Étale homotopy theory

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.

...
Wikipedia