Johnstone's topological topos


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Let Top be the category of topological spaces, and let Σ\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 JJ be the canonical Grothendieck topology on Σ\Sigma.

Johnstone’s topological topos (specifically, the one described in the eponymous paper referenced below) is the topos of canonical sheaves Sh J(Σ)Sh_J(\Sigma) on Σ\Sigma. The functor

TopSet Σ op:XTop(,X)Top \to Set^{\Sigma^{op}}: X \mapsto Top(-, X)

is faithful and factors through Sh J(Σ)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 :=Sh J(Σ)\mathcal{E}:=Sh_J(\Sigma) allows one to represent the geometric realization functor as the inverse image of a geometric morphism from \mathcal{E} to the topos of simplicial sets.


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

Revised on January 28, 2013 06:39:24 by Bas Spitters (