# Rotman's Homological Algebra Theorem 5.68

This is the Theorem 5.68 of Rotman's "An introduction to Homological Algebra". I found this theorem while studying sheaves. This is the theorem: My question is more about category theory and functors than about sheaves. We have a fixed topological space and we consider the categories of presheaves, sheaves and etale-sheaves over this topological space. In the second point of the theorem, is given a functor $$\Phi$$ which is, in some way, "the inverse" of the functor defined by the sheaf of section $$\Gamma$$. Here again, we see in brackets this functor is INJECTIVE on objects. In the third point we see the restriction of this functor $$\Phi$$ to the subcategory of sheaves is an isomorhpism of categories between sheaves and etale-sheaves.

I can't understand how a functor can be at the same time injective on objects and its restriction to a subcategory be an isomorphism. For sure this doesn't work for a classical injective function.

Moreover, the functor $$\Phi$$ gives the associated etale-sheaf of a presheaf. I'm not sure if it can be injective on objects at all. The associated etale-sheaf is built from the stalks, and we have more than one non-zero presheaf with zero stalks.

I hope I explained myself. Thank you in advance to those who can answer me.

• Rotman says in (ii) that there is a natural transformation which is an isomorphism for sheaves. So perhaps its a mistake in (iii) and he actually meant that these categories are equivalent, not isomorphic. Jan 13 at 11:16
• I'm actually convinced that this is a mistake. He uses the same term "isomorphism of categories" for Theorem 5.55 (Morita), which is not true. The conclusion of the theorem is that these categories are naturally equivalent. And in fact it follows from the proof. Jan 13 at 11:23
• Thank you @freakish! I totally agree with replacing with "natural equivalence". In your opinion, is it also false that the functor $\Phi$ is injective on objects? Jan 13 at 14:27
• Sorry, that one is beyond my understanding. Can't help you with that. If I have some time I'll try to read Rotman carefully, but can't tell you when that happens. Jan 13 at 17:56