Conjuncted: Of Topos and Torsors

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The condition that each stalk Fx be equivalent to a classifying space BG can be summarized by saying that F is a gerbe on X: more precisely, it is a gerbe banded by the constant sheaf G associated to G.

For larger values of n, even the language of stacks is not sufficient to describe the nature of the sheaf F associated to the fibration X~ → X. To address the situation, Grothendieck proposed (in his infamous letter to Quillen; see [35]) that there should be a theory of n-stacks on X, for every integer n ≥ 0. Moreover, for every sheaf of abelian groups G on X, the cohomology group Hn+1sheaf(X;G) should have an interpreation as sheaf classifying a special type of n-stack: namely, the class of n-gerbes banded by G. In the special case where the space X is a point (and where we restrict our attention to n-stacks in groupoids), the theory of n-stacks on X should recover the classical homotopy theory of n-types: that is, CW complexes Z such that the homotopy groups πi(Z, z) vanish for i > n (and every base point z ∈ Z). More generally, we should think of an n-stack (in groupoids) on a general space X as a “sheaf of n-types” on X.

When n = 0, an n-stack on a topological space X simply means a sheaf of sets on X. The collection of all such sheaves can be organized into a category ShvSet(X), and this category is a prototypical example of a Grothendieck topos. The main goal of this book is to obtain an analogous understanding of the situation for n > 0. More precisely, we would like answers to the following questions:

(Q1) Given a topological space X, what should we mean by a “sheaf of n-types” on X?

(Q2)  Let Shv≤n(X) denote the collection of all sheaves of n-types on X. What sort of a mathematical object is Shv≤n(X)?

(Q3)  What special features (if any) does Shv≤n(X) possess?

Our answers to questions (Q2) and (Q3) may be summarized as follows:

(A2)  The collection Shv≤n(X) has the structure of an ∞-category.

(A3)  The ∞-category Shv≤n(X) is an example of an (n+1)-topos: that is, an ∞-category which satisfies higher categorical analogues of Giraud’s axioms for Grothendieck topoi.

Grothendieck’s vision has been realized in various ways, thanks to the work of a number of mathematicians (most notably Jardine), and their work can also be used to provide answers to questions (Q1) and (Q2). Question (Q3) has also been addressed (at least in limiting case n = ∞) by Toën and Vezzosi

To provide more complete versions of the answers (A2) and (A3), we will need to develop the language of higher category theory. This is generally regarded as a technical and forbidding subject. More precisely, we will need a theory of (∞, 1)-categories: higher categories C for which the k-morphisms of C are required to be invertible for k > 1.

Classically, category theory is a useful tool not so much because of the light it sheds on any particular mathematical discipline, but instead because categories are so ubiquitous: mathematical objects in many different settings (sets, groups, smooth manifolds, etc.) can be organized into categories. Moreover, many elementary mathematical concepts can be described in purely categorical terms, and therefore make sense in each of these settings. For example, we can form products of sets, groups, and smooth manifolds: each of these notions can simply be described as a Cartesian product in the relevant category. Cartesian products are a special case of the more general notion of limit, which plays a central role in classical category theory. 

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