TITLE: Heuristic explanation of why we lose projectives in sheaves.
QUESTION [34 upvotes]: We know that presheaves of any category have enough projectives and that sheaves do not, why is this, and how does it effect our thinking?
This question was asked(and I found it very helpful) but I was hoping to get a better understanding of why.
I was thinking about the following construction(given during a course);
given an affine cover, we normally study the quasi-coherent sheaves, but in fact we could study the presheaves in the following sense:
Given an affine cover of X,
$Ker_2\left(\pi\right)\rightrightarrows^{p_1}_{p_2} U\rightarrow X$
then we can define $X_1:=Cok\left(p_1,p_2\right)$, a presheaf, to obtain refinements in presheaves where we have enough projectives and the quasi-coherent sheaves coincide. Specifically, if  $X_1\xrightarrow{\varphi}X$ for a scheme $X$, s.t. $\mathcal{S}\left(\varphi\right)\in Isom$ for $\mathcal{S}(-)$ is the sheaffication functor, then for all affine covers $U_i\xrightarrow{u_i}X$ there exists a refinement $V_{ij}\xrightarrow{u_{ij}}U_i$ which factors through $\varphi$.
This hinges on the fact that $V_{ij}$ is representable and thus projective, a result of the fact that we are working with presheaves. In sheaves, we would lose these refinements. Additionally, these presheaves do not depend on the specific topology(at the cost of gluing).
In this setting, we lose projectives because we are applying the localization functor which is not exact(only right exact). However, I don't really understand this reason, and would like a more general answer. 
A related appearance of this loss is in homological algebra. Sheaves do not have enough projectives, so we cannot always get projective resolutions. They do have injective resolutions, and this is related to the use of cohomology of sheaves rather than homology of sheaves. In paticular, in Rotman's Homological Algebra pg 314, he gives a footnote;

In The Theory of Sheaves, Swan writes "...if the base space X is not discrete, I know 
  of no examples of projective sheaves except the zero sheaf." In Bredon, Sheaf Theory: 
  on locally connected Hausdorff spaces without isolated points, the only 
  projective sheaf is 0

addressing this situation.

In essence, my question is for a
  heuristic or geometric explanation of
  why we lose projectives when we pass
  from presheaves to sheaves.

Thanks in advance!

REPLY [4 votes]: We can turn the question around to ask: why do we have projectives in module categories? One answer is that we know we have a plentiful supply of projectives because free modules are projective. Abstractly, we have a forgetful functor $U : \text{Mod}(R) \to \text{Set}$ with left adjoint $F : \text{Set} \to \text{Mod}(R)$. Then we have the following two results:

By the axiom of choice, every set is projective (in the sense that homs out of it preserve epimorphisms).
If a functor $U$ with a left adjoint preserves epimorphisms, then its left adjoint $F$ preserves projectives.

Finally, it is straightforward to verify that $U$ in fact preserves epimorphisms (that is, epimorphisms of $R$-modules are surjective on underlying sets). 
Now what is the analogous situation for sheaves? We still have a forgetful functor $U : \text{Sh}(X) \to \text{Psh}(X)$, and it still has a left adjoint, namely sheafification. However, $U$ no longer preserves epimorphisms (this is exactly Dinakar's observation that an epimorphism of sheaves need not be an epimorphism of presheaves), so the above argument doesn't go through. 
For sheaves what we can instead do is the following. There is a different forgetful functor sending a sheaf on $X$ to its stalks; it can be thought of as pullback $p^{\ast}$ along the map $p : X_d \to X$ where $X_d$ denotes $X$ with the discrete topology. As a pullback, this functor has a right adjoint, namely pushforward $p_{\ast}$. The composite $p_{\ast} p^{\ast}$ is the Godement construction. In any case, a dual argument to the above shows that because pullback $p^{\ast}$ preserves monomorphisms, its right adjoint $p_{\ast}$ preserves injectives. So now instead of a plentiful supply of projectives we have a plentiful supply of injectives.