Johnstone, P.T., Topos Theory, L.M.S. Monographs, vol. 10, Academic Press, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Proof: Since Top is a full sub 2 category of Top S , and Top S is extensive, it will su#ce to show that Top S is closed in Top S under finite coproducts and coproduct summands. Top S is closed under finite coproducts by Section 2.2 of [31] and coproduct summands by Proposition 4. 47 of [23] # Corollary 4.6 The 2 category LTop of locally connected toposes bounded over an elementary topos S is extensive. Proof: Once again, it su#ces to observe that LTop S is closed in Top S (and so in Top S ) under finite coproducts and coproduct summands. # 11 5 The van Kampen theorems for ....

P.T. Johnstone, Topos Theory, Academic Press, London, 1977.

....p (# Y ) is compact and if p is perfect, then p (# Y ) is compact and regular. He also establishes the converse to each of these statements in the case where T is a TD space in the sense of [A] i.e. points of T are locally closed. For more on internal locales, the reader is referred to [J1], J2] and [JT] 4. Exponentiability of Perfect Maps In this section, we recall the characterization of exponentiable morphisms in Top presented in [N1,N2] and apply it to obtain the exponentiability of perfect maps. We conclude with a description of the relevant exponentials so that the ....

P. T. Johnstone, Topos Theory, Academic Press, 1977.

.... identi ed in practice [19, 13] This brings into consideration the desirability of assuming that the base (elementary) topos S satis es an axiom of (small) stack completions , which was suggested by Lawvere in 1974 and which is known to hold at least of all Grothendieck toposes S , as shown in [16, 23]. We prove in section 4 (Galois groupoids and Galois toposes) that there is indeed an equivalence (not just a Morita equivalence) between the Galois groupoid of automorphisms of a universal cover of a locally connected and locally simply connected topos e : E S , and the Galois groupoid of the ....

.... cover; this is in line with the view advocated by Grothendieck [20] that rather than xing a single base point, one ought work with a suitable bag of points ( paquet des points ) Pointed connected Galois toposes over S have been investigated by Moerdijk [27] following Grothendieck [18] see also [23]) We obtain here characterization theorems in a manner parallel to [27] in particular we show that Galois toposes in our sense (and which correspond, modulo the intervention of locales in the subject, to the multi Galois toposes of Grothendieck [19] are precisely the classifying toposes of ....

[Article contains additional citation context not shown here]

P. T. Johnstone, Topos Theory, Academic Press, 1977.

....FIX= and FIX correspond in a precise way to FIX categories and FIX hyperdoctrines respectively. We shall now define the categorical structure which corresponds to FIX = such structures will be called FIX categories with attributes. Useful background information can be found in [Pit87] and [Joh77]. Definition 9.2.1 Let C be a category with attributes where for each object X of C, Fib(X) is regarded as a category with objects the fibrations over X and 142 morphisms given by Fib(X) F; F = C=X( F ; F 0 ) Then C is a FIX categorywith attributes if it satisfies the following ....

P.T. Johnstone. Topos Theory. Academic Press, 1977.

....this end, we shall examine here for morphisms of bundles the notion of descent equivalence, which was introduced in the rst author s Ph.D. thesis [3] and study its properties. We formulate this notion in the (essentially equivalent) language of internal categories and of indexed categories (see [5,6,7]) rather than that of brations, making extensive use of some of the results of [5] which we recall here in sucient detail. After some preliminary observations concerning descent equivalences and their comparison with e ective descent morphisms, in Theorem 1 we give a somewhat surprising ....

....descent equivalences whose domain or codomain is given by an e ective descent morphism. Acknowledgement: We thank the anonymous referee for a suggestion which led to an improved exposition of the rst part of the proof of Theorem 1. 1. Internal categories. Recall that an internal category D (cf. [6]) in C is given by a diagram D 2 D 1 D 0 m e in C, which satis es I1. de = 1 D 0 = ce; I2. dm = d 2 ; cm = c 1 ; I3. m(1D1 m) m(m 1 D1 ) I4. m 1 D 1 ; ed = 1 D 1 = m ec; 1 D 1 ; where D 2 ; 1 ; 2 are given by the following pullback diagram in C: between two ....

