preuniform convergence spaces are established and relations between the different subcategories are presented. We conclude that stratified L-preuniform convergence structures introduced here could play a role in the framework of lattice-valued asymmetric space structures in lattice-valued topology.

This paper presents a definition of (L,M)-fuzzy convergence spaces. It is shown that the category (L,M)-FC of (L,M)-fuzzy convergence spaces, which embeds the category (L,M)-FTop of (L,M)-fuzzy topological spaces as a reflective subcategory, is a Cartesian closed topological category. Further, it is proved that the category of (topological) pretopological (L,M)-fuzzy convergence spaces is isomorphic to the category of (topological) (L,M)-fuzzy quasi-coincident neighborhood spaces. Moreover, the relations among (L,M)-fuzzy convergence spaces, pretopological (L,M)-fuzzy convergence spaces and topological (L,M)-fuzzy convergence spaces are investigated in the categorical sense.

This paper presents a definition of enriched (L, M)-fuzzy convergence spaces. It is shown that the resulting category E(L, M)-FC is a Cartesian closed topological category, which can embed the category E(L, M)-FTop of enriched (L, M)- fuzzy topological spaces as a reflective subcategory. Also, it is proved that the category of topological enriched (L, M)-fuzzy convergence spaces is isomorphic to E(L, M)-FTop and the category of pretopological enriched (L, M)-fuzzy convergence spaces is isomorphic to the category of enriched (L, M)-fuzzy quasi-coincident neighborhood spaces.

Directed topology is a refinement of standard topology, where spaces may have non-reversible paths. It has been put forward as a candidate approach to the analysis of concurrent processes. Recently, a wealth of different frameworks for, i.e., categories of, directed spaces have been proposed. In the present work, starting from Grandis's notion of directed space, we propose an additional condition of saturation for distinguished sets of paths and show how it allows to rule out exotic examples without any serious collateral damage. Our saturation condition is local in a natural sense, and is satisfied by the directed interval (and the directed circle). Furthermore we show in which sense it is the strongest condition fulfilling these two basic requirements. Our saturation condition selects a full, reflective subcategory of Grandis's category of d-spaces, which is closed under arbitrary limits of d-spaces, has arbitrary colimits (obtained by saturating the corresponding colimits of d-spaces), and has nice cylinder and cocylinder constructions. Finally, the forgetful functor to plain topological spaces has both a right and a left adjoint., Comment: 8 pages

We define a category of stratified L-generalized convergence spaces for the case where the lattice is an enriched cl-premonoid. We then investigate some of its categorical properties and those of its subcategories, in particular the stratified L-principal convergence spaces and the stratified L-topological convergence spaces. For some results we need to introduce a new condition on the lattice (which is always true in the case where the lattice is a frame, but not always true in the more general case). As examples where we may apply the more general lattice context we examine the stratified L-topological spaces and probabilistic limit spaces. We show that the category of stratified L-topological spaces is a reflective subcategory of our category and that the category of probabilistic limit spaces under a T-norm is both a reflective and a coreflective subcategory of our category if we choose the lattice context appropriately.

This paper presents a definition of (L,M)-fuzzy nets and the corresponding (L,M)-fuzzy generalized convergence spaces. It establishes a Moore-Smith convergence in (L,M)-fuzzy topology. It is shown that the category (L,M)-GConv of (L,M)-fuzzy generalized convergence spaces is topological, which embeds the category of (L,M)-fuzzy topological spaces as a reflective subcategory. It also defines a strong (L,M)-fuzzy generalized convergence space and shows that the resulting category S(L,M)-GConv is topological and Cartesian-closed, which also embeds the category of (L,M)-fuzzy topological spaces as a reflective subcategory and can be embedded in (L,M)-GConv as a coreflective subcategory. As a special case, (2,M)-GConv is cartesian-closed.

