**Title:** Cocycles and pro-objects

**Speaker:** Rick Jardine

**Speaker Info:** University of Western Ontarios

**Brief Description:**

**Special Note**:

**Abstract:**

For spaces (or simplicial presheaves) $X$ and $Y$, the cocycle category $h(X,Y)$ is a translation category for a functor from the weak equivalences over $X$ to sets, which takes a weak equivalence $U \to X$ to the set $hom(U,Y)$ of maps from $U$ to $Y$. The set of path components of this category can be identified with the set of morphisms $[X,Y]$ from $X$ to $Y$ in the homotopy category.If $Y$ is locally fibrant, or a Kan complex in stalks, then the Verdier hypercovering theorem says that the set $[X,Y]$ can be identified with a filtered colimit of sets of simplicial homotopy classes of maps $\pi(U,Y)$, indexed over simplicial homotopy classes of hypercovers $U \to X$. This theorem is used in \'etale homotopy theory to define \'etale homotopy types as pro-objects in simplicial sets, and to calculate \'etale cohomology with constant coefficients.

These results will be discussed. I shall also present a description of the category of small diagrams of spaces, and of pro-weak equivalences in that context. A proper analysis of the homotopy theory for this framework is still work in progress, but it specializes to both the cocycle and hypercover descriptions of morphisms in the homotopy category. It can also be used to define analogues of \'etale homotopy types for arbitrary Grothendieck topologies, in a way which makes no use of either pro-objects or the theory of hypercovers.

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