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)