Post #1905742

@maxsnew I tend to think of sheaves in terms of converging processes. To back up, think of a category as a collection of "interfaces" and ways of moving/translating information from one interface to another. Then a presheaf X can by thought of as assigning a set X(I) of "systems with interface I" to each object I of the category, which is compatible with the structure of translations in that you if you have a system with interface J and a translation f : I -> J, you can construct a system with interface I. Now suppose you additionally have a coverage on your category. Informally, what a covering family f_i : U_i -> U tells you is the U_i "converge to" U via the f_i, in the sense that they jointly account for all the "information" in U. Then the sheaf condition tells you that, if you have a collection of systems s_i : X(U_i) that are suitably compatible with such a covering family, then there is a unique system in X(U) which they "converge to" under that coverage.