While studying topos theory I was wondering if there is something like internal logic of an abelian category. Aparently the answer is yes 
(by 7º slide in <https://www.mimuw.edu.pl/~gael/xxi/files/slides_posters/Blechschmidt.pdf>) 
but I can't see what this weaker variant could be.


Besides that, is there relations between this internal logic and the internal logic of a topos? 
This relations would be interesting for topos theory? Maybe for homological algebra?


Is there more relations between homological algebra and topos theory beyond Chapther 8 (Cohomology) of "Topos Theory" Johnstone's book? I know that SGA4 is a reference in this topic to. 


It's my first time making a question here, sorry for any mistake and thanks in advance



Is there something like internal language of an abelian category?
I don't know exactly what Blechschmidt had in mind, but an abelian category is in particular a [regular category](https://ncatlab.org/nlab/show/regular%20category) (indeed, a [Barr-exact category](https://ncatlab.org/nlab/show/exact+category)), and hence it has an [internal logic](https://ncatlab.org/nlab/show/internal%20logic) that is a [regular logic](https://ncatlab.org/nlab/show/regular+logic), having $\exists,\wedge,\top$ but no other connectives (except "$\forall$ and $\Rightarrow$ at top-level" in the sense of an entailment). This regular logic doesn't include the "additiveness" of an abelian category, but you could augment it by axioms making every object an abelian group object and every morphism an abelian group homomorphism.


