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October 2, 2006

Topos Theory at Chicago

Posted by John Baez

Tom Fiore of the University of Chicago writes:

The Category Theory Seminar will meet Thursdays 3-4 in Eckhart 203. This quarter we will have a series of talks on topos theory, model categories, and quasicategories. These current topics lie at the intersection of algebraic geometry, topology, logic, and higher category theory, and may be of interest to a wider audience. We will begin with an introduction to topos theory this Thursday, October 5th with a talk by Mike Shulman. Our website is below:

Here’s the abstract for Shulman’s talks…

I’m on mailing lists for lots of seminars. Sometimes when I hear of an interesting talk I feel like mentioning it on this blog. So far I haven’t, because that might make me feel obliged to be systematic about it - which would take too much work. And, I don’t want people to feel that talks I don’t mention are uninteresting to me.

But what the heck! Even a small biased sampling might give a sense of how n-categories are developing. Here are two lectures on topos theory, part of the University of Chicago’s Fall 2006 Category Theory Seminar, and serving as a warmup for ∞-topoi.

3:00 pm Thursday, October 5, 2006

A Friendly Introduction to Topos Theory

by Michael Shulman (University of Chicago) in Eckhart 203

A topos is many different things to different people. To an algebraic geometer, a topos may be a setting in which to study cohomology. To a topologist, a topos may be a generalization of a topological space. To a logician, a topos may furnish a semantics for constructive logic. To a differential geometer, a topos may be a universe of smooth spaces with infinitesimal objects. To a set theorist, a topos may be used for independence proofs. Topos theory, the unification of this diversity of viewpoints, is a beautiful field of mathematics, but one which it can be difficult to get a handle on. In this sequence of two talks, I’ll give an introduction to a few aspects of topos theory, assuming only a basic background in category theory (through limits, adjunctions, and universal properties). In the first talk, I’ll start out with the logical and topological points of view: what a topos is, how we get examples from logic and from topology, and the relationship between the two. I’ll explain in what sense the “logic of topology” is constructive, and what implications this has for “parametrized mathematics”. In the second talk, I’ll say more about Grothendieck topoi, sheaves, cohomology, and stacks, in preparation for talks later this quarter on the “higher topos theory” of Toen & Vezzosi, Lurie, and others.

If this piques your curiosity, you can dip your toe into the pool:

take a swim using these online texts:

or take a deep dive into the icy waters:

You can get a lot more to read online by looking under “References” here.

Posted at October 2, 2006 5:55 PM UTC

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Re: Topos Theory at Chicago

Here’s the next talk in this series:

3:00 pm Thursday, October 19, 2006

Subobject Classifiers and Higher Topoi

by Thomas Fiore (University of Chicago)

Classifying objects can be found in many categories. For example, the space BGBG classifies principal GG-bundles: homotopy classes of maps from a space XX into BGBG are in bijective correspondence with isomorphism classes of principal GG-bundles on XX. Similarly, maps from an object YY into a subobject classifier are in bijective correspondence with subobjects of YY. An important property of a topos, i.e. a generalized space, is the presence of a subobject classifier. In this talk I will begin with the easy example of a subobject classifier in Set\mathrm{Set}, then move to a subobject classifier in the strict 2-topos Cat\mathrm{Cat}, and finish the talk with a discussion of subobject classifiers in \infty-topoi. This expository talk complements the previous two talks on topoi, so I will not use any explicit results from those talks.

Posted by: John Baez on October 19, 2006 7:01 AM | Permalink | Reply to this

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