I came across a video the other day which discussed practical applications of sheaves. However, rather than defining a sheaf on the open sets of a traditional topological space, the lecturer outlined how you would define them on a simplicial complex, assigning sections to each n-simplex with the restriction maps commuting between across the faces to higher dimensional simplices.
It was nice to have a simple "finite" presentation of a sheaf to get an idea for what the gluing condition says and when it might fail.
Because the lecture was focused on applications, no formal definition was given. I was wondering if this kind of construction is common, and what the formal definition would look like. Also, I would like to know if there are any written resources (papers, textbooks, articles) that discuss it.
The closest I was able to find on my own is that in category theory, there is a notion of a simplicial sheaf, which is defined on a simplicial set. I have a very vague understanding of simplicial sets, and I'm wondering if perhaps sheaves on a simplicial complex are a special case of these.