Serre classes for toposes

Adelman, Murray; Johnstone, Peter

doi:10.1017/s0004972700005086

Cited by 6 publications

(8 citation statements)

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“…(ii) Given a filter O on a locale A, we can modify the above example if we set 3PI = A 1 as before, but i>\-jf iff A the remaining structure being defined as in (i). Because <t > is a filter, it is not hard to see that \-j is a preorder on A 1 , and the conditions of 1-2 are again easy to verify. More generally, if O is a filter on any tripos SP (i.e.…”

Section: J M E H Y L a N D P T J O H N S T O N E And A M mentioning

confidence: 98%

“…If we think of subsets of A as 'propositions', then elements of A are ' proofs' of these propositions, with a ep read as ' a proves p'.) Now for any set / let 0*1 be PA 1 , with | -7 denned by (j) h-7 ir if and only if there is aeA with ae(<f>(i)->ft{i)) for all iel. Given/:/ > J, we define 0>f to be composition with / , whilst V/ sends $ e PA 1 to the map…”

Section: J M E H Y L a N D P T J O H N S T O N E And A M mentioning

confidence: 99%

“…Suppose 0 is ^-standard. Then the Heyting algebra A = "L/-W-is complete, and the maps q 1 :0I-+A 1 and e 1 : A 1 > 01 define a geometric morphism from the canonical tripos of A to 0.…”

Section: Sub ( 1 X 7 )mentioning

confidence: 99%

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Tripos theory

Hyland

Johnstone

Pitts

1980

Math. Proc. Camb. Phil. Soc.

129

144

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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 ‘presheaves on A satisfying a gluing condition’; but, as Higgs(7) and Fourman and Scott(5) have observed, they may also be regarded as ‘sets structured with an A-valued equality predicate’ (briefly, ‘A-valued sets’). From the latter point of view, it is an inessential feature of the situation that every sheaf has a canonical representation as a ‘complete’ 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.

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Section: J M E H Y L a N D P T J O H N S T O N E And A M mentioning

confidence: 98%

Section: J M E H Y L a N D P T J O H N S T O N E And A M mentioning

confidence: 99%

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Tripos theory

Hyland

Johnstone

Pitts

1980

Math. Proc. Camb. Phil. Soc.

129

144

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“…Specifically, we shall consider the idea of the 'germ' of a smooth manifold M at a point p. Clearly, given M and p, it would be useful to have a 'space' representing the notion of the germ at p of a smooth map (and the study of Synthetic Differential Geometry was largely motivated by the desire to have such 'spaces' available). If we wish this 'space' to be a topos, it is no use trying Loc p (Sh(M)), since by 3.8 this contains no information about M or p. Another possibility which suggests itself is to take the filterpower Sh(M)/jV p> where M p is the filter of open neighbourhoods of p in M (regarded as a filter of subobjects of 1 in Sh(M)); this is always a topos, but never a Grothendieck topos unless p is an isolated point [2]. However, we may define another point p of MlyM, corresponding to the flat K continuous functor on Mf/M which sends (N -> M) to the set of germs at p of (smooth) sections of n. Of course, evaluation at p defines a natural transformation from this functor to the previous one, and hence a 2-cell p => pc in £op.…”

Section: Theoremmentioning

confidence: 99%

“…x the germ at p of the identity map on U, is co-initial in Nbd(p); so we may replace the inverse limit in the statement of 3.7 by an inverse limit over this subcategory. It is now not difficult to see, using the techniques of [25], that a site for M p may be obtained by taking the category whose objects are maps (/: N^>M) in Mf, and whose morphisms {f x : N^M)-*(f 2 : N 2 -+M) are equivalence classes of smooth maps f\\V)-*N 2 over M, where U is an open neighbourhood of p in M (the equivalence relation being the obvious one which identifies such a map with its restriction iof^\V)…”

Section: Perhaps a Better Way Of Representing A Manifold M By A Toposmentioning

confidence: 99%

Local Maps of Toposes

Johnstone

Moerdijk

1989

Proceedings of the London Mathematical Society

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IntroductionThe notion of local (Grothendieck) topos was introduced by Grothendieck and Verdier in SGA 4 [11, VI, 8.4]; a Grothendieck topos % is said to be local if the global sections functor <£-» $f has a right adjoint as well as its usual left adjoint, i.e. if it is the inverse image part of a geometric morphism 5^-> %, as well as the direct image of the unique morphism %^>& > . The example which the authors of [11] particularly had in mind was the topos of sheaves on a space (such as the Zariski spectrum of a local ring) containing a point whose only neighbourhood is the whole space. (Such points have been called focal by Freyd [9]; the similarity between 'focal' and 'local' is deliberate.) They also showed that, for a locally coherent topos %, there is a process of 'localization' which mirrors the passage from the Zariski spectrum of a ring to the spectrum of one of its localizations.Since then, the notion of local topos has not attracted much attention, apart from the occasional passing reference in papers about other things. We believe that the time is now ripe, if not overdue, for a more detailed investigation of the notion, for a number of reasons which we shall now describe briefly.In the first place, the property of being local, like many of the properties of toposes studied in [11], is not so much a property of the topos % itself as of the geometric morphism %-> tf; thus there is scope for studying toposes which are local over some base topos other than the classical topos of sets, or (to change our terminology slightly) studying local maps between toposes. (Note: throughout this paper we shall restrict ourselves to maps of toposes (i.e. geometric morphisms) which are bounded in the sense of [13, Definition 4.43]; we make this restriction in order to be assured of the existence of all the pullbacks which we shall wish to consider, but many of our results are not crucially dependent on it. To emphasize the unimportance of the restriction, we shall denote the 2-category of toposes and bounded geometric morphisms simply by Sop.) Thus, in addition to the properties of local toposes studied in [11], there is scope for studying properties of local maps (stability under composition, pullback, etc.) which cannot even be conveniently formulated in terms of the Grothendieck-Verdier definition.Secondly, the restriction to the locally coherent case in the construction of localizations in [11] seems to have been a matter of necessity rather than desirability (despite the remarks on p. 321 of [11]): Grothendieck and Verdier simply did not have techniques available for handling filtered inverse limits in the non-locally-coherent case. With the more powerful techniques available now, we shall show (in § 3 of this paper) that the notion of localization makes perfectly good sense for arbitrary (bounded) pointed toposes over a base, and we shall also The research of the second author was supported by a Huygens fellowship of the Z.W.O. A.M.S. (1980) subject classification: 18B25. Proc. London Math. Soc. (3) 58 (1989) 28...

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