nLab generic interval

Idea

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

Properties

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

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 $S_I \,\colon\, \mathcal{E}\rightarrow sSet$ with direct image part $S_I_* \,\colon\, \mathcal{E}\rightarrow sSet$ the singular functor and inverse image part the geometric realization functor $|\quad|_I \,\colon\, sSet\rightarrow \mathcal{E}$.

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).

Remark

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)

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

Related Concepts

• topological topos

• Aufhebung

• interval object

• sSet

References

• Peter Johnstone, On a topological topos , Proc. London Math. Soc. 3 38 (1979) pp.237-271. (cf. this n-café blog (link))

• F. William Lawvere, Qualitative Distinctions between some Toposes of Generalized Graphs, Contemporary Mathematics 92 (1989) pp.261-299 (doi:10.1090/conm/092).

• 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)

• Gavin C. Wraith, Using the Generic Interval, Cah. Top. Géom. Diff. Cat. XXXIV 4 (1993) pp.259-266. (pdf)

• Saunders Mac Lane, Ieke Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994.

• F. William Lawvere, Core Varieties, Extensivity, and Rig Geometry, TAC 20 no.14 (2008) pp.497-503. (pdf)

• F. William Lawvere, Ross Street, Todd Trimble, catlist exchange October 2011 , (link) .

• Marco Grandis, F. William Lawvere, catlist exchange November 2013 , (link).