##### Department of Mathematics,

University of California San Diego

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### Algebraic Geometry Seminar

## Michael McQuillan

#### Universita di Roma Tor Vergata/IHES

## 2-Galois theory

##### Abstract:

A theorem of Whitehead asserts that the topological 2-type of a (connected) space is uniquely characterised by the triple ($\pi_1, \pi_2, k_3$), where the $\pi_i, i\leq 2$ are the homotopy groups $\pi_i, i\leq 2, k_3$ is the Postnikov class $\in H^3$($pi_1, \pi_2$), and, indeed all such triples may be realised. Such triples are synonymous with a 2-group, $\Pi_2$, i.e. a group `object' in the category of categories, which plays the same role for 2-types as the fundamental group does for 1-types. In particular, there is a 2-Galois correspondence between the 2-category of champs which are etale fibrations over a space and $\Pi_2$ equivariant groupoids generalising the usual 1-Galois correspondence between spaces which are etale fibrations over a given space and $\pi_1$ equivariant sets. The talk will explain the pro-finite analogue of this correspondence, so, albeit only for the 2-type, a much simpler and more generally valid description of the etale homotopy than that of Artin-Mazur.

Host: James McKernan

### May 16, 2014

### 2:30 PM

### AP&M 7218

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