We consider the following two properties of a functor F from a presheaf topos to the category of sets: (a) F preserves connected limits, and (b) the Artin glueing of ā¦
We consider the following two properties of a functor F from a presheaf topos to the category of sets: (a) F preserves connected limits, and (b) the Artin glueing of F is again a presheaf topos. We show that these two properties are in fact equivalent. In the process, we develop a general technique for associating categorical properties of a category obtained by Artin glueing with preservation properties of the functor along which the glueing takes place. We also give a syntactic characterization of those monads on Set whose functor parts have the above properties, and whose units and multiplications are cartesian natural transformations.
One of the most important constructions in topos theory ia that of the category Shv ( A ) of sheaves on a locale (= complete Heyting algebra) A. Normally, the ā¦
One of the most important constructions in topos theory ia that of the category Shv ( A ) of sheaves on a locale (= complete Heyting algebra) A. Normally, the objects of this category are described as āpresheaves on A satisfying a gluing conditionā; but, as Higgs(7) and Fourman and Scott(5) have observed, they may also be regarded as āsets structured with an A -valued equality predicateā (briefly, ā A -valued setsā). From the latter point of view, it is an inessential feature of the situation that every sheaf has a canonical representation as a ācompleteā A -valued set. In this paper, our aim is to investigate those properties which A must have for us to be able to construct a topos of A -valued sets: we shall see that there is one important respect, concerning the relationship between the finitary (propositional) structure and the infinitary (quantifier) structure, in which the usual definition of a locale may be relaxed, and we shall give a number of examples (some of which will be explored more fully in a later paper (8)) to show that this relaxation is potentially useful.
Category theory and related topics of mathematics have been increasingly applied to computer science in recent years. This book contains selected papers from the London Mathematical Society Symposium on the ā¦
Category theory and related topics of mathematics have been increasingly applied to computer science in recent years. This book contains selected papers from the London Mathematical Society Symposium on the subject which was held at the University of Durham. Participants at the conference were leading computer scientists and mathematicians working in the area and this volume reflects the excitement and importance of the meeting. All the papers have been refereed and represent some of the most important and current ideas. Hence this book will be essential to mathematicians and computer scientists working in the applications of category theory.
Proceedings of the London Mathematical SocietyVolume s3-38, Issue 2 p. 237-271 Articles On a Topological Topos P. T. Johnstone, P. T. Johnstone Department of Pure Mathematics, University of Cambridge, 16 ā¦
Proceedings of the London Mathematical SocietyVolume s3-38, Issue 2 p. 237-271 Articles On a Topological Topos P. T. Johnstone, P. T. Johnstone Department of Pure Mathematics, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SBSearch for more papers by this author P. T. Johnstone, P. T. Johnstone Department of Pure Mathematics, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SBSearch for more papers by this author First published: March 1979 https://doi.org/10.1112/plms/s3-38.2.237Citations: 26AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes3-38, Issue2March 1979Pages 237-271 RelatedInformation
Abstract Our principal goal in this chapter is to establish the characterization of exponentiable objects in the 2-category ā¬āā“š /š®, for a general base topos š®, as those which are ālocally ā¦
Abstract Our principal goal in this chapter is to establish the characterization of exponentiable objects in the 2-category ā¬āā“š /š®, for a general base topos š®, as those which are ālocally compactā in an appropriate sense. We have already made some observations about exponentiability in ā¬āā“š /š® in Section B4.3; and we shall assume that the reader of this chapter is familiar with them. In this and the next section we shall take our base topos to be Set, in order to simplify the notation; but virtually all that we do can be extended to a general base topos S with a natural number object, using an appropriate internal finiteness notion in S to interpret the finiteness conditions which appear below.
In a recent paper, Steven Vickers introduced a 'generalized powerdomain' construction, which he called the (lower) bagdomain, for algebraic posets, and argued that it provides a more realistic model than ā¦
In a recent paper, Steven Vickers introduced a 'generalized powerdomain' construction, which he called the (lower) bagdomain, for algebraic posets, and argued that it provides a more realistic model than the powerdomain for the theory of databases (cf. Gunter). The basic idea is that our 'partial information' about a possible database should be specified not by a set of partial records of individuals, but by an indexed family (or, in Vickers' terminology, a bag) of such records: we do not want to be forced to identify two individuals in our database merely because the information that we have about them so far happens to be identical (even though we may, at some later stage, obtain the information that they are in fact the same individual).