The paper provides variety-based extensions of the concepts of (lattice-valued) interchange system and space, introduced by Denniston, Melton and Rodabaugh, and shows that variety-based interchange systems incorporate topological systems of Vickers, state property systems of Aerts, Chu spaces (over the category of sets in the sense of Pratt) of P.-H. Chu and contexts (of formal concept analysis) of Wille. The paper also provides an explicit description of (co)limits in the category of variety-based topological systems and applies the obtained results to extend the claim of Denniston et al. that the category of topological systems of Vickers is small initially topological over the category of sets.

We characterize abstract Morita contexts in several bicategories. In particular, we use heteromorphisms for the bicategory \(\mathbb{Prof} \) of categories and profunctors and coreflective subcategories for \(\mathbb {Cat}\) (categories and functors). In addition, we prove general statements concerning strict Morita contexts, and we give new equivalent forms to the standard notions of adjointness, category equivalence and Morita equivalence by studying the collage of a profunctor.

We investigate universal algebra over the category Nom of nominal sets. Using the fact that Nom is a full reflective subcategory of a monadic category, we obtain an HSP-like theorem for algebras over nominal sets. We isolate a ‘uniform’ fragment of our equational logic, which corresponds to the nominal logics present in the literature. We give semantically invariant translations of theories for nominal algebra and NEL into ‘uniform’ theories, and systematically prove HSP theorems for models of these theories.

We characterize nearness frames whose completions are fine (we call them quotient-fine), and show that the subcategory QfNFrm they form is reflective in the category of strong nearness frames. The resulting functor commutes with the completion functor. QfNFrm is isomorphic to the subcategory of the functor category (RegFrm)2 given by the dense onto \(h\colon M\to L\), where 2 denotes the category with only two objects and exactly one morphism between them.

We investigate a categorial duality between quasi MV-algebras (a variety of algebras arising from quantum computation and tightly connected with fuzzy logic) and a reflective subcategory of l-groups with strong units.

We study a category of lattice-valued uniform convergence spaces where the lattice is enriched by two algebraic operations. This general setting allows us to view the category of lattice-valued uniform spaces as a reflective subcategory of our category, and the category of probabilistic uniform limit spaces as a coreflective subcategory.

We study topological spaces with a distinguished set of paths, called directed paths. Since these directed paths are generally not reversible, the directed homotopy classes of directed paths do not assemble into a groupoid, and there is no direct analog of the fundamental group. However, they do assemble into a category, called the fundamental category. We define models of the fundamental category, such as the fundamental bipartite graph, and minimal extremal models which are shown to generalize the fundamental group. In addition, we prove van Kampen theorems for subcategories, retracts, and models of the fundamental category., Comment: 20 pages, 10 figures

Based on complete, residuated lattices, the relations between stratified L-fuzzy convergence spaces and enriched L-fuzzy topological spaces are studied. It is shown that, as a reflective subcategory, the category of enriched L-fuzzy topological spaces can embedded in the category of stratified L-fuzzy convergence spaces. Also the specialization L-preorder of stratified L-fuzzy convergence spaces is investigated.

We give a new sufficient condition for the normal extensions in an admissible Galois structure to be reflective. We then show that this condition is indeed fulfilled when X is the (protomodular) reflective subcategory of S-special objects of a Barr-exact S-protomodular category C, where S is the class of split epimorphic trivial extensions in C. Next to some concrete examples where the criterion may be applied, we also study the adjunction between a Barr-exact unital category and its abelian core, which we prove to be admissible., Comment: 22 pages

Let frA denote the category of f-rings which are reduced and Archimedean, and let Φ be the (nonfull) subcategory of such rings with identity (each with the natural morphisms). Some time ago, the second author showed, using his representation theory, that for each A ∈ | frA| there is a certain minimal embedding u A :A→ uA ∈ | Φ|. More recently, he has revisited the representation theory, expanding it to include the representation of morphisms. Based upon this, the present article analyzes the operator u:| frA| → Φ: the construction of uA is tidied, several characterizations of the pair (u A , uA) are given, and the relation between the maximal ideal structures of A and uA is described. Membership in the class U of frA-morphisms that are “u-extendable” is characterized and it is shown that U = (| frA|,U) is a category in which Φ is a full essentially-reflective subcategory. The frA-objects are characterized for which, respectively, ∀ B(frA(A, B) = U (A, B)), and, ∀ B ≠ 0(frA(B, A) = U(B, A)).

