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2 changed order - credits to Dinakar first

Module sheaves can be seen as module objects in the category of sheaves.

This category is pretty much Dinakar's answer from a different view point: He says that it is too easy for a topos and can sheaf morphism to be seen as an intuitionistic set-theoretic universe (in a precise sense: epi, so, since there are so many epis, it is now a sound and complete topos semantics stronger requirement that for intuitionistic logic, see e.gevery 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 book)

Now 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.

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, i.esee e.g. this book). Now in an intuitionistic universe of sets, the axiom of choice is almost never not valid and we may already fail in step onegeneral; there might not be a "set-theoretic" section of the epimorphism!

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 every epi has a section - so strong that is not satisfied most of the times. I just wanted to call attention to the fact that this problem has nothing to do with module sheaves but is about sheaves of sets - and has such has the nice interpretation I gave above.

1

Module sheaves can be seen as module 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)

Now the condition of being projective 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, i.e. an intuitionistic universe of sets, the axiom of choice is almost never valid and we may already fail in step one; there might not be a "set-theoretic" section of the epimorphism!

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 every epi has a section - so strong that is not satisfied most of the times. I just wanted to call attention to the fact that this problem has nothing to do with module sheaves but is about sheaves of sets - and has such has the nice interpretation I gave above.