It has been known for some time (( 6 ), p. 270; ( 4 ), theorem 9Ā·19) that if is a Boolean topos, then the full subcategory Kf of Kuratowski-finite ā¦
It has been known for some time (( 6 ), p. 270; ( 4 ), theorem 9Ā·19) that if is a Boolean topos, then the full subcategory Kf of Kuratowski-finite objects in is again a topos. For a non-Boolean topos , however, Kf need not be a topos, as can be seen when is the Sierpinski topos (( 1 ), example 7Ā·1); on the other hand, two other full subcategories of , coinciding with Kf when is Boolean, suggest themselves as candidates for a subtopos of finite objects. Of one of these, the category dKf of decidable K -finite objects in , the Main Theorem of ( 1 ) asserts that it is always a (Boolean) topos. The other is the category sKf of -subobjects of K -finite objects. The inclusions dKf ā Kf ā sKf are clear. are clear.
Since the publication of the paper Carboni and Johnstone (1995), we have become aware of two independent errors in it. Although neither of them has any effect on the main ā¦
Since the publication of the paper Carboni and Johnstone (1995), we have become aware of two independent errors in it. Although neither of them has any effect on the main results of the paper, concerning when a category obtained by Artin glueing is a topos, we feel it is appropriate to publish this correction in the hope that it may prevent future readers of Carboni and Johnstone (1995) from being misled. We are grateful to Tom Leinster and to Marek Zawadowski for drawing our attention to the two errors.
In (5), Peter Freyd recently raised the question of whether every Grothendieck topos could be obtained from the topos of sets by means of the two constructions of taking sheaves ā¦
In (5), Peter Freyd recently raised the question of whether every Grothendieck topos could be obtained from the topos of sets by means of the two constructions of taking sheaves on a locale and of taking continuous actions of a topological group (i.e. the topos-theoretic analogues of the set-theorists' techniques of forcing extensions and permutation models). He showed that these two constructions do suffice to within epsilon; provided we allow ourselves the freedom to take exponential varieties (4) (which do not change the internal logic of the topos) we can obtain every Grothendieck topos in this way.
Abstract In this paper we study the weak versions of the fibrewise separation axioms for locales over a base locale, whose introduction has been made possible by the development, in ā¦
Abstract In this paper we study the weak versions of the fibrewise separation axioms for locales over a base locale, whose introduction has been made possible by the development, in a previous paper by the author, of the fibrewise notion of closure for sublocales of locales over a base. We establish the implications which hold between these axioms and the traditional separation axioms for locales, and give counter-examples to show that some of these implications are irreversible.
(1979). Another condition equivalent to de morgan's law. Communications in Algebra: Vol. 7, No. 12, pp. 1309-1312.
(1979). Another condition equivalent to de morgan's law. Communications in Algebra: Vol. 7, No. 12, pp. 1309-1312.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Abstract We provide a new, unified approach to the necessary and sufficient conditions found by Mal'cev (1939) and by Lambek (1951) for embeddability of a semigroup in a group, and ā¦
Abstract We provide a new, unified approach to the necessary and sufficient conditions found by Mal'cev (1939) and by Lambek (1951) for embeddability of a semigroup in a group, and also show that each provides a necessary and sufficient set of conditions for the embeddability of a category in a groupoid. We show that all such conditions, and more besides, may be derived in a uniform way from a particular class of directed graphs which we call quadrangle clubs, and we prove a number of results (extending those of Mal'cev, Lambek, Bush and KrstiÄ) on which families of quadrangle clubs provide sufficient conditions for embeddability.
Some years ago, at the end of a survey article on locales, this chapter expresses an opinion that the theory of uniform locales was ripe for further development. The time ā¦
Some years ago, at the end of a survey article on locales, this chapter expresses an opinion that the theory of uniform locales was ripe for further development. The time thus seems ripe at last for developing a constructive theory of uniform locales ā even though it still cannot be claimed that the theory is as simple as one would have wished. The chapter mentions that J. J. C. Vermeulen has in-depndently been investigating the constructive theory of uniform locales; full details of his results have not reached the author at the time of writing, but he understands that his has covered at least part of the same ground as myself.