We show that in a locally \(\lambda\)-presentable category, every \(\lambda_m\)-injectivity class (i.e., the class of all the objects injective with respect to some class of \(\lambda\)-presentable morphisms) is a weakly reflective subcategory determined by a functorial weak factorization system cofibrantly generated by a class of \(\lambda\)-presentable morphisms. This was known for small-injectivity classes, and referred to as the ‘small object argument.’ An analogous result is obtained for orthogonality classes and factorization systems, where \(\lambda\)-filtered colimits play the role of the transfinite compositions in the injectivity case. \(\lambda\)-presentable morphisms are also used to organize and clarify some related results (and their proofs), in particular on the existence of enough injectives (resp. pure-injectives). Finally, locally \(\lambda\)-presentable categories are shown to be cellularly generated by the set of morphisms between \(\lambda\)-presentable objects.

The category-theoretic nature of general frames for modal logic is explored. A new notion of "modal map" between frames is defined, generalizing the usual notion of bounded morphism/p-morphism. The category Fm of all frames and modal maps has reflective subcategories CHFm of compact Hausdorff frames, DFm of descriptive frames, and UEFm of ultrafilter enlargements of frames. All three subcategories are equivalent, and are dual to the category of modal algebras and their homomorphisms. An important example of a modal map that is typically not a bounded morphism is the natural insertion of a frame A into its ultrafilter enlargement EA. This map is used to show that EA is the free compact Hausdorff frame generated by A relative to Fm. The monad E of the resulting adjunction is examined and its Eilenberg-Moore category is shown to be isomorphic to CHFm. A categorical equivalence between the Kleisli category of E and UEFm is defined from a construction that assigns to each frame A a frame A* that is "image-closed" in the sense that every point-image {b : aRb} in A is topologically closed. A* is the unique image-closed frame having the same ultrafilter enlargement as A. These ideas are connected to a category 2U shown by S. K. Thomason to be dual to the category of complete and atomic modal algebras and their homomorphisms. 2U is the full subcategory of the Kleisli category of E based on the Kripke frames.

If (R, M) and (S, N) are quasi-local domains and f : R → S is a ring homomorphism, then f is said to be a strong local homomorphism if f(M) = N. Let PVD be the category whose objects are the pseudo-valuation domains that are not fields and whose morphisms are the strong local homomorphisms; let VD be the full subcategory of PVD whose objects are all the valuation domains that are not fields. Then VD is shown to be a reflective subcategory of PVD. This result is extended by obtaining a reflective conclusion for a category whose class of objects properly contains all pseudo-valuation domains.

Let $\mathcal{A}$ be an epireflective subcategory of the category Top of topological spaces that is not bireflective (e.g., the category of Hausdorff spaces, the category of Tychonoff spaces) and ℬ be a coreflective subcategory of $\mathcal{A}$ . Extending the corresponding result obtained for coreflective subcategories of Top we prove that ℬ is hereditary if and only if it is closed under the formation of prime factors. As a consequence we obtain that every hereditary coreflective subcategory ℬ of $\mathcal{A}$ containing a non-discrete space is generated by a class of prime spaces and if $\mathcal{A}$ is a quotient-reflective subcategory of Top, then the assignment $\mathcal{B}\mapsto \mathcal{B}\cap \mathcal{A}$ gives a bijection of the collection of all hereditary coreflective subcategories of Top that contain the class FG of all finitely generated spaces onto the collection of all hereditary coreflective subcategories of $\mathcal{A}$ that contain $\mathbf{FG}\cap \mathcal{A}$ . Some applications of these results in the categories of Hausdorff spaces, Tychonoff spaces and zero-dimensional Hausdorff spaces are presented.

