nLab
Johnstone's topological topos

Definition

Let Top be the category of topological spaces, and let $\Sigma$ be the full subcategory whose only two objects are a one-point space and ${\mathbb{N}}^{+}$, the one-point compactification of the discrete space of natural numbers. Let $J$ be the canonical Grothendieck topology on $\Sigma$.

Johnstone’s topological topos (specifically, the one described in the eponymous paper referenced below) is the topos of canonical sheaves ${\mathrm{Sh}}_J(\Sigma )$ on $\Sigma$. The functor

$$\mathrm{Top}\to {\mathrm{Set}}^{{\Sigma }^{\mathrm{op}}}:X\mapsto \mathrm{Top}(-,X)$$Top \to Set^{\Sigma^{op}}: X \mapsto Top(-, X)

is faithful and factors through ${\mathrm{Sh}}_J(\Sigma )$, and its restriction to the category of sequential spaces is full.

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 $ℰ:={\mathrm{Sh}}_J(\Sigma )$ allows one to represent the geometric realization functor as the inverse image of a geometric morphism from $ℰ$ to the topos of simplicial sets.

Reference

• Peter Johnstone, On a topological topos, Proc. London Math. Soc. (3) 38 (1979) 237–271. (PDF)