On sheaf cohomology and natural expansions

Sheaves on semicartesian monoidal categories and applications in the quantalic case

“…The reader may consult the definitions in [Wei94] and [Bor94b], here we only observe that having an abelian category with enough injective objects gives enough structure to define (co)homology as right/left derived functors of a left/right exact functor. This is true even for the "abelian form" of Grothendieck toposes, as one may check in [Gro63], [Joh77], on in our survey [TM21] about sheaf cohomology.…”

“…This can be done by brute force or by pretending that  is 𝑆𝑒𝑡. Since there is an equivalence 𝐴𝑏(𝑆𝑒𝑡) ≃ 𝐴𝑏, the proof follows from the fact that 𝐴𝑏 is an abelian category, see [Joh77] or [TM21]). Thus, the Soundness Theorem exempts us from 10 pages of calculations, which are done in [Şte81].…”