Let Top be the category of topological spaces, and let be the full subcategory whose only two objects are a one-point space and , the one-point compactification of the discrete space of natural numbers. Let be the canonical Grothendieck topology on .
The category of subsequential spaces can also be found as a full subcategory of this topos (in fact, it consists of the separated objects for a Lawvere-Tierney topology). A general object of the topos can be thought of as like a subsequential space, but such that a given sequence can converge to a given point in “more than one way.”
Importantly, following an idea by Joyal, the topological topos allows one to represent the geometric realization functor as the inverse image of a geometric morphism from to the topos of simplicial sets. As Johnstone points out, this approach fails for the big topos on and also for Lawvere’s topos for continuum mechanics.
n-cafe blog about this paper.