From the topos theory perspective it is a general motto that sheaves are continuously variable sets. The internal logic of sheaf toposes justifies this motto, but I would like an additional "local" justification from the bundle picture.
It's known the category of sheaves on a topological space is equivalent to the category of local homeomorphisms to $X$, so it seems a local homeomorphism to $X$ captures the correct notion of a "set continuously varying over $X$". I don't understand "why" the variation of the fibers of a local homeomorphism deserves to be identified with "continuous variation".
Intuitively, I feel a continuously variable set might just be a continuous open map (see below) with discrete fibers (to ensure they are sets and not cohesive spaces). As this answer points out, these properties do not imply local homeomorphy since e.g $z\mapsto z^2$ as a function $\mathbb C\to \mathbb C$ satisfies them all but is not homeomorphic about zero. In addition to continuity and openness, local injectivity is needed (it implies discreteness of fibers).
As a simpler example, the function $\mathbb R\to [0,\infty)$ defined by $x\mapsto x^2$ is continuous and open with discrete fibers, and the variation of fibers seems continuous to me. I would like to understand why the "collapse of fibers" at zero should be viewed as pathological/discontinuous. Before comparing the variation of fibers at zero with that of a local homeo to $[0,\infty)$, here's my intuition for the properties involved.
So a function $f:X\to Y$ is a local homeomorphism iff it's a continuous open local injection.
- Continuity means close points have close images.
- Openness is equivalent to $C\mapsto \forall_f(C)= \left\{ y\in Y\mid f^{-1}(y)\subset C \right\}$ preserves closed sets. I interpret this as saying that a net of fibers always converges "onto" a limit fiber.
- Local injectivity means "unramified" (in $\mathsf{Top}$ it is equivalent to having an open diagonal).
For a concrete comparison of local homeomorphism vs continuous open map with discrete fibers consider:
- $x\mapsto x^2$ as a function $\begin{smallmatrix}\mathbb R\\ \downarrow\\ [0,\infty) \end{smallmatrix}$.
- Let $U\subset X$ be an open subset and consider the local homeomorphism $\begin{smallmatrix}X\amalg U\\ \downarrow\\ X \end{smallmatrix}$ defined by component-wise inclusion. Particularly consider $\begin{smallmatrix}[0,\infty)\amalg (0,1)\\ \downarrow\\ [0,\infty) \end{smallmatrix}$.
I have tried to draw the variation of fibers below.
Question. Why should I think of the upper drawing, where the double fibers "collapse" into a point" as "discontinuous variation"?
Perhaps the "justification" is to simply avoid duplicate copies of the same variable set, where one copy is the unramified version and the other is the ramified version.
I think presheaves correspond to bundles with covering local sections, whose total space topology is generated by its sections. If so, then the second picture in the question has topologically distinct representation of the same fibers: ramify the two branches to have them both converge at the endpoints.
Restricting to local injections excludes ramified representations, so passing from presheaves to sheaves "kills duplicates".