Why Is The Category Of Sets A Grothendieck Topos?

Question

Why Is The Category Of Sets A Grothendieck Topos? How can just object and relation lead to a Grothendieck Topos which {!} 'suggests the possibility of synthesis of algebraic geometry, topology, and arithmetic' [ReS, The vision]?

Answer 1

Your question is wholly unclear.

However, based on your comment on your previous posting of this question, it seems you're wholly misunderstanding the comment.

That Set is a sheaf category is supposed to be obvious. This fact alone doesn't "suggest the possibility...".

Instead, what the comment is presumably referring to is the whole collection of facts:

• Set is a topos
• Every (sober) topological space can be identified with a topos
• Every scheme can be identified with a topos with extra structure (a locally ringed topos)

So the point is, by studying toposes, you are learning about all of these subjects simultaneously. Conversely, each of these different subjects has practices that can be adapted to the study of toposes.

(disclaimer: this is based on my knowledge of the subjects, not on my knowledge of Grothendieck's writings specifically)