We show that there are infinitely many distinct closed classes of colimits (in the sense of the Galois connection induced by commutation of limits and colimits in Set) which are ā¦
We show that there are infinitely many distinct closed classes of colimits (in the sense of the Galois connection induced by commutation of limits and colimits in Set) which are intermediate between the class of pseudo-filtered colimits and that of all (small) colimits. On the other hand, if the corresponding class of limits contains either pullbacks or equalizers, then the class of colimits is contained in that of pseudo-filtered colimits.
We show that there are infinitely many distinct closed classes of colimits (in the sense of the Galois connection induced by commutation of limits and colimits in Set) which are ā¦
We show that there are infinitely many distinct closed classes of colimits (in the sense of the Galois connection induced by commutation of limits and colimits in Set) which are intermediate between the class of pseudo-filtered colimits and that of all (small) colimits. On the other hand, if the corresponding class of limits contains either pullbacks or equalizers, then the class of colimits is contained in that of pseudo-filtered colimits.
We show that the classifying topos for the theory of fields does not satisfy De Morgan's law, and we identify its largest dense De Morgan subtopos as the classifying topos ā¦
We show that the classifying topos for the theory of fields does not satisfy De Morgan's law, and we identify its largest dense De Morgan subtopos as the classifying topos for the theory of fields of nonzero characteristic which are algebraic over their prime fields.
Abstract We provide a new, unified approach to the necessary and sufficient conditions found by Mal'cev (1939) and by Lambek (1951) for embeddability of a semigroup in a group, and ā¦
Abstract We provide a new, unified approach to the necessary and sufficient conditions found by Mal'cev (1939) and by Lambek (1951) for embeddability of a semigroup in a group, and also show that each provides a necessary and sufficient set of conditions for the embeddability of a category in a groupoid. We show that all such conditions, and more besides, may be derived in a uniform way from a particular class of directed graphs which we call quadrangle clubs, and we prove a number of results (extending those of Mal'cev, Lambek, Bush and KrstiÄ) on which families of quadrangle clubs provide sufficient conditions for embeddability.
Since the publication of the paper Carboni and Johnstone (1995), we have become aware of two independent errors in it. Although neither of them has any effect on the main ā¦
Since the publication of the paper Carboni and Johnstone (1995), we have become aware of two independent errors in it. Although neither of them has any effect on the main results of the paper, concerning when a category obtained by Artin glueing is a topos, we feel it is appropriate to publish this correction in the hope that it may prevent future readers of Carboni and Johnstone (1995) from being misled. We are grateful to Tom Leinster and to Marek Zawadowski for drawing our attention to the two errors.
Abstract The definition and some of the basic properties of natural number objects in a topos were introduced in Section A2.5. However, there were some important properties which could not ā¦
Abstract The definition and some of the basic properties of natural number objects in a topos were introduced in Section A2.5. However, there were some important properties which could not be developed there because they required the ability to interpret higher-order logic in a topos. Having developed the techniques for this in the previous chapter, we are now ready to return to the study of natural number objects and complete what we left unfinished in Section A2.5.
Abstract We are now ready to present the definition of an (elementary) topos, which is the central notion of this book: it is a category having all the good āset-likeā ā¦
Abstract We are now ready to present the definition of an (elementary) topos, which is the central notion of this book: it is a category having all the good āset-likeā properties discussed in the previous chapter, except (possibly) for Booleanness. It is a remarkable fact that all these properties are implied by the conjunction of just two of them:
Abstract In this chapter we shall study the remarkable 2-categorical structure of the 2-category BTop/S of bounded S-toposes, where S is a fixed topos. We shall restrict ourselves entirely to ā¦
Abstract In this chapter we shall study the remarkable 2-categorical structure of the 2-category BTop/S of bounded S-toposes, where S is a fixed topos. We shall restrict ourselves entirely to bounded geometric morphisms throughout the chapter: although some of our results can be extended to unbounded morphisms, the structure of Top/S is much more delicate, and the gain in generality does not seem worth the extra trouble that would be involved in formulating all our results so as to apply to unbounded morphisms. In this first section, our aim is to construct finite weighted limits (in the sense of 1.1.5) in BTop/S.
Abstract For many of the topics we wish to study in topos theory, it is impossible to ignore the fact that toposes and geometric morphisms form not just a category ā¦
Abstract For many of the topics we wish to study in topos theory, it is impossible to ignore the fact that toposes and geometric morphisms form not just a category but a 2-category; that is, we have to take account also of the geometric transformations between geometric morphisms, as introduced in A4.1.1(b). Even amongst experienced category-theorists, there is often a good deal of reluctance to take 2-categorical (or, still worse, bicategorical) notions seriously; so the purpose of this section is to review as much of the basic theory as we shall need for subsequent developments.