P.T. Johnstone, Topos Theory, Academic Press, New York, 1977.

....topoi by topological groupoids Carsten Butz and Ieke Moerdijk, Utrecht Abstract It is shown that every topos with enough points is equivalent to the classifying topos of a topological groupoid. 1 Definitions and statement of the result We recall some standard definitions ([1, 5, 9]) A topos is a category E which is equivalent to the category of sheaves of sets on a (small) site. Equivalently, E is a topos iff it satisfies the Giraud axioms ( 1] p. 303) The category of sets S is a topos, and plays a role analogous to that of the one point space in topology. In ....

....points. We recall the definition of the space X = X E from [2] x2, and show that it is part of a groupoid G X. First, although the collection of all points of E is in general a proper class, there will always be a set of points p for which the functors p are already jointly conservative [5], Corollary 7.17. Fix such a set, and call its members small points of E. Next, let S be an object of E with the property that the subobjects of powers of S, i.e. all sheaves B ae S for n 0, together generate E. For example, S can be the disjoint sum of all the objects in some small site for ....

P. T. Johnstone. Topos Theory. Academic Press, New York, 1977.

....However, the exposition should serve more as a reminder and a statement about notation than a first introduction to the concepts just listed. For that a much longer text is needed, and we refer the reader to the following list of core references. Fourman Scott 77] Mac Lane Moerdijk 92] Johnstone 77] Troelstra van Dalen 88] Wyler 91] Fourman 74] and also [Fourman 77] Rosolini 80] Ambler 92] and [Nawaz 85] The intention behind the notion of an m set is to model sets in a constructive universe with truth values in . Thus operations like equality (between members of sets) and ....

....on or internally, using logical connectives instead. We use the logic of forcing over , that is, the logic in h( For a good reference to forcing over a cHa, see [Troelstra van Dalen 88] volume II, and also in general for internal logic in a topos, for instance [Mac Lane Moerdijk 92] Johnstone 77] or [Fourman 74] 69 Definition 4.18 For an atomic formula with value I ] P c [2 we define q k if q p. For composite formulae we define forcing as follows (taken from [Troelstra van Dalen 88] p. 720 with minor typographical changes) By A(q) we denote the set of sections of the sheaf A ....

Johnstone, P.T., Topos Theory, L.M.S. Monographs, vol. 10, Academic Press, 1977.

....;g; fn 2 INj B 6= g) KE (A; B) B; B) KM (A; B) A Now clearly, in S , the maps KE and (A; B) 7 ( IN) A; B) are isomorphic, whence kE is internally given as w :u ) w, u being the point of Omega which classifies the inclusion U 1. By the definition of open subtoposes ([Joh76]) Eff is the open subtopos determined by U . Likewise, the map KM is isomorphic (in S ) to (A; B) 7 ( IN) A; B) so kM is internally the topology w :u w, which is the complement (in the lattice of internal topologies) of kE . This proves statement 1. For the second statement, since ....

P.T. Johnstone, Topos Theory, Academic Press 1976

....and finite limits, satisfying certain other conditions that (i) make the colimits and limits behave like those in S, and (ii) ensure that it can be presented by a small (in the set theoretic sense) theory presentation. The conditions are exactly those set out for Giraud s theorem in Johnstone [4]. One aspect of Giraud frames being similar to S is that constructive set theory can be interpreted in them, and we can talk about models of a theory T in a Giraud frame in fact, a model of Tin S[T 0 ] is just an interpretation of T in T 0 . This is in accordance with what we originally ....

Peter Johnstone, Topos theory, Academic Press, London, 1977.

....i # I , C i in C. The following calculation shows that # weakly represents Prf . Prf (X) Prf ( # i#I C i ) # = # i#I Prf (C i ) ## # i#I G(C i , #) # = G( # i#I C i , #) G(X, #) 5.2.1. G sets. For any group G it is possible to consider the presheaf topos Sets G of G sets [12, 18, 1]. It is well known that the indecomposable G sets are the nonempty ones with only one orbit and that every G set is a small coproduct of these. Moreover, every indecomposable is isomorphic to a G set given by a coset space in G (see Proposition 4 in Section 3 of Chapter 1 in [1] By Proposition ....

