The generic interval is the standard 1-simplex $\Delta_1$ in $sSet$.
What makes this interval ‘generic’ is the following result of Joyal^{1}:
Theorem. 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 $\Delta_1=Hom_{\Delta}(\quad,[1])$.
If $I$ is an interval in a topos $\mathcal{E}$, then it corresponds to a geometric morphism $S_I:\mathcal{E}\rightarrow sSet$ with direct image part $S_I_*:\mathcal{E}\rightarrow sSet$ the singular functor and inverse image part the geometric realization functor $|\quad|_I:sSet\rightarrow \mathcal{E}$.
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)
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 $sSet$ and the category of groupoids $Grpd$ as reflective subcategories (cf. Lawvere 1989, pp.273-275).
Peter Johnstone, On a topological topos , Proc. London Math. Soc. 3 38 (1979) pp.237-271. (cf. this n-café blog (link))
Marco Grandis, F. William Lawvere, catlist exchange November 2013 , (link).
F. William Lawvere, Ross Street, Todd Trimble, catlist exchange October 2011 , (link) .
F. William Lawvere, Categories of Spaces may not be Generalized Spaces as Exemplified by Directed Graphs, Revista Colombiana de Matemáticas XX (1986) pp.179-186. Reprinted with commentary in TAC 9 (2005) pp.1-7. (pdf)
F. William Lawvere, Qualitative Distinctions between some Toposes of Generalized Graphs, Contemporary Mathematics 92 (1989) pp.261-299.
F. William Lawvere, Core Varieties, Extensivity, and Rig Geometry, TAC 20 no.14 (2008) pp.497-503. (pdf)
Saunders Mac Lane, Ieke Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994.
Gavin C. Wraith, Using the Generic Interval, Cah. Top. Géom. Diff. Cat. XXXIV 4 (1993) pp.259-266. (pdf)
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.