Abstract In this book, we start from the assumption that the reader is already familiar with the basics of category theory: specifically, we assume that he has already seen the ā¦
Abstract In this book, we start from the assumption that the reader is already familiar with the basics of category theory: specifically, we assume that he has already seen the definitions of the terms category, functor, natural transformation, adjunction, limit and monad. These definitions can, of course, be found in any of the standard textbooks on category theory (for example, [742], [381] or [147]). However, there are a number of conventions on which the standard texts differ, and we need to state our position on these before proceeding further.
Abstract We are interested in this chapter in studying the 2-category Top/S of toposes over a fixed topos S, through the medium of indexed category theory. We recall the precise ā¦
Abstract We are interested in this chapter in studying the 2-category Top/S of toposes over a fixed topos S, through the medium of indexed category theory. We recall the precise definition of this 2-category from Section A4.1: its objects (which we call toposes defined over S, or simply S-toposes) are geometric morphisms p: E ā S with codomain S, its morphisms (q: F ā S) ā (p: E ā S) are pairs (f, a) where f: F ā E is a geometric morphism and a : q = pf a geometric transformation, and its 2-cells (f, a) ā (g, fi) are geometric transformations f ā g compatible in the obvious sense with a and fi. However, we shall almost invariably abuse notation by suppressing any mention of the 2-isomorphism a when specifying 1-cells of Top/S; we shall also tend to suppress the structural morphism p when specifying objects of Top/S, and simply write āE is an S-toposā. (The justification for the latter abuse is contained in Theorem 3.1.2 below.)
Abstract The theory of enriched categories is normally studied in the context of a base category V with a monoidal structure (usually, but not always, symmetric). However, we are primarily ā¦
Abstract The theory of enriched categories is normally studied in the context of a base category V with a monoidal structure (usually, but not always, symmetric). However, we are primarily interested in studying categories enriched in a topos, where the monoidal structure is taken to be the cartesian one (i.e. that induced by finite products), and so we shall save ourselves a bit of extra work by developing the theory only for cartesian monoidal structures. Throughout this section, therefore, our base category V will be assumed to have finite products.
Abstract Our principal goal in this chapter is to establish the characterization of exponentiable objects in the 2-category ā¬āā“š /š®, for a general base topos š®, as those which are ālocally ā¦
Abstract Our principal goal in this chapter is to establish the characterization of exponentiable objects in the 2-category ā¬āā“š /š®, for a general base topos š®, as those which are ālocally compactā in an appropriate sense. We have already made some observations about exponentiability in ā¬āā“š /š® in Section B4.3; and we shall assume that the reader of this chapter is familiar with them. In this and the next section we shall take our base topos to be Set, in order to simplify the notation; but virtually all that we do can be extended to a general base topos S with a natural number object, using an appropriate internal finiteness notion in S to interpret the finiteness conditions which appear below.
Abstract In this chapter, our main aim is to describe the āgeometricā origins of topos theory, by setting up the topos of sheaves on a topological space, and indicating how ā¦
Abstract In this chapter, our main aim is to describe the āgeometricā origins of topos theory, by setting up the topos of sheaves on a topological space, and indicating how topological properties of spaces and continuous maps may be translated into properties of toposes of sheaves and geometric morphisms between them. However, there is a significant āgeneralizationā of the notion of topological space, namely the notion of locale, which is of importance in the topos-theoretic context, and almost everything that can be said about toposes of sheaves on spaces carries over without extra effort to toposes of sheaves on locales. Since the notion of locale may not be familiar to the reader, we therefore begin with a couple of sections setting up the category of locales, and describing its relation to the category of spaces.
Abstract The title of this chapter perhaps needs a few words of explanation. Our general thesis throughout Part C has been that the notion of (Grothendieck) topos is a generalization ā¦
Abstract The title of this chapter perhaps needs a few words of explanation. Our general thesis throughout Part C has been that the notion of (Grothendieck) topos is a generalization of the notion of topological space (or, better, of the notion of locale). In particular, toposes (or geometric morphisms between them) can be ascribed topological properties such as (local) compactness or (local) connectedness, as we saw in Chapter C3; and the theory of exponentiability for toposes, which we explored in Chapter C4, also has close parallels with that for spaces (or locales). Nevertheless, we know from the examples of toposes we have seen (in Section A2.1 and elsewhere) that, even if we restrict our attention to Grothendieck toposes, there are many which (in their origins, at least) do not appear at all āspatialā in character. So one is tempted to ask whether this generalization is perhaps too large to be genuinely useful.
