# Intuition behind exact sequences as extensions in algebraic geometry

Introduction: There are certain categories of group schemes or group varieties in which one can define exact sequences. For instance, the categories of commutative group schemes of finite type over a field $$k$$ and of commutative affine group schemes over $$k$$ are abelian (see for instance here). So let's say that we have a short exact sequence $$0 \to A \to B \to C \to 0$$ in such a suitable category (bonus question: examples of abelian categories of group varieties?). There is, a priori, no reason why the sequence should be split. Yet people call such a $$B$$ (together with the maps) fitting in the above exact sequence an extension of $$C$$ by $$A$$. Note that this is true also for other categories and does not have to do with algebraic geometry specifically, but I am most interested in this setting.

My question: What kind of information can one deduce, in general, about $$B$$, knowing only $$A$$ and $$C$$? Does one know something about the underlying topological space of $$B$$ or of $$B(k)$$ up to homotopy, or even up to homeomorphism? Is it any better if we consider maps up to isogeny? Feel free to restrict to some specific category where there are better results.

I am asking this because the terminology seems to suggest, and it seems to me that people often assume, that $$B$$ is some kind of direct sum between $$A$$ and $$C$$, even though there is a priori no reason for this to be the case. The isomorphism theorem does imply that $$B/A \cong C$$, but this does not imply that we know how to obtain $$B$$ knowing $$A$$ and $$C$$, since there might be many possibilities for doing so.

Approximate paraphrase 1: A paraphrase of my question is: what do all possible $$B$$'s that fit in the above sequence have in common, topologically?

Approximate paraphrase 2: Is there some exact functor $$F$$ from some "geometric" category $$\mathcal{I}$$ to some "topological" category $$\mathcal{J}$$ such that in $$\mathcal{J}$$ exact sequences are always split or $$F(B)$$ is at least somehow isomorphic to $$F(A) \oplus F(C)$$ or something similar?

Note: I am aware of the fact that, at least for modules, the set of the $$B$$'s fitting in the above exact sequence, up to some equivalence and with some sum operation, is isomorphic to $$Ext^1(C,A)$$, but I think that the term extension gave rise to the notation $$Ext$$ rather than the converse, so this fact does not provide any useful insight.