Topos, Being, And Existence


The concept of (Grothendieck) topos was introduced by Alexandre Grothendieck in the early 1960s, in order to provide a mathematical underpinning for the ‘exotic’ cohomology theories needed in algebraic geometry. Every topological space $X$ gives rise to a topos (the category of sheaves of sets on $X$ ) and every topos in Grothendieck’s sense can be considered as a ‘generalized space’.

A topos is a category with finite limits and power objects. A powerobject of an object $X$ , is an object Z along with a mono $m : ∈ \rightarrow X \times Z$ such that for any mono $r : R \rightarrow X×Y$ , there is a unique morphism $\chi : Y \rightarrow Z$ making this a pullback square:

The Bloch Sphere