Topos nlab

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Let us concentrate on Grothendieck topoi. As mentioned in earlier posts, these are those topoi which arise as the category of sheaves for a category equipped with a Grothendieck topology. These are those topoi which "behave the most like sheaves of sets on a topological space". Let me try to explain to what extent Grothendieck topoi are topological in nature.

First, given a continuous map f:X->Y, it produces a geometric morphism Sh[X]->Sh[Y]. If X and Y are sober, then there is a bijection between Hom[X,Y] and Hom[Sh[X],Sh[Y]] [where the later is again geometric morphisms]. This means, if we restrict to sober spaces, we get a fully faithful functor Sh:SobTop->topoi. [Recall via stone duality that the category of sober spaces is equivalent to the category of locales with enough points].

More generally, if G is a topological groupoid [a groupoid object in Top], we can construct its classifying topos. This can be defined as follows: Take the enriched nerve of G to obtain a simplicial space, applying the functor "Sh" [viewing the nerve as a diagram of space] and obtain a simplicial topos. Now take the [weak] colimit of the diagram to obtain a topos BG. This topos can be described concretely as equivariant sheaves over G_0.

Geometrically, BG is a model for the topos of "small sheaves" over the topological stack associated to G. In fact, on etale topological stacks* [this include all orbifolds], we also have an equivalence between maps of stacks and geometric morphisms between their categories of sheaves, so, there is a subcategory [sub-2-category] of Grothendieck topoi which is equivalent to etale topological stacks. These Grothendieck topoi are called topological etendue.

*[over sober spaces]

It turns out that a large class of topoi can be obtained as BG for some topological groupoid. In fact, every Grothendick topos "with enough points" is equivalent to BG for some topological groupoid. The more general statement is that EVERY Grothendieck topos is equivalent to BG for some localic groupoid [a groupoid object in locales]. Since locales are a model for "pointless topology", we see in some sense, every Grothendieck topos is "topological". You can make sense of the statement that every Grothendieck topos is equivalent to the category of small sheaves on a localic stack.

Welcome to the research homepage of the Topos Institute. For information on the goals and vision of the institute, see topos.institute ↗︎.

Here are some other places you can find us:

About us

The Topos Institute works to shape technology for public benefit by advancing sciences of connection and integration. Our goal is a world where the systems that surround us benefit us all. Our approach is two-fold. First, we believe that large-scale systems — some natural, such as the climate, and others constructed, like the internet — are increasingly influential, and that a deep and powerful mathematical understanding of the common principles behind these systems is critical to ensuring they benefit the public. Second, we believe that technologies, including mathematics, are not value-neutral, and ethical considerations must be engaged with from the very beginning, including within the fundamental research process itself. We thus work across fundamental research, tool-building, education, and public engagement to achieve our mission.

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion [and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards] is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

The nLab was born out of conversations at the Café back in 2008. Over the past 12 years it has grown as a wiki to over 15000 pages.

For many years it was funded personally by Urs Schreiber, until in the last few years when it relied on a fund kindly provided by Steve Awodey.

But now we have new arrangements in place, and are looking to its users to help fund the nLab:

In 2021, the nLab will move to the cloud. To fund the running of the nLab in the cloud, we have decided to rely upon funding by donations. In the autumn of 2020, at the kind initiative of Brendan Fong, the nLab decided to collaborate with the Topos Institute for the practical side of this: the Topos Institute is legally able to handle donations, and the financing of the nLab will be handled by the Topos Institute. The Topos Institute owns the cloud account in which the nLab will be run.

Please do consider making a donation here.

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Donation sent. I’m not a heavy nLab user, but I do sometimes enjoy reading the articles there, and so I’m happy to make a donation.

Finances are now looking good! Here is Richard Williamson’s report.

Thanks to all the donors.

The migration of the nnLab is afoot, thanks to Richard Williamson. You’ll notice impaired functionality for a time, including the ability to edit pages. On the other hand, the rendering speed has improved noticeably.

For discussion of the migration, see here.

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