The purpose of this talk is to give a very general definition of what a ``space'' should be. The motivation comes from the idea of a sheaf on a topological space. After reviewing such classical sheaves, I will explain how to generalize the notion of a sheaf to a category with a topology, and why we should consider these sheaves as generalized spaces. In the process I will need to explain some fundamental ideas in category theory, like the Yoneda lemma. I will give examples to try to convince you that this general notion of space is helpful and interesting. In the end we will talk about what a topos is, relations to algebraic geometry, and maybe comment on higher category and topos theory.
The talk should be accessible and does not assume any real knowledge. Some familiarity with categories would be helpful.