Abstract Our aim in this chapter is to revisit some of the ātopologically inspiredā classes of geometric morphisms which we introduced in Section C1.5, in the context of morphisms between ā¦
Abstract Our aim in this chapter is to revisit some of the ātopologically inspiredā classes of geometric morphisms which we introduced in Section C1.5, in the context of morphisms between localic toposes, and to investigate their properties in the more general context of morphisms between Grothendieck toposes. As in Section C2.4, we shall henceforth interpret āGrothendieck toposā loosely as meaning any topos defined and bounded over a base to pos š® having a natural number object; but we shall often treat š® notationally as if it were the classical category Set of sets, relying on the reader to translate our arguments as required into the language of š®-indexed categories as developed in Part B.
Abstract We saw in Section D1.4 that, for each cartesian (resp. regular, coherent, geometric) theory š, there is a category Cš of the appropriate kind containing a model of š ā¦
Abstract We saw in Section D1.4 that, for each cartesian (resp. regular, coherent, geometric) theory š, there is a category Cš of the appropriate kind containing a model of š which is generic, in the sense defined there. Since we are mainly interested in models which live in toposes, it would be convenient to be able to construct a topos containing a model of a given theory which is generic amongst models in toposes. There are various methods for doing this. One, which was sketched in Section B4.2, relies on the 2-categorical structure of (Grothendieck) toposes, in order to build up (the topos containing) the generic model by an inductive process. Here we shall give a more āpedestrianā, but also more explicit, method, which has the advantage that the explicit description of the generic model which we obtain allows us to establish results about models of a theory by first proving them in the generic case, and then transferring them along inverse image functors.
Abstract Finitary sketches, i.e., sketches with finite-limit and finite-colimit specifications, are proved to be as strong as geometric sketches, i.e., sketches with finite-limit and arbitrary colimit specifications. Categories sketchable by ā¦
Abstract Finitary sketches, i.e., sketches with finite-limit and finite-colimit specifications, are proved to be as strong as geometric sketches, i.e., sketches with finite-limit and arbitrary colimit specifications. Categories sketchable by such sketches are fully characterized in the infinitary first-order logic: they are axiomatizable by Ļ -coherent theories, i.e., basic theories using finite conjunctions, countable disjunctions, and finite quantifications. The latter result is absolute; the equivalence of geometric and finitary sketches requires (in fact, is equivalent to) the non-existence of measurable cardinals.
By a classifying topos for a first-order theory T, we mean a topos<br />E such that, for any topos F, models of T in F correspond exactly to<br />open geometric ā¦
By a classifying topos for a first-order theory T, we mean a topos<br />E such that, for any topos F, models of T in F correspond exactly to<br />open geometric morphisms F ! E. We show that not every (infinitary)<br />first-order theory has a classifying topos in this sense, but we<br />characterize those which do by an appropriate `smallness condition',<br />and we show that every Grothendieck topos arises as the classifying<br />topos of such a theory. We also show that every first-order theory<br /> has a conservative extension to one which possesses<br /> a classifying topos, and we obtain a Heyting-valued completeness<br /> theorem for infinitary first-order logic.
We consider the following two properties of a functor F from a presheaf topos to the category of sets: (a) F preserves connected limits, and (b) the Artin glueing of ā¦
We consider the following two properties of a functor F from a presheaf topos to the category of sets: (a) F preserves connected limits, and (b) the Artin glueing of F is again a presheaf topos. We show that these two properties are in fact equivalent. In the process, we develop a general technique for associating categorical properties of a category obtained by Artin glueing with preservation properties of the functor along which the glueing takes place. We also give a syntactic characterization of those monads on Set whose functor parts have the above properties, and whose units and multiplications are cartesian natural transformations.
In a recent paper, Steven Vickers introduced a 'generalized powerdomain' construction, which he called the (lower) bagdomain, for algebraic posets, and argued that it provides a more realistic model than ā¦
In a recent paper, Steven Vickers introduced a 'generalized powerdomain' construction, which he called the (lower) bagdomain, for algebraic posets, and argued that it provides a more realistic model than the powerdomain for the theory of databases (cf. Gunter). The basic idea is that our 'partial information' about a possible database should be specified not by a set of partial records of individuals, but by an indexed family (or, in Vickers' terminology, a bag) of such records: we do not want to be forced to identify two individuals in our database merely because the information that we have about them so far happens to be identical (even though we may, at some later stage, obtain the information that they are in fact the same individual).
