Why exact categories? history

Why exact categories? history

[Colin McLarty]

	Myles Tierney has told me about a perspective on exact categories,
in the 1960s, that I had not understood. I probably still do not see it
very much the way it looked then. So I ask about it here. What were the
reasons for studying exact categories in the 1960s?

	Here is what I used to think: Every additive category with a
generator has a faithful functor to the category of Abelian groups.
MacLane had explored this idea in 1950. Then Grothendieck's Tohoku paper
axiomatized Abelian categories in a more useful way for homological
algebra, and showed that all sheaf categories satisfied the axioms (i.e.
sheaves of Abelian groups on topological spaces, and the key theorem says
they have enough injectives). That created two reasons to look for a
non-additive generalization. First, to extend from Abelian groups to all
groups, for use in non-Abelian cohomology. (MacLane had already hinted at
replacing Abelian groups by all groups in 1950). And second to axiomatize
sheaves of sets. The exact category axioms were a promising non-additive
analogue to the Abelian category axioms.

	And I have always thought of the Abelian category embedding
theorems as proving that, if you want to, you can think of Abelian
categories as concrete categories with the natural limits and colimits.

	Myles did not disagree with any of that but he put it this way:
Not all categories enriched in Abelian groups are so nicely embeddable in
the category of Abelian groups, but the Abelian categories are.

	This suggested a general question, when does an enriched category
embed nicely in the enriching category? And Myles had a good description
of which Abelian-group enriched categories are Abelian: the exact ones. So
the exact category axioms became an approach to this problem.


	To me this question seems very different from looking for a
non-additive analogue of Abelian categories. Am I wrong about that? How
did this question look in, say, 1970? How did it look at Dalhousie?

Thanks, Colin