The gluing axiom states that given an open cover $\{U_i\}$ of some open set $U$, if I have a section $s_i$ on each $U_i$ such that the restrictions agree on intersections, then there exists a unique section of $U$ such that its restriction to each $U_i$ is $s_i$.

Does dropping "unique" from the gluing axiom lead to anything mathematically interesting?

I think a trivial example of this sort of sheaf is a topological space $X$ where $F(U)$ for any proper subset of $X$ is the trivial group, while $F(X)$ is any nontrivial group. Then if I take an open cover of $s$, I can glue all of those trivial sections into any element of $F(X)$.

  • $\begingroup$ I suspect that any such 'pseudo-sheaf' can be turned into a separated sheaf by 'remembering' the source of every restriction map, which can then be further extended into a true sheaf by sheafification. For instance, in the case of your example the first step will likely yield a constant presheaf, which, upon sheafifying, yields a constant sheaf. $\endgroup$ – Sofie Verbeek Sep 23 '18 at 7:58
  • $\begingroup$ It might be hard to come up with real-life scenarios in which such 'pseudo-sheaves' would pop up. This is because most sheaves (and indeed presheaves) we care about are defined by taking some collection of functions, and functions tend to satisfy the uniqueness property. $\endgroup$ – Sofie Verbeek Sep 23 '18 at 8:01

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