Category theory and related topics of mathematics have been increasingly applied to computer science in recent years. This book contains selected papers from the London Mathematical Society Symposium on the ā¦
Category theory and related topics of mathematics have been increasingly applied to computer science in recent years. This book contains selected papers from the London Mathematical Society Symposium on the subject which was held at the University of Durham. Participants at the conference were leading computer scientists and mathematicians working in the area and this volume reflects the excitement and importance of the meeting. All the papers have been refereed and represent some of the most important and current ideas. Hence this book will be essential to mathematicians and computer scientists working in the applications of category theory.
Abstract In this paper we study the weak versions of the fibrewise separation axioms for locales over a base locale, whose introduction has been made possible by the development, in ā¦
Abstract In this paper we study the weak versions of the fibrewise separation axioms for locales over a base locale, whose introduction has been made possible by the development, in a previous paper by the author, of the fibrewise notion of closure for sublocales of locales over a base. We establish the implications which hold between these axioms and the traditional separation axioms for locales, and give counter-examples to show that some of these implications are irreversible.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
It has been known for some time (( 6 ), p. 270; ( 4 ), theorem 9Ā·19) that if is a Boolean topos, then the full subcategory Kf of Kuratowski-finite ā¦
It has been known for some time (( 6 ), p. 270; ( 4 ), theorem 9Ā·19) that if is a Boolean topos, then the full subcategory Kf of Kuratowski-finite objects in is again a topos. For a non-Boolean topos , however, Kf need not be a topos, as can be seen when is the Sierpinski topos (( 1 ), example 7Ā·1); on the other hand, two other full subcategories of , coinciding with Kf when is Boolean, suggest themselves as candidates for a subtopos of finite objects. Of one of these, the category dKf of decidable K -finite objects in , the Main Theorem of ( 1 ) asserts that it is always a (Boolean) topos. The other is the category sKf of -subobjects of K -finite objects. The inclusions dKf ā Kf ā sKf are clear. are clear.
Augustfunctor oĆ®3/t\n if; also, we shall say that Fis a space-preserving functor from 38 toĀ«".It turns out that Spec can be inverted on surprisingly large subcategories of Ā”f.At this point, ā¦
Augustfunctor oĆ®3/t\n if; also, we shall say that Fis a space-preserving functor from 38 toĀ«".It turns out that Spec can be inverted on surprisingly large subcategories of Ā”f.At this point, the results of Ā§7 should be read as part of this introduction.In fact, after looking at Ā§1, Ā§2 and the results of Ā§7, the reader can continue and finish the paper (except for the proof of Theorem 9 in Ā§16), and then go back to the technical Ā§ Ā§3-6 if he wishes.We note that Ā§16 uses the functorial nature of our constructions to characterize the underlying spaces of preschemes and schemes.
Various aspects of the prime spectrum of a distributive continuous lattice have been discussed extensively in Hofmann-Lawson [ 7 ]. This note presents a perhaps optimally direct and self-contained proof ā¦
Various aspects of the prime spectrum of a distributive continuous lattice have been discussed extensively in Hofmann-Lawson [ 7 ]. This note presents a perhaps optimally direct and self-contained proof of one of the central results in [ 7 ] (Theorem 9.6), the duality between distributive continuous lattices and locally compact sober spaces, and then shows how the familiar dualities of complete atomic Boolean algebras and bounded distributive lattices derive from it, as well as a new duality for all continuous lattices. As a biproduct, we also obtain a characterization of the topologies of compact Hausdorff spaces. Our approach, somewhat differently from [ 7 ], takes the open prime filters rather than the prime elements as the points of the dual space. This appears to have conceptual advantages since filters enter the discussion naturally, besides being a well-established tool in many similar situations.
We consider the following two properties of a functor F from a presheaf topos to the category of sets: (a) F preserves connected limits, and (b) the Artin glueing of ā¦
We consider the following two properties of a functor F from a presheaf topos to the category of sets: (a) F preserves connected limits, and (b) the Artin glueing of F is again a presheaf topos. We show that these two properties are in fact equivalent. In the process, we develop a general technique for associating categorical properties of a category obtained by Artin glueing with preservation properties of the functor along which the glueing takes place. We also give a syntactic characterization of those monads on Set whose functor parts have the above properties, and whose units and multiplications are cartesian natural transformations.