Let L be a meet continuous lattice. It is proved that the category Top of topological spaces can be embedded in the category of stratified L-topological spaces as a concretely both reflective and coreflective full subcategory if and only if L is a continuous lattice.

The aim of this paper is to establish some Cartesian closed categories which are between the two Cartesian closed categories: SLP (the category of L-domains and stable functions) and DI (the full subcategory of SLP whose objects are all dI-domains). First we show that the exponentials of every full subcategory of SLP are exactly the spaces of stable functions. Then we prove that the full subcategories SDMBC, SDCBC and SDABC of SLP which contain DI are all Cartesian closed, where the objects of SDMBC (resp., SDCBC, SDABC) are all distributive bc-domains which are meet-continuous (resp., continuous, algebraic). We also obtain many non-Cartesian closed full subcategories of SLP and present some reflective relations between those categories concerned.

A collection of results are presented which are loosely centered around the notion of reflective subcategory. For example, it is shown that reflective subcategories are orthogonality classes, that the morphisms orthogonal to a reflective subcategory are precisely the morphisms inverted under the reflector, and that each subcategory has a largest “envelope” in the ambient category in which it is reflective. Moreover, known results concerning the envelopes of the category of sober spaces, spectral spaces, and jacspectral spaces, respectively, are summarized and reproved. Finally, attention is focused on the envelopes of one-object subcategories, and examples are considered in the category of groups.

Several methods for constructing left determined model structures are expounded. The starting point is Olschok's work on locally presentable categories. We give sufficient conditions to obtain left determined model structures on a full reflective subcategory, on a full coreflective subcategory and on a comma category. An application is given by constructing a left determined model structure on star-shaped weak transition systems., 19 pages; v2: typos fixed, in particular a very confusing one between \gamma_i and \sigma_i; v3: reachable transition system replaced by star-shaped transition system + 2 new theorems in the toolbox; v4 new title + minor revisions (final version)

A semi-localization of a category is a full reflective subcategory with the property that the reflector is semi-left-exact. In this article we first determine an abstract characterization of the categories which are semi-localizations of an exact Mal'tsev category, by specializing a result due to S. Mantovani. We then turn our attention to semi-abelian categories, where a special type of semi-localizations are known to coincide with torsion-free subcategories. A new characterisation of protomodular categories in terms of binary relations is obtained, inspired by the one discovered in the pointed context by Z. Janelidze. This result is useful to obtain an abstract characterization of the torsion-free and of the hereditarily-torsion-free subcategories of semi-abelian categories. Some examples are considered in detail in the categories of groups, crossed modules, commutative rings and topological groups. We finally explain how these results extend similar ones obtained by W. Rump in the abelian context., 30 pages. v2: introduction and references updated

It is proved that the category Top of topological spaces is a mono-reflective and epi-coreflective subcategory of fuzzy topological spaces in the sense of Hutton.

The categories of sober spaces and generalized (i.e., not necessarily T 0 ) Scott spaces are adjoint. We define upper - and lower-sober spaces; lower-sober spaces are a reflective subcategory of TOP that is intermediate to that of sober spaces. We establish why the sobrifications of Alexandrov spaces and of certain Scott spaces described by Mislove (1981) are Scott.

