Grothendieck Topoi (topoi in Geometry)
Since the introduction of sheaves into mathematics in the 1940s a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological intuition is very effective but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the intuition. The single greatest success of this programmatic idea to date has been the introduction of the étale topos of a scheme.
Read more about this topic: Topos