....not a set, a contradiction. Notice that this also shows that even for the restricted class of presheaf toposes, the condition that Prf takes values in Sets does not imply the existence of a weak proof classifier. Actually, it is possible to extend the characterization of boolean presheaf toposes [12, 10, 18] as follows. In [20] it is shown that for any small category C, the presheaf topos Sets C op has a weak proof classifier if and only if C is a groupoid. We still do not know of a non boolean topos with a weak proof classifier. 7. Generic monos In this short section we identify the structure ....

P. T. Johnstone. Topos theory. Academic Press, 1977.

....cartesian closed category C , all existing sums are universal. This follows, as for every morphism u : I J the functor u : C =J C =I has a right adjoint Q u and thus preserves arbitrary coproducts. If, moreover, C is a topos, then all sums existing are disjoint (a proof can be found in [John77]) 4.2. Extensive Fibrations. We are going to define extensivity, universality and disjointness for fibrations with internal coproducts. We do this in a way such that FinSets(C ) has universal disjoint coproducts or is extensive as a fibration if and only if C has finite universal disjoint ....

....of examples is the most general possible. Theorem 9 (Diaconescu) A geometric morphism F a U : E S between toposes is bounded if and only if there exists an internal site (C ; J) in S such that F a U is equivalent to the canonical geometric morphism from Sh(C ; J) to S in Top=S. Proof. see [John77] Thus, a topos over Sets is bounded if and only if it is cocomplete and has a small generating family. For the if part take the coproduct of a generating family as a bound, for the only if part use Theorem 9) Nevertheless, neither cocompleteness nor the existence of a small generating ....

P.T. Johnstone. Topos Theory. Academic Press, London, 1977

....= a U 1 X U ; which in presence of the law of excluded middle is X 1 tX 0 = X 1 again. Now suppose our category S is a non Boolean elementary topos; then, the coproduct above still exists, and is different from X 1 it is in fact X , the partial map classifier of X (see e.g. [9]) X is characterized by a universal property: morphisms from any Y to it are in one to one correspondence with partial maps Y X , i.e. morphisms from subobjects of Y to X . Note that X 1 has a similar universal property, but with all subobjects of Y replaced by the complemented ones ....

....strict. Since surjective geometric morphisms reflect isomorphisms, i is strict iff the topology that forces it to be iso is the largest one or, the sublocale Omega i of Omega corresponding to this topology is degenerate. There is an explicit formula for such forcing topology (see e.g. [9]) which gives an explicit description of that sublocale: OE 2 Omega i iff 8 x2X OE : OE : x 2 X 0 ) hence X 0 ae X is strict iff 8 OE2 Omega OE :8 x2X OE : OE : x 2 X 0 ) Comparing this with our expression for I reveals the following Proposition 4 I is isomorphic to the set of ....

P. T. Johnstone, Topos Theory, Academic Press 1977.

....9h 2 P (h = sup(p) holds in the internal logic of E , where h = sup(p) stands for (8j 2 e (a) 1 (i) p(j) h) 8k 2 P (8j 2 e (a) 1 (i) p(j) k) h k) An analogous characterisation is available for S completeness replacing suprema by in ma. Proof. Topos semantics [13] applied in this case for arbitrary a: J I in S shows that the validity of the formula 8i 2 e (I) 8p 2 P e (a) 1 (i) 9h 2 P (h = sup(p) is equivalent to the requirement that E(d; P ) have a left adjoint for all d which are pullbacks of e (a) along some morphism in E satisfying ....

....which the connected components of the inverse images of opens of X form a base for Y . The corresponding notion of an S spread over a topos E bounded over S was introduced in [7] as that of a (bounded) geometric morphism : F E with a generating family F (E) of F over E in the sense of [13], which is S de nable in the sense of [1] It is shown in [7] that a geometric morphism : F E over S is a spread if and only if it is localic and 16 BUNGE FUNK JIBLADZE STREICHER furthermore the morphism ( f ( S ) F ) join generates the frame ( F ) We now review ....