Higher dimensional central extensions of groups were introduced by G. Janelidze as particular instances of the abstract notion of covering morphism from categorical Galois theory. More recently, the notion has been extended to and studied in arbitrary semi-abelian categories. Here, we further extend the scope to exact Mal'tsev categories and beyond. For this, we consider conditions on a Galois structure G = (C, X, I, H, e, E) which insure the existence of an induced Galois structure G_1 = (C_1, X_1, I_1, H_1, e_1, E_1) such that C_1 and X_1 are full subcategories of the arrow category Arr(C) consisting, respectively, of all morphisms in the class E, and of all covering morphisms with respect to G. Moreover, we prove that G_1 satisfies the same conditions as G, so that, inductively, we obtain, for each n, a Galois structure G_n = (G_n+1)_1, whose coverings we call n + 1-fold central extensions.

We construct the analogue of Takeuchi's free Hopf algebra in the setting of Poisson Hopf algebras. More precisely, we prove that there exists a free Poisson Hopf algebra on any coalgebra or, equivalently that the forgetful functor from the category of Poisson Hopf algebras to the category of coalgebras has a left adjoint. In particular, we also prove that the category of Poisson Hopf algebras is a reflective subcategory of the category of Poisson bialgebras. Along the way, we describe coproducts and coequalizers in the category of Poisson Hopf algebras, therefore showing that the latter category is cocomplete., Comment: to appear in J. Math. Phys

Some old and new constructions of free categories with good properties (regularity, exactness, etc.) are investigated, consistently showing their role in proof theory and in realizability theory, and in particular in the construction of the “effective topos” of M. Hyland. The subject of “small complete categories” is discussed, with a proposed new definition of what “complete” should mean for a full reflective subcategory of a topos.

We investigate a Galois connection in poset enriched categories between subcategories and classes of morphisms, given by means of the concept of right-Kan injectivity, and, specially, we study its relationship with a certain kind of subcategories, the KZ-reflective subcategories. A number of well-known properties concerning orthogonality and full reflectivity can be seen as a particular case of the ones of right-Kan injectivity and KZ-reflectivity. On the other hand, many examples of injectivity in poset enriched categories encountered in the literature are closely related to the above connection. We give several examples and show that some known subcategories of the category of T 0 -topological spaces are right-Kan injective hulls of a finite subcategory.

A full reflective subcategory E of a presheaf category [C*,Set] is the category of sheaves for a topology j on C if and only if the reflection preserves finite limits. Such an E is called a Grothendieck topos. More generally, one can consider two topologies, j contained in k, and the category of sheaves for j which are separated for k. The categories E of this form, for some C, j, and k, are the Grothendieck quasitoposes of the title, previously studied by Borceux and Pedicchio, and include many examples of categories of spaces. They also include the category of concrete sheaves for a concrete site. We show that a full reflective subcategory E of [C*,Set] arises in this way for some j and k if and only if the reflection preserves monomorphisms as well as pullbacks over elements of E., Comment: v2: 24 pages, several revisions based on suggestions of referee, especially the new theorem 5.2; to appear in the Journal of Algebra

We answer a question of J. Rosický and W. Tholen by showing that the class of complete Boolean algebras (which is a prereflective subcategory of the category of frames) is not an intersection of reflective subcategories of the category of frames. In order to deduce this result, we first prove the following observation (using some basic facts about congruence frames): the three-element chain belongs to every reflective subcategory of the category of frames which contains the class of complete Boolean algebras.

A quasiordered set Q (qoset for short) is ideal-distributive iff Q I , the lattice of ideals in the sense of Frink, is distributive. The principal ideal embedding of Q in Q I is characterized by certain density properties, by extremal conditions and by a universal property. The reflector I from the category of qosets and ideal-continuous maps (where inverse images of ideals are ideals) to the category of algebraic lattices and join-preserving maps has several interesting subreflectors, for example, from the category of certain ideal-distributive qosets (including all bounded distributive lattices) to the subcategory of algebraic frames. Generalizing the classical Stone duality for distributive (semi-)lattices, we establish a dual equivalence between the category of ideal-distributive posets with so-called ∧-stable ideal-continuous maps and the category of pairs ( X , B ) where B is a base of a sober topology on X and B is meet-dense in the collection of all compact open sets; morphisms in this category preserve the distinguished bases under inverse images. We also study a self-dual notion of distributivity for qosets, compare it with ideal-distributivity and determine the corresponding Stone representation.

