This is a sequel to a previous paper, developing an intrinsic, combinatorial
homotopy theory for simplicial complexes; the latter form the cartesian closed
subcategory of 'simple presheaves' in !Smp, the topos of symmetric simplicial
sets, or presheaves on the category of finite, positive cardinals.
We show here how this homotopy theory can be extended to the topos itself,
!Smp. As a crucial advantage, the fundamental groupoid functor !Smp --> Gpd is
left adjoint to a natural functor Gpd --> !Smp, the symmetric nerve of a
groupoid, and preserves all colimits - a strong van Kampen property. Similar
results hold in all higher dimensions. Analogously, a notion of
(non-reversible) directed homotopy can be developed in the ordinary simplicial
topos Smp, with applications to image analysis as in the previous paper. We
have now a 'homotopy n-category functor' Smp --> n-Cat, left adjoint to a nerve
functor.
This construction can be applied to various presheaf categories; the basic
requirements seem to be: finite products of representables are finitely
presentable and there is a representable 'standard interval'.Comment: Revised version with minor changes 36 pages, 428
