TITLE: Abstract Relation between Presehaves and Simplicial Sets
QUESTION [6 upvotes]: 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)

REPLY [3 votes]: 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.