Site characterizations for geometric invariants of toposes

📝 Abstract

We discuss the problem of characterizing the property of a Grothendieck topos to satisfy a given ‘geometric’ invariant as a property of its sites of definition, and indicate a set of general techniques for establishing such criteria. We then apply our methodologies to specific invariants, notably including the property of a Grothendieck topos to be localic (resp. atomic, locally connected, equivalent to a presheaf topos), obtaining explicit site characterizations for them.

💡 Analysis

We discuss the problem of characterizing the property of a Grothendieck topos to satisfy a given ‘geometric’ invariant as a property of its sites of definition, and indicate a set of general techniques for establishing such criteria. We then apply our methodologies to specific invariants, notably including the property of a Grothendieck topos to be localic (resp. atomic, locally connected, equivalent to a presheaf topos), obtaining explicit site characterizations for them.

📄 Content

arXiv:1112.2542v1 [math.CT] 12 Dec 2011 Site characterizations for geometric invariants of toposes Olivia Caramello DPMMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK O.Caramello@dpmms.cam.ac.uk December 12, 2011 Abstract We discuss the problem of characterizing the property of a Grothen- dieck topos to satisfy a given ‘geometric’ invariant as a property of its sites of definition, and indicate a set of general techniques for estab- lishing such criteria. We then apply our methodologies to specific invariants, notably including the property of a Grothendieck topos to be localic (resp. atomic, locally connected, equivalent to a presheaf topos), obtaining explicit site characterizations for them. 1 Introduction In [8] we advocate that Grothendieck toposes can effectively act as unifying spaces in Mathematics serving as ‘bridges’ for transferring information be- tween distinct mathematical theories. The transfer of information between theories classified by the same topos, represented by different sites of defi- nition (C, J) and (C′, J′) of their classifying topos, takes place by expressing topos-theoretic invariant properties (resp. constructions) on the topos in terms of properties (resp. constructions) of its two different sites of defini- tion. For each invariant, we thus have a ‘bridge’ 1 Sh(C, J) ≃Sh(C′, J′) % ❯ ❘ ◆ ❑ (C, J) 3 t ♣ ♠ ✐ (C′, J′) whose central part is the equivalence of toposes Sh(C, J) ≃Sh(C′, J′) and whose ‘legs’, represented by the dashed arrows, are given by the site charac- terizations corresponding to the given invariant. In light of this view, it becomes essential to be able to establish, for relevant classes of topos-theoretic invariant properties I, criteria of the kind ‘a topos Sh(C, J) satisfies the property I if and only if the site (C, J) satisfies a property P(C,J) (explicitly written in the language of the site (C, J))’, holding for any site (C, J) or for appropriate classes of sites. Indeed, any such criterion gives us the possibility, in presence of any Morita-equivalence, to operate an automatic transfer of information between the two theories, leading to concrete mathematical results of various nature. Particular cases of these general results have already been applied by the author in several different contexts (cf. for example [5] and [7]), and in fact the primary aim of this paper is to make a systematic investigation of the problem of obtaining site characterizations which can be conveniently applied in connection to our general philosophy ‘toposes as bridges’. Throughout the past years, site characterizations have been established for several important geometric invariants of toposes, including the ones con- sidered in the present paper (cf. in particular [2] and [3], and [9] as a general reference); however, all of these characterizations are of form ‘A Grothendieck topos satisfies an invariant I if and only if there exists a site of definition (C, J) of it satisfying a certain property P(C,J)’; as such, they cannot be di- rectly applied in connection to the philosophy ‘toposes as bridges’, since the criteria that they give rise to only allow one to enter a given bridge (i.e., to pass from the property P(C,J) of the site (C, J) to the invariant I) and not to exit from it. In fact, not even the proofs of these results provide information which one can exploit to obtain site characterizations going in the other direction. Therefore, to achieve our goal, the problem needs to be completely reconsidered and approached from a different angle; we do so in this paper, by adopting the point of view of separating sets of toposes. In fact, it turns out that most of the geometric invariants of toposes consid- ered in the literature, notably including the property of a topos to be localic (resp. atomic, locally connected, equivalent to a presheaf topos, coherent), can be expressed in terms of the existence of a separating set of objects for the topos satisfying some invariant property. In this paper, we show that ex- 2 pressing topos-theoretic invariants in terms of the existence of a separating set of objects of the topos satisfying some property paves the way for natural ‘unravelings’ of such invariants as properties of the sites of definition of the topos, and hence for criteria of the desired form. The paper is organized as follows. In section 2, we make some general remarks about the problem of obtaining site characterizations for geometric invariants of toposes, while in the following sections we apply these consid- erations to the specific invariants mentioned above, obtaining natural site characterizations for them of the desired kind. Besides their technical in- terest, these results are meant to provide the reader with a general idea of how the technique ‘toposes as bridges’ introduced in [8] actually works in a variety of different cases. This work should be considered as a companion to [7], where several syntactic characterizations of geometric invariants on t

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