We introduce a special type of order-preserving maps between quasiordered sets, the so-called cut-stable maps. These form the largest morphism class such that the corresponding category of quasiordered sets contains the category of complete lattices and complete homomorphisms as a full reflective subcategory, the reflector being given by the Dedekind-MacNeille completion (alias normal completion or completion by cuts). Suitable restriction of the object class leads to the category of separated quasiordered sets and its full reflective subcategory of completely distributive lattices. Similar reflections are obtained for continuous lattices, algebraic lattices, etc.

A protolocalisation of a homological (resp. semi-abelian) category is a regular full reflective subcategory, whose reflection preserves short exact sequences. We study the closure operator and the torsion theory associated with such a situation. We pay special attention to the fibered, the regular epireflective and the monoreflective cases. We give examples in algebra, topos theory and functional analysis. http://www.sciencedirect.com/science/article/B6V0K-4RN48M4-2/1/8f47e542c1c6a989386a75a9efb80428

As is well-known topological spaces can be related to each other by means of continuous maps. More generally, objects in a category can be related to each other by means of morphisms. There is an analogous relationship between categories via so-called functors. The classical definition of universal maps in the sense of N. Bourbaki [18] corresponds to a categorical one which utilizes a functor. The existence of all universal maps with respect to a given functor F is related to a pair of adjoint functors (G, F), where G (resp. F) is called a left adjoint (resp. right adoint). The relationships between these functors are described by means of natural transformations u and v (which occur as “maps” between functors). Thus, an adjoint situation (G, F, u, v) is obtained. In the first part of this chapter adjoint situations are studied together with some examples. In the second part an important special case of adjoint situations (G, F, u, v) is investigated, namely the case where F is an inclusion functor I from a subcategory A of a category C to C (the notion of inclusion functor corresponds to the notion of inclusion map in classical mathematics). Then G is called a reflector from C to A and A is called reflective. If the morphisms belonging to all universal maps with respect to I are epimorphisms, extremal epimorphisms or bimorphisms, then G is called an epireflector, extremal epireflector or bireflector respectively and we say epireflective, extremal epireflective or bireflective subcategory rather than reflective subcategory. The famous characterization theorem for epireflective (and extremal epireflective) subcategories is proved and the results are applied to bire-flective subconstructs of topological constructs.

The categorySob of sober spaces is a full isomorphismclosed reflective subcategory of the categoryTo of To-spaces (and continuous maps).Sob↪To is characterized by: (i) every universal morphism of the adjunction is aTo-epic embedding, and (ii) everyTo-epic embedding, whose domain is sober, is a homeomorphism.Sob is the epi-reflective hull of the Sierpinski space D inTo. A subspace of a sober space is sober, iff it is “b-closed”. A space is sober, iff it is a b-closed subspace of a power of D.

In a category K with finite limits, the exponentiability of a morphisms s is (rather easily) characterised in terms of K admitting partial products (essentially those of Pasynkov) over s; and that of a monomorphism is characterised in terms of the new concept of a pullback complement (a universal construction of a pullback diagram whose top and right sides are given). Then, characterisations, previously given by the first author for the category Sp of topological spaces, of the notions of totally reflective subcategory and of hereditary factorisation system are shown to be instances of simple results on adjointness and factorisations.

In this paper we obtain a reflective subcategory C of the category FTS of fuzzy topological spaces. The associated reflection β has properties similar to those of the ‘Stone-Cech’ compactification β and, in effect, is an extension of it. We study relations between β and β in particular subcategories of FTS; β is completely determined in the case of fuzzy topological spaces topologically generated.

