I am trying to wrap my mind around Tannaka-Krein duality and it seems quite mysterious for me, as well, as its history. So let me ask:

Question: What was the motivation and historical context for works of major contributors to the "Tannaka-Krein theory" (in a broad sense) ? Just to name a few names: Tannaka, Krein, Saavedra, Deligne, Milne, Lurie, (it seems Grothendieck should also be in this list (?) ).

Let me explain some points in the history which seems to me puzzling:

1) Mark Krein was a famous Soviet mathematician, but he was an expert in analysis, it seems it is the only paper by him devoted to algebra. How did he come to it ? Why did not continue ?

2) Similar question about
Tadao Tannaka. "His interest in mathematics lied mainly in algebraic number theory" , And it seems similar to Krein, it is the only work by devoted to group theory. (See his publication list).

3) P. Deligne seems devoted quite much efforts on "Tannakian formalism" and more generally on tensor categories. What was his motivation ? He is leading algebraic geometer. So probably the subj. should be quite important in algebraic geometry ? What is its importance ?

4) Wikipedia article starts with a sentence: "... natural extension to the non-Abelian case is the Grothendieck duality theory." What is the role of Grothendieck in this history ? And what is "Grothendieck duality theory" - wikipedia links to something not related.

5) Important work was done by Saavedra. Seems not so much is known about him, his motivation, his other works.

6) J. Lurie seems to develop the theory further (see e.g. MO question Tannakian formalism). What is motivation ?