example of a gluable unseparated presheaf

Question

Let $X$ be a topological space. Is there an expample of a presheaf $\mathcal{F}$ which satisfies the glueing axiom, but is not separated?

Note that I use the terminology of wikipedia, see wikipedia article about presheaves

Answer 1

Here is a class of examples:

Take the presheaf $\mathcal F$ on $X$ such that $\mathcal F(X)=\mathbb Z$ and such that $\mathcal F(U)=0$ for all open $U\subsetneq X$.
Then $\mathcal F$ is gluable:
1) Gluability in the case of a covering $U_i\subset U$ of $U\subsetneq X$ is completely trivial.
2) If $U=X$ and if $(U_i)$ is a covering of $X$ by strict open subsets $U_i\subsetneq X$ the only datum $z_i=0\in \mathcal F(U_i)$ is gluable to any $z\in \mathcal F(X)=\mathbb Z$.
So gluability is possible in that case but clearly not unique since $z$ is arbitrary.

In conclusion, the presheaf $\mathcal F$ is gluable but not separated.

Answer 2

A presheaf is separated if equality of its sections is a local property. So a presheaf is non-separated if it has distinct sections which are locally equal.

Fiber bundles with fiber $F$ over a base space $Y$ all look the same locally, but can be distinct.

To formalize this, consider the presheaf $\mathrm{FibBun}_{/Y}$ of isomorphism types of fiber bundles over $Y$:

• its value on an open $V\subset Y$ is the set of isomorphism types of fiber bundles on $V$,
• its restriction maps are given by pullback,
• functoriality follows from the pullback pasting lemma.

Suppose $Y$ admits a non-trivial fiber bundle with fiber $F$. Then the presheaf $\mathrm{FibBun}_{/Y}$ is not separated, because the sections given by the (isomorphism type of the) trivial bundle and the non-trivial bundle are locally equal, but not globally equal.