P. T. Johnstone, Topos Theory, Academic Press (1977).

.... [87] For more on realizability toposes and applications of these ideas to the semantics of programming languages, logics and computation see for example [40, 43, 100, 92, 66, 69, 85, 84, 88, 68, 67, 65, 11, 12] Although we are not going to be particularly interested in the class of pretoposes [74, 45] as such, we are going to encounter them (apart from dealing with toposes) from time to time, so we might as well define them. Definition 2.5.4. A pretopos is an exact lextensive category. Chapter 3 Completions The main purpose of this chapter is to introduce four free constructions that are ....

....classifier # : 1 # ## is exponentiating if# is. Proposition 4.3.3. In a category with finite limits and an exponentiating strongsubobject classifier every regular equivalence relation is a kernel pair. Proof. This is the argument to prove that in toposes every equivalence relation is e#ective [45, 51]. For any regular equivalence relation e = #e 0 , e 1 # on X consider its classifying map # e : X X # ## It is then proved that e 0 , e 1 is the kernel pair of the transposition # e : X ## X . So we easily obtain the following corollary. Corollary 4.3.4. Let C be a lextensive category ....

[Article contains additional citation context not shown here]

P. T. Johnstone. Topos theory. Academic Press, 1977.

.... It is of interest to note that well pointed topoi arise independently, both as models of Lawvere s categorical set theory [18] and as models of a certain well known fragment Z Gamma of Zermelo Fraenkel set theory, called variously bounded (or weak) Zermelo set theory or Mac Lane set theory ([19, 11, 21] and the references there) The classical part of the foregoing strong completeness theorem can therefore also be stated in terms of models (of theories) in models of Z Gamma : a sentence in the language of a classical theory T is provable if it is true in every T model in every model of Z ....

P. T. Johnstone. Topos Theory. Academic Press, London, 1977.

....Germany, by a DAAD NUFFIC grant, by a scholarship of the University of Utrecht, and by a grant from a project funded by the Netherlands Organization for Scientific Research NWO. 2 Review of Sheaf Models We assume that the reader is familiar with (Grothendieck) toposes, see for example [9] [6] or [8] For sheaves over topological spaces we refer as well to [10] In particular, we assume that the reader has seen the basic definition of a site (C ; J) sheaves on a site and maps between sheaves. These data together yield the Grothendieck topos Sh(C ; J) Maps between Grothendieck toposes ....

P. T. Johnstone. Topos Theory. Academic Press, New York, 1977.

....Equality is part of the geometric logic, but inequality is not (because there is no negation) Nonetheless, certain decidable sets come equipped with inequality, a relation complementary to equality two good examples are N and Q. Finiteness is as remarked above Kuratowski finiteness [5]: X is Kuratowski finite iff the free semilattice F X has an element T such that x T for every x. This notion can sometimes behave surprisingly from the point of view of classical mathematics: for instance, subsets of finite sets, or intersections of finite subsets, need not themselves be ....

....has been lacking and we take the opportunity to present one here. The reader who is more interested in the localic account of completion is invited to skip this section and admire the audacity of the subsequent treatment. The notion of geometric theory can be found in standard texts such as [5] and [12] it is a firstorder, many sorted theory in which the axioms are of the form f S y, where f and y are geometric formulae (the connectives allowed are finite conjunction, arbitrary disjunction, equality and existential quantification) all of whose free variables are in the finite set S. ....

[Article contains additional citation context not shown here]

P.T. Johnstone, Topos Theory (Academic Press, 1977).

No context found.

Johnstone, P.T., Topos Theory, L.M.S. Monographs, vol. 10, Academic Press, 1977.

No context found.

P.T. Johnstone. Topos Theory. Academic Press, 1977.

No context found.

P.T. Johnstone, Topos Theory, Academic Press (1977). 13

No context found.

P.T. Johnstone. Topos Theory, volume 10 of L.M.S. Monographs. Academic Press, 1977.

No context found.

P.T. Johnstone. Topos Theory. Academic Press, 1977.

No context found.

P. Johnstone. Topos Theory. Academic Press, London, 1977.

No context found.

P. T. Johnstone, Topos Theory, Academic Press, 1977.

No context found.

P.T. Johnstone, Topos Theory (Academic Press, London, 1977).

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC