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Johnstone's topological topos

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Topos Theory

topos theory

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Definition

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 sSetsSet \to \mathcal{E} as the inverse image of a geometric morphism from \mathcal{E} to the topos of simplicial sets. As Johnstone points out, this approach fails for the big topos on TopTop and also for Lawvere’s topos for continuum mechanics.

Reference

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

n-cafe blog about this paper.

Revised on July 30, 2015 22:24:41 by David Roberts (58.179.231.138)