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](https://mathoverflow.net/questions/5378/when-are-there-enough-projective-sheaves-on-a-space-x) 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!



Heuristic explanation of why we lose projectives in sheaves.
This is pretty much Dinakar's answer from a different view point: He says that it is too easy for a sheaf morphism to be an epi, so, since there are so many epis, it is now a stronger requirement that for *every* epi we find a lift - so strong that is not satisfied most of the times. I just want to call attention to the fact that this problem has nothing to do with module sheaves but is about sheaves of sets - and as such has the following nice interpretation:


The condition of being a projective module sheaf can be split in two conditions: That of existence of the lifting map as a *morphism of sheaves of sets* and that of it being a *morphism of module sheaves*.


In the category of sets the first condition is always satisfied; we have the axiom of choice which says that every epimorphism has a section and composing the morphism from our would-be projective with this section produces a lift - set-theoretically. Then one has to establish that one such lift is a module homomorphism.


But in a sheaf category step one can fail. Sheaves (of sets) are objects in the category of sheaves. This category is a topos and can be seen as an intuitionistic set-theoretic universe (in a precise sense: there is a sound and complete topos semantics for intuitionistic logic, see e.g. [this book](http://books.google.com/books?id=SGwwDerbEowC&pg=PA454&dq=moerdijk+maclane&client=firefox-a&hl=de#v=onepage&q=moerdijk%20maclane&f=false)). Now in an intuitionistic universe of sets, the axiom of choice is not valid in general; there might not be a *"set-theoretic"* section of the epimorphism!


