Do you know math.univ-lille1.fr/~fresse/OperadHomotopyBook?

archive.numdam.org/ARCHIVE/BSMF/BSMF_1962__90_/… may help. There is a nice section about quotient categories including treatment about essential extensions and injective envelopes.

Cross posted math.stackexchange.com/questions/499525/…

@arsmath Indeed, you can. Sheaves of abelian groups are the quotient of the presheaves by the kernel of the sheafification-functor. This Serre subcategory is localizing, from which one can conclude, that the sheaves are a even a Grothendieck category. Especially they must have injective envelopes, are bicomplete and can be realized (via Gabriel-Popescu theorem) as a quotient of a category of modules over a ring by a localizing subcategory.

Nice. Thanks! I didn't know Grothendieck categories without enough projectives. Thank you. Meanwhile, I solved more of my points. An update of the question will follow.

Isn't this notation quite common? $Lex(A,B)$ denotes the category of all left−exact functors from $\mathcal{A}$ to $\mathcal{B}$ and $Func(\mathcal{A},\mathcal{B})$ the category of all functors.

I like the fact, that its fundamental group is uncountable. This is a vivid example for showing students, which are new to algebraic topology, that the fundamental group are not just "some" loops in the space.