Abstract Relation between Presehaves and Simplicial Sets Every presheaf (let's say on a topological space) comes with restriction maps. The open sets of a topological space are ordered by inclusion and these inclusions yield the restrictions. Now a sheaf satisfies a gluing condition: that you can glue along elements which coincide on common restrictions.
Every simplicial object (let's say a simplicial set) comes with face maps. The simplex category is ordered by faces & degeneracies and these maps yield simplicial maps. Now a Kan complex satisfies a gluing* condition: that you can glue along simplices which coincide on common faces.
Is there a deeper theoretical framework to relate these 2 notions? I guess that this is the case, and that it is rather trivial.
Side-question: Can we define "degeneracies" for presheaves?
Ideas?
*it's not a gluing condition, but "somehow similar" (see answers below)
 A: I don't really get how you see the Kan Horn filling condition as a gluing condition.
But sheaves and Kan simplicial sets play parallel roles in their categories if you look at it through model category theory: In both situations you have an endofunctor replacing a presheaf by a sheaf, a simplicial set by a Kan set respectively. Both categories have a model structure - that is a bunch of data allowing to handle the formal inversion of morphisms which are called weak equivalences.Both times you have a morphism from the old to the new object which is a weak equivalence, this process is called fibrant replacement and is formalized in the theory of model categories.
In the presheaf case the weak equivalences are those morphisms which become isomorphisms after applying the sheafification functor. If you formally invert these, the resulting category is equivalent to the category of sheaves.
In the simplicial set case the weak equivalences are those maps which induce isomorphisms of homotopy groups after applying geometric realization. If you formally invert those you get a category equivalent to the homotopy category of spaces.
A: The Kan condition isn't exactly like the sheaf condition: the Kan condition allows you to "glue" (as you put it) in certain cases, but the result is not unique.
A better analogy to the Kan condition in sheaf theory might be the notion of a flasque sheaf: a sheaf F is flasque if for all subsets V of U, all sections of F over V extend to sections over U.
A: Yes, so:
A simplicial set is indeed precisely a presheaf on the simplex category.
There are various model category structures on categories of presheaves in general and on simplicial sets in particular.
With respect to the standard model structure on simplicial sets the Kan complexes are precisely the fibrant-and-cofibrant objects.
With respect to the local model structure on presheaves on a site the sheaves are precisely the fibrant-and-cofibrant objects.
There is a very useful combination of these two statements:
A simplicial presheaf is a presheaf on the product category of the simplex category and some site.
in the local projective model structure on simplicial presheaves the fibrant objects are precisely those simplicial presheaves that are Kan-complexes over each object of the site and that satisfy the  oo-version of the sheaf condition ("descent"): these are the (hypercomplete) oo-stacks on the given site.
