nLab
generic interval

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topos theory

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Idea

The generic interval is the standard 1-simplex Δ 1\Delta_1 in sSetsSet.

What makes this interval ‘generic’ is the following result of Joyal1:

Theorem. The topos sSetsSet of simplicial sets is the classifying topos for linear orders with distinct top and bottom elements, i.e. intervals (aka nontrivial totally distributive lattices) with generic interval Δ 1=Hom Δ(,[1])\Delta_1=Hom_{\Delta}(\quad,[1]).

If II is an interval in a topos \mathcal{E}, then it corresponds to a geometric morphism S I:sSetS_I:\mathcal{E}\rightarrow sSet with direct image part S I*:sSetS_I_*:\mathcal{E}\rightarrow sSet the singular functor and inverse image part the geometric realization functor || I:sSet|\quad|_I:sSet\rightarrow \mathcal{E}.

The classifying topos point of view is helpful, because it is the choice of an algebraic structure, of the kind classified, in a topological topos that gives rise to an exact singular/realization pair. (Lawvere 2013, link)

Remark

The generic interval and the results of Wraith (1993) play a role in Lawvere’s attempt to view classical combinatorial topology as part of greater landscape situated in the presheaf topos on the category of nonempty finite sets, which is the classifying topos for nontrivial Boolean algebras and contains sSetsSet and the category of groupoids GrpdGrpd as reflective subcategories (cf. Lawvere 1989, pp.273-275).

References


  1. The result was announced by Joyal in 1974 at the Isle of Thorns, the first published proof appears in Johnstone (1979) and in book form in MacLane-Moerdijk (1994, sec. VIII.8).

Last revised on June 6, 2015 at 20:59:58. See the history of this page for a list of all contributions to it.