Mac Lane and Abelian Categories

Mac Lane and Abelian Categories

[Colin McLarty]

	This is to correct some remarks I made at the History of Categories
meeting in Montreal in August, and to substantiate what I have written in
the on-line Encyclopedia Britannica entry on Saunders Mac Lane. Mac Lane
deserves credit for both the name "Abelian category" and the concept, in
his paper "Duality for groups" (BAMS 1950, 485-516). 

	Mac Lane 1950 defines "Abelian categories" in section III. He says an
"Abelian category" is a category with a generator and a cogenerator and
with zero object and biproducts (which he calls "free-and-direct products).
 He proves that in such a category, the arrows from any object A to another
B form an additive commutative monoid (he actually says "semigroup",
p.511); and that any such category is isomorphic to a category of
commutative monoids (p.512).

	Of course this is far from our current definition of Abelian categories.
But Mac Lane comes very close to the current definition with his "Abelian
bicategories" on p.513 (using terms from 503). The definition makes heavy
use of "submaps" and "supermaps". 

	First, let me simplify by taking "submaps" to be monics, and "supermaps"
to be epics. In these simplified terms Mac Lane's axioms for an "Abelian
bicategory" say it is an "Abelian category" such that every arrow has an
epic-monic factorization and:

LC-1 the subobjects of any object A form a complete lattice.

LC-2 sups of subobjects are preserved by direct images.

and furthermore:

ABC-1 The arrows from any A to another B form a commutative group.

ABC-3 every subobject is a kernel and dually.


These imply that every arrow has a kernel (sup of all subobjects killed by
the arrow) and dually.

Mac Lane's other axioms ABC-2 and ABC-4 and 5, are redundant in this
simplified case.

	So in this simplified form an "Abelian bicategory" for Mac Lane in
1950 is what we would call an Abelian category with generator and
cogenerator and with sups of subobjects (preserved by direct image). Mac
Lane talks about pre-images of sups also, in terms not exactly ours today,
and I have not worked out the connection.


	Now, to stop simplifying, Mac Lane's "submaps" and "supermaps" are not
defined but axiomatized. A category with submaps is a category with a
distinguished class of monics satisfying certain axioms (which imply that
every split monic is in the class, p.499). "Supermaps" are axiomatized
dually. And the axioms imply that every arrow has a "supermap-submap"
factorization.

	Actually the axioms begin with a much narrower class of selected monics
called "injections", and corresponding epics called "projections", such
that every arrow factors uniquely as a projection followed by an
isomorphism followed by an injection. The axioms pose further conditions,
apparently all based on the idea that injections should correspond to
certain set theoretic inclusions. I have not examined them closely.

	A submap is defined to be any isomorphism followed by an injection, and
dually for supermaps and projections.

	Clearly the motivation for using "submaps" rather than monics was
that Mac Lane hoped to extend this approach to include the category of all
groups, with normal monics as the "submaps". See the footnote on p.513
saying "proofs of the first and second isomorphism theorems for all groups
can be based on 'categorical' axioms".

	I think it fair to count this business of submaps and supermaps as an
inessential complication.

	Grothendieck radically simplified the axioms, extended them, and
found far more penetrating applications than Mac Lane had in mind in 1950.
I am sure Grothendieck did not know of Mac Lane's paper directly, and I
suspect that once he heard something like this was possible he worked it
out for himself. One key to Grothendieck's improvements is that
Grothendieck proceeds in more purely categorical terms. But the name and
the concept are from Mac Lane