University of Chicago
Department of Mathematics

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.

Last week, I talked about topoi mostly as generalized universes of sets in which to do (constructive) mathematics. This week, I'll talk about how we can view a topos as a generalized topological space. We'll talk first about `locales', or "topological spaces without points", in what ways they're better than topological spaces, and how to represent them with their topoi of sheaves. Then we'll generalize to sheaves over `sites' to obtain the full generality of topoi as generalized spaces. I'll mention how to define the cohomology and homotopy groups of a topos, and also "Giraud-type" theorems for recognizing when a category is a topos. At the end, I'll say a little about stacks and homotopy theory, which leads us towards higher topoi and later talks this quarter.

Classifying objects can be found in many categories. For example, the space BG classifies principal G-bundles: homotopy classes of maps from a space X into BG are in bijective correspondence with isomorphism classes of principal G-bundles on X. Similarly, maps from an object Y into a subobject classifier are in bijective correspondence with subobjects of Y. 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, then move to a subobject classifier in the strict 2-topos Cat, and finish the talk with a discussion of subobject classifiers in infinity-topoi. This expository talk complements the previous two talks on topoi, so I will not use any explicit results from those talks.

The definitions of topological categories, simplicial categories, Segal categories, complete Segal categories, and quasicategories will be given. The last of these are Lurie's infinity categories. Some idea of why these are all of interest will be given, and the web of Quillen equivalences of model categories that relate them will be sketched.

Today in category theory seminar I will discuss how the theory of higher topoi arises naturally in questions stable homotopy theory and some properties of this theory. Cohomology theories arising from moduli problems are among the most interesting and computable. We will try to use methods from algebraic geometry to build a general context for building such cohomology theories. In the process, we will see what we will need to make such a theory work. These needs are satisfied by Lurie's theory of higher topoi which can be interpreted as giving a good theory of sheaves taking values in infinity categories. I'll give a brief exposition about his theory of infinity categories and how it relates to our problem. Time (and preparation) permitting I will try to state some of the properties of Lurie's theory. This is only one perspective on Lurie's theory of higher topoi. His theory can also be used in a purely algebro-geometric way to solve various moduli problems in that setting. Since I am usually unable to attend the seminar, this talk will not depend on any previous talks this quarter. And since this is just a survey of a massive theory many statements will have to be given in a fuzzy form.

We look at the basic theory of quasi-categories (a model of weak $\infty$-categories with $k$-morphisms invertible for all $k>1$), inspired by A. Joyal's founding observation that "most concepts and results of category theory can be extended to quasi-categories." We'll look at Hom spaces, functors, functor quasi-categories, equivalences, (homotopy) limits and colimits, and adjunctions. In addition, we will discuss constructions of the $\infty$-topoi of spaces and stacks and the quasi-category of quasi-categories, using the language of fibred quasi-categories (also known as right fibrations). The geometric (which is to say Grothendieckian) preference for this formalism will be explained.

The Stabilization Hypothesis of Baez and Dolan states that certain structured kinds of monoidal weak $n$-categories are stable in some sense. I will discuss an alternate approach to these same structures, and show that commutative monoids are stable. The 2-dimensional case is still work in progress, but there is an obvious conjecture for a statement of stability.