Étale homotopy type

In mathematics, especially in algebraic geometry, the étale homotopy type is an analogue of the homotopy type of topological spaces for algebraic varieties.

Roughly speaking, for a variety or scheme X, the idea is to consider étale coverings ${\displaystyle U\rightarrow X}$ and to replace each connected component of U and the higher "intersections", i.e., fiber products, ${\displaystyle U_{n}:=U\times _{X}U\times _{X}\dots \times _{X}U}$ (n+1 copies of U, ${\displaystyle n\geq 0}$) by a single point. This gives a simplicial set which captures some information related to X and the étale topology of it.

Slightly more precisely, it is in general necessary to work with étale hypercovers ${\displaystyle (U_{n})_{n\geq 0}}$ instead of the above simplicial scheme determined by a usual étale cover. Taking finer and finer hypercoverings (which is technically accomplished by working with the pro-object in simplicial sets determined by taking all hypercoverings), the resulting object is the étale homotopy type of X. Similarly to classical topology, it is able to recover much of the usual data related to the étale topology, in particular the étale fundamental group of the scheme and the étale cohomology of locally constant étale sheaves.

• Artin, Michael; Mazur, Barry (1969). Etale homotopy. Springer.
• Friedlander, Eric (1982). Étale homotopy of simplicial schemes. Annals of Mathematics Studies, PUP.

• http://ncatlab.org/nlab/show/étale+homotopy