(1979). Another condition equivalent to de morgan's law. Communications in Algebra: Vol. 7, No. 12, pp. 1309-1312.
(1979). Another condition equivalent to de morgan's law. Communications in Algebra: Vol. 7, No. 12, pp. 1309-1312.
In a recent paper, Steven Vickers introduced a 'generalized powerdomain' construction, which he called the (lower) bagdomain, for algebraic posets, and argued that it provides a more realistic model than ā¦
In a recent paper, Steven Vickers introduced a 'generalized powerdomain' construction, which he called the (lower) bagdomain, for algebraic posets, and argued that it provides a more realistic model than the powerdomain for the theory of databases (cf. Gunter). The basic idea is that our 'partial information' about a possible database should be specified not by a set of partial records of individuals, but by an indexed family (or, in Vickers' terminology, a bag) of such records: we do not want to be forced to identify two individuals in our database merely because the information that we have about them so far happens to be identical (even though we may, at some later stage, obtain the information that they are in fact the same individual).
The diagrams for symmetric monoidal closed categories proved commutative by Mac Lane and the author in [19] were diagrams of (generalized) natural transformations. In order to understand the connexion between ā¦
The diagrams for symmetric monoidal closed categories proved commutative by Mac Lane and the author in [19] were diagrams of (generalized) natural transformations. In order to understand the connexion between these results and free models for the structure, the author introduced in [13] and [14] the notion of club, which was further developed in [15] and applied later to other coherence problems in [16] and elsewhere.The club idea seemed to apply to several diverse kinds of structure on a category, but still to only a restricted number of kinds. In an attempt to understand its natural limits, the author worked out a general notion of āclubā, as a monad with certain properties, not necessarily on Cat now, but on any category with finite limits. A brief account of this was included in the 1978 Seminar Report [17], but was never published; the author doubted that there were enough examples to make it of general interest.During 1990 and 1991, however, we were fortunate to have with our research team at Sydney Robin Cockett, who was engaged in applying category theory to computer science. In lectures to our seminar he called attention to certain kinds of monads involved with data types, which have special properties : he was calling them shape monads, but in fact they are precisely examples of clubs in the abstract sense above.
The following paper is a study of abstract algebras qua abstract algebras. As no vocabulary suitable for this purpose is current, I have been forced to use a number of ā¦
The following paper is a study of abstract algebras qua abstract algebras. As no vocabulary suitable for this purpose is current, I have been forced to use a number of new terms, and extend the meaning of some accepted ones.
F. Hausdorff and D. Montgomery showed that a subspace of a completely metrizable space is developable if and only if it is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript sigma"> ā¦
F. Hausdorff and D. Montgomery showed that a subspace of a completely metrizable space is developable if and only if it is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript sigma"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>Ļ<!-- Ļ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{F_\sigma }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript delta"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>Ī“<!-- Ī“ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{G_\delta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This extends to arbitrary metrizable locales when "<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript sigma"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>Ļ<!-- Ļ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{F_\sigma }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>" and "<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript delta"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>Ī“<!-- Ī“ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{G_\delta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>" are taken in the localic sense (countable join of closed, resp. meet of open, sublocales). In any locale, the developable sublocales are exactly the complemented elements of the lattice of sublocales. The main further results of this paper concern the strictly pointless relative theory, which exists becauseāalways in metrizable localesā there exist nonzero pointless-absolute <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G prime Subscript delta Baseline s"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>Ī“<!-- Ī“ --></mml:mi> </mml:msub> </mml:mrow> <mml:mo>ā²</mml:mo> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>s</mml:mtext> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{G_\delta }ā{\text {s}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript delta"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>Ī“<!-- Ī“ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{G_\delta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in every pointless extension. For instance, the pointless part <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pl left-parenthesis bold upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>pl</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {pl}}({\mathbf {R}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the real line is characterized as the only nonzero zero-dimensional separable metrizable pointless-absolute <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript delta"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>Ī“<!-- Ī“ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{G_\delta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. There is no nonzero pointless-absolute <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript sigma"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>Ļ<!-- Ļ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{F_\sigma }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The pointless part of any metrizable space is, if not zero, second category, i.e. not a countable join of nowhere dense sublocales.