It is well known that there is a one to one correspondence between idempotent monads in a category and reflective subcategories. In this paper it is examined what replaces the reflective subcategory if the idempotent monad is replaced (a) by a monad and (b) by a symmetric unad. It is shown that in case (a) one obtains the weakly reflective subcategory of objects injective relative to the functor part of the monad. In case (b) one obtains a proto-reflection and it is shown that (for complete categories) the associated orthogonal subcategory is reflective if and only if there exists a free monad associated to the unad.

We show that if S is a cocontinuous triple on a full reflective subcategory of a functor category then the category of S-algebras is again a full reflective subcategory of a functor category. This note should be considered an addendum to [41, and definitions for all of the terminology and concepts we use can be found there. We shall also fix V to be a closed bicomplete category and, in this note, all of the category theory is done relative to V. In [41, we have shown the following: (I) If C is a small category and T is a cocontinuous triple on the functor category VC, then there is a small category C' and a functor /: C -, C' so that (a) T is the triple induced by the adjoint pair (/*, /): VC' VC, where /*: Vc ) Vc is the functor induced by / and /1 is the left adjoint of / *; (b) the adjoint pair (/ *, /1) is tripleable, so that there is an equivalence of categories vC (VC)T, where (VC)T is the category of Talgebras. (II) If C is a small category and T is any triple on Vc, there is a unique cocontinuous triple T on V0 and a map of triples r: T so that, if R: Co )VC denotes the right Yoneda embedding of C into the representable functors of VC, rR is the identity (we shall refer to T as the cocontinuous approximation to T). In this paper, we shall prove the following 1. Theorem. Suppose C is a small category, A is a full reflective subcategory of VC (i.e. the inclusion functor of A to VC has a left adjoint) and S is a cocontinuous triple on A. Then there is a small categoPresented to the Society January 25, 1973; received by the editors August 16, 1972 and, in revised form, August 6, 1973. AMS(MOS) subject classifications (1970). Primary 18C15; Secondary 18E99, 18F 20. Copyright C 1975, American Mathematical Societv

The aim is to construct the free topos generated by a category. Up to equivalence, one may assume that each topos is equipped with a canonical choice of representative subobjects satisfying certain obvious conditions, and one insists that morphisms between toposes preserve these exactly. Between the category of small categories and that of toposes one inserts the category of “dogmas” (Volger's closed logical categories). Dog is known to be equational over Cat, and it is here shown to be a reflective subcategory of Top. A dogma is a category with canonical finite products, it has a Heyting algebra object Ω which admits arbitrary objects as exponents such that, for each object A, the canonical morphism Ω → ΩA has a left adjoint ∃A and a right adjoint ∀A, and it satisfies the usual axiom of extensionality (interpreted in the obvious way). The construction of the topos generated by a dogma follows Volger: its objects are “sets” 1 → ΩA and its morphisms are “relations” between sets which are universally defined and single valued. Inasmuch as a topos consists of sets, a dogma consists of types, and we find here much of traditional type theory in a categorical setting, which incorporates both Frege's process of set abstraction and something like Russell's theory of description.

In [2] Lawvere characterized the category of sets by elementary axioms. using a language, with one sort of variable symbols (mappings) and two unary functions symbols (domain and codomain 1) and one ternary relation. composition. A r >B means f is a map with domain A and codomain B. Schlomiuk [5], presented a method of characterizing the category of topological spaces by using the full subcategory of discrete spaces; the fact that the functor, inclusion, from sets to topological spaces has a left adjoint, together with additional axioms on a certain constant ~the two-point space ({a, b}, {a}, {a, b}). Lawvere also characterized algebraic categories [1], using the special properties of a certain constant, the free object on one generator. and an adjoinmess condition. It is conjectured that what one needs is a previously characterized reflective subcategory cg and finitely many objects at . . .a . determined up to isomorphism, such that the full subcategory on cg u ~ ai, is in the transitive closure of" adequate". We shall adopt this technique i in the present paper to the category of relational systems~