Local objects and projective objects

Let 𝒞 be a category and let J be a subcanonical Grothendieck topology on 𝒞. A J-local object in 𝒞 is an object X in 𝒞 such that the only J-covering sieve on X is the maximal sieve. For example, if 𝒞op is the category of commutative rings and J is the Zariski topology, then the J-local objects are precisely the local rings; or if 𝒞 is the category of T₁-spaces and J is the open cover topology, then J-local objects are precisely the points.

It is easy to see that representable J-sheaves on 𝒞 corresponding to J-local objects in 𝒞 are projective (with respect to sheaf epimorphisms). However, the converse is not true in general; after all, the class of projective objects is closed under coproduct, whereas the class of J-local objects in 𝒞 may fail to be closed under coproduct. For example, if 𝒞 is the category of (all) topological spaces and J is the open cover topology, then any representable sheaf corresponding to a discrete space is projective. In fact, a representable sheaf is projective if and only if every open cover of the corresponding topological space can be refined by a cover of clopen subsets.

So, whereas J-local objects in 𝒞 should be thought of as being point-like, it would seem that objects in 𝒞 such that the corresponding representable sheaf is projective should be thought of as being “highly atomised”, but not necessarily a coproduct of J-local objects. This is one heuristic explanation for why we should not expect sheaf toposes to have enough projectives.

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