A Question on a claim regarding the notion of "space" in "Indiscrete Thoughts" I'm reading Gian-Carlo Rota's book "Indiscrete Thoughts".
In page 220 I came across a strange quotation with very few explanations:

We thought that the generalizations of the notion of space had ended with topoi, but we were mistaken. We probably know less about space now than we pretended to know fifty years ago. As mathematics progresses, our understanding of it regresses!

Question: What is the most generalized and widely accepted definition of the notion of "space" in category theory, particularly I would like to know the generalizations of the notion of topos? 
 A: Perhaps the best known generalization of a topos is simpler a higher topos, in particular, an $(\infty,1)$-topos as developed most thoroughly in Jacob Lurie's Higher Topos Theory. To give a vague sense of the purpose of the generalization, an ordinary topos is usually a category of sheaves of sets on some kind of space, i.e. a small category with a Grothendieck topology (this excludes elementary toposes which aren't Grothendieck, but these arguably no longer model spaces.) This is one sense in which a topos is a generalized space. 
Sheaves of sets are great and permit globalizations of all kinds of classical abstract algebra, e.g. sheaves of abelian groups and modules, and homological algebra, e.g. sheaves of chain complexes. But if we're concerned with globalizing more modern homotopical algebra, then we need to generalize sheaves of sets to sheaves whose values "cohere" better. It's roughly sufficient to think about sheaves of topological spaces. Now a category of sheaves of topological spaces is no longer a topos, but it's possible to formalize the notion of its being a topos "up to homotopy", and that is more or less what higher toposes do.
