Geometric interpretation of sheaves defined with equalisers

Question

In their book "Sheaves in geometry and logic", Mac Lane and Moerdijk gave the following definition of sheaves of sets:

The definition makes "categorical" sense to me, but how do I interpret it geometrically like with definitions given in say, Hartshorne's "Algebraic Geometry" ?

Answer 1

Forgive me if this seems a little low brow, I'm not very well versed with sheaves or category theory, but the equalizer in this case $FU$ is the subset of $\prod_i F U_i$ such that if $x \in \prod_i F U_i$ then $p(x) = q(x)$. To me it is simply saying that there is a map $e$ between the parts of the functor that "glue together correctly" when one restricts to intersections and the more general images of the functor on each separate part of the cover.

Answer 2

That $FU\to \prod_iFU_i$ is the equalizer of these two maps means that given a family of sections $(s_i)_{i\in I}, s_i\in FU_i$, if they are compatible, i.e. $s_{i\mid U_i\cap U_j} = s_{j\mid U_i\cap U_j}$ for all $i,j$ (this is the equalizing condition, $p=q$), then there is a unique section $s\in FU$ that lifts them, that is, such that $s_{\mid U_i} = s_i$ for all $i$.

Indeed recall that the equalizer of two maps $f,g : A\to B$ in $\mathbf{Set}$ is $\{x\in A\mid f(x)=g(x)\}$, so saying that $FU\to \prod_i FU_i$ is the equalizer of $p=q$ means that

$FU\to \{(s_i)_{i\in I} \mid s_{i\mid U_i\cap U_j} = s_{j\mid U_i\cap U_j}$ for all $i,j\}$, $s\mapsto (s_{\mid U_i})_{i\in I}$ is a bijection (isomorphism in $\mathbf{Set}$)

So essentially this is a glueing condition : if my sections $s_i$ are compatible, then I can glue them (the map above is surjective), and I can only glue them in one way (the map above is injective).

For instance, if the map is only injective (that is, I can't necessarily glue compatible things together, but when I can, there's a unique way to do so), the presheaf is called separated.