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In topos theory, the notion of “being geometric” often comes up. Some examples are: geometric morphisms, geometric logic, and geometric theories. For instance, here's a quote from Steve Vickers' Locales and Toposes as Spaces:

... the logic is not at all ordinary classical logic. It is an infinitary positive logic known as geometric logic.

I have a vague intuition on this notion of geometricity: in locale theory one sees open sets as observable properties, and my understanding of geometricity is that when we categorify from opens to sheaves, the notion of observability generalises to geometricity. In this context, “geometric morphisms” make sense as they respect the geometric structure as a continuous function respects the observability structure. However, I don't quite understand the use of the adjective “geometric” for this.

How do I connect the intuition of observability with the notion of geometricity? If this approach doesn't make much sense, what is a good way to think about geometricity?


PS: I am aware that the paper of Vickers I linked to essentially aims to explain the spatial intuition in detail. However, it seems to take for granted the notion of being “geometric” without a familiarity with which I find hard to follow the paper.

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  • $\begingroup$ Why look to grammar, but not to content? The reference clears up what is being discussed. $\endgroup$
    – Narasimham
    Commented Nov 15, 2020 at 19:21
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    $\begingroup$ Geometric logic is precisely the fragment of logic preserved by (the inverse image part of) geometric morphisms. Geometric morphisms are contrasted with logical morphisms, which preserve all higher order logic. $\endgroup$
    – Zhen Lin
    Commented Nov 15, 2020 at 22:10
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    $\begingroup$ I think @ZhenLin hits the nail on the head, pushing the question back one step to "why are geometric morphisms called geometric?". Here the answer is clear: geometric morphisms are the functors between toposes which generalize continuous maps of topological spaces, in the sense that a geometric morphism between locales is precisely a morphism of locales, and a morphism of spatial locales corresponds precisely to a continuous map of the corresponding topological spaces. $\endgroup$
    – tcamps
    Commented Nov 15, 2020 at 22:46
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    $\begingroup$ Just a note that the reason why “geometric” was chosen rather than “topological” is that toposes originated in algebraic geometry, whereas locale theory came later. $\endgroup$ Commented Nov 16, 2020 at 15:06
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    $\begingroup$ This question has already been answered, but let me mention a very nice exposition of Blass. $\endgroup$ Commented Nov 27, 2020 at 8:07

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To summarize the comments: geometric logic constitutes the logic, models of whose theories are preserved by geometric morphisms between topoi. Geometric morphisms are those appropriate to toposes viewed as generalized spaces, for instance, identifying the topos of sheaves on a topological space, or on a locale, with the space itself. Historically, toposes were first introduced by Grothendieck's school to model generalized algebro-geometric spaces, which explains why these morphisms are called geometric, rather than topological.

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