nLab generic interval



Topos Theory

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Cohomology and homotopy

In higher category theory




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


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


The topos sSet 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 \,\colon\, \mathcal{E}\rightarrow sSet with direct image part S I*:sSetS_I_* \,\colon\, \mathcal{E}\rightarrow sSet the singular functor and inverse image part the geometric realization functor || I:sSet|\quad|_I \,\colon\, sSet\rightarrow \mathcal{E}.

This result was announced by André 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).


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)


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 simplicial complexes and the category of groupoids Grpd as reflective subcategories (cf. Lawvere 1989, pp.273-275).


Last revised on December 1, 2021 at 06:03:49. See the history of this page for a list of all contributions to it.