Motivation/intuition behind the definition of delta-functors and related concepts I originally posted this on Maths SE, but then realised that the question probably fits MO better, as my objective was to gain different perspectives regarding the matter.


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*Why are $\delta$-functors between abelian categories $\mathfrak{A}$ and $\mathfrak{B}$ defined as a collection of (right) derived functors, and connecting morphisms $\delta^i: T^i(A'') \rightarrow T^{i+1}(A')$ (given an original short exact sequence $0 \rightarrow A' \rightarrow A \rightarrow A'' \rightarrow 0$ in $\mathfrak{A}$)
$$(T^*, \delta^*) := \{(T^i, \delta^i)\}_{i \geq 0}$$
Particularly, in the context of algebraic geometry, what about this definition is geometric ?

*What is a geometric interpretation of the universality of $\delta$-functors ?

*What is a geometric interpretation of effaceable $\delta$-functors, and by extension, theorem 2.2.2 in Grothendieck's Tohoku paper stating that effaceability implies universality ?

 A: To be honest, the question is a bit borderline for MO, but I'll make a few comments anyway. I would argue that there is nothing inherently geometric about the definition of (universal) delta functors, but the end result is  certainly very useful in geometry, and algebraic geometry in  particular. Prior to Grothendieck's Tohoku paper, sheaf cohomology was defined as Cech cohomology. This is very nice and concrete, and works well in most cases, but it does have pathologies. Grothendieck, in Tohoku, realized that the right definition for sheaf cohomology was via derived functors invented a few years before by Cartan-Eilenberg. But he added many insights of his own. 
To be specific, one would like a short exact sequence of sheaves to give a long exact sequence of cohomologies, i.e. the global sections functor should extend to a $\delta$-functor. But perhaps  there are  many ways to do this, or no ways at all-- so pick the one that dominates all the others, which would be the universal $\delta$-functor, provided it exists. Grothendieck went on to show that it did  exist (for sheaf cohomology), and that it also coincided with the right derived functor for global sections.
Hope that helps.
A: I am not an expert on the subject but looking at the definition I would dare to say that these are basically the axioms that characterize an homology theory, if it weren't for the fact the $\delta$ maps usually go in the opposite directions.
The family of functors play the role of the various homology functors (one for each degree) and the $\delta$-maps are required to extend the family of mappings
$$T^n(A') \to T^n(A) \to T^n(A'')$$ 
(obtained applying the functors $T^n$'s to the two non trivial morphism of the short exact sequence)
to a long exact sequence.
The additional axioms on the data for a $\delta$-functor are required so that the long exact construction can be extended to a functor from the category of short exact sequences in the domain category $\mathcal{A}$ to a category of long exact sequences in the category $\mathcal B$.
If you take a look these are some of the Eilenberg-Steenrod's axioms for an homology theory: you are basically dropping the axioms for the excision which could be meaningless in a generic abelian category.
The universal property capture another property of homology theory: the fact that it is completely determinated by its 0-degree functor, indeed two homology theories satisfying the Eilenberg-Steenrod axioms that have naturally isomorphic 0-degree functors are isomorphic as homology theories.
Basically a universal $\delta$-functor seems to be a sort of homology that is completely determinated by its 0-degree part.
This should address part 1 and 2. Unfortunately I am not familiar with the Tohoku, so I can't address part 3, but hopefully someone better qualified than myself will be able to answer that.
Meanwhile I wish this was of some help.
