Free abelian stuff

Free abelian stuff

[Peter Freyd]

Given an additive category  *C*, its finite co-completion, R(*C*), may
be constructed as the full subcategory of finitely presented objects
(see footnote for a definition of f.p.) in the category of abelian-
group-valued presheaves on  *C*. The finite completion, L(*C*)  may
therefore be constructed as  (R(*C*`))` (using ` to denote here the
dual category).

Remarkably:
                       R(L(*C*))  =  L(R(*C*))
Moreover:
          It is the free abelian category generated by  *C*.

To be precise, it is an abelian category and every additive functor
from  *C*  to an abelian category extends to an exact functor on
R(L(*C*)), uniquely up to natural equivalence.

(If the target of the additive functor is not abelian then there are
extensions, but not necessarily any that preserve both finite limits
and co-limits.)

If one starts with a one-object additive category -- that is, a ring
-- then its finite co-completion, the category of f.p. presheaves,
is, of course, just the category of f.p. modules (as usually
understood). So the free abelian category generated by a ring, R, may
be constructed as the full category of f.p. covariant abelian-group-
valued functors on the category of f.p. R-modules.

If  R  is commutative (or if it has an anti-involution) then the
freeness means this category of f.p. functors is self-dual.
Moreover:
                    It is a *-autonomous.category.

There are a lot of functors in this category. Besides being closed
under the abelian operations it is closed under composition. (Any
group-valued functor lifts canonically to an R-module-valued functor,
hence composition is defined.) The dual of  Hom(A,--)  is  A * --
(using  *  for tensor product) and the dual of Ext^n (A,--) is
Tor_n (A,--). The dual of a composition  ST  is the composition
of their duals (do not reverse the order of composition).

I found all this when trying recently to simplify the proofs for the
symmetric construction of free abelian categories (Murray Abelian,
Categories Over Additive Ones. J. Pure Appl. Algebra 3 (1973),
103--117). The point of departure was the recognition that free
abelian categories always have enough projectives (and, dually, enough
injectives). Since at least the world's first category theory
conference it has been known that an abelian category with enough
projectives is the finite co-completion of its full subcat of
projectives. (Representations in abelian categories. Proc. Conf.
Categorical Algebra (La Jolla, Calif., 1965) pp. 95--120 Springer, New
York)

Herein is the outline of this alternate approach. I find it hard to
believe that it is new.

We will assume that  *C*  is already additive in the stronger sense,
that is, it has finite direct sums.

For a first-order construction of  R(*C*)  start with the category
of maps in  *C*  (that is, the category of functors from the ordinal
2  to  *C* ) and then reduce by a notion of "homotopy": a map from
A' --> A  to  B' --> B,
                                  a
                               A'--> A

                             f'|     | f

                               B'--> B
                                  b

(all vertical arrows point down) is null-homotopic if a map  A --> B'
can be inserted making the _lower_ of the two triangles commute.

The functor  *C* --> R(*C*)  sends  A  to  O --> A.

A cokernel of the displayed  f-map is constructible as:

                               B' ---> B

                               |       |

                             A + B'--> B

where each map is given by a matrix in which each entry is either 1
or a single letter (b,f), if such fits, else 0.

Given a functor, T, to a finitely co-complete category, define
T(A'--> A)  by choosing a cokernel of  T(A') --> T(A). It is
easy to verify that  T  preserves cokernels.

LEMMA: If  *C*  is finitely complete, then  R(C)  is abelian and
       *C* --> R(*C*) preserves finite limits.

BECAUSE: An additive category is abelian if each map can be factored
as a cokernel (of something) followed by a kernel. The  f-map
displayed map above factors as:
                                  a
                               A'--> A

                               |     | 1

                               P --> A

                               |     | f

                               B'--> B
                                  b

with the lower square is a pullback diagram. It is a good finger-
exercise to see that the upper square is a cokernel of the pullback
square:
                               P'--> P

                               |     |

                               A'--> A

(which being monic is therefore the kernel of the given  f-map) and
the lower square is a kernel of the already mentioned cokernel:

                               B' ---> B

                               |       |

                             A + B'--> B

Finally then, given a functor, T, from a finitely complete  *C*  to an
abelian category it is easy to verify that if  T  preserves finite
limits then the right-exact extension of  T  to  R(*C*)  is not just
right exact but exact.

When  *C*  is finitely complete one may verify that every projective
in  R(*C*)  is isomorphic to an object in the image of  *C* --> R(*C*)
Because that functor preserves kernels we see easily that  R(*C*)  has
global dimension at most two. One may prove that if a free abelian
category has global dimension at most one then it has global dimension
zero. Hence the only global dimensions that can occur are 0 and 2.
(Every abelian category of global dimension zero is a free abelian
category -- of itself. All additive functors therefrom preserve finite
limits and co-limits.)

The dual of the free abelian category generated by  *C*  is
necessarily isomorphic to the free abelian category generated by the
dual of  *C*. Hence (still using ` to denote duals) we have
(R(L(*C*`)))` = R(L(*C*)). But (R(L(*C*`))` = L(L(*C*`)`) = L(R(*C*)).

The *-autonomous structures must wait.

Footnote:

By a FINITELY PRESENTED object in a co-complete category is meant an
object whose corresponding covariant representable functor preserves
filtered co-limits. If  *C*  has finite direct sums then a presheaf
(all presheaves will be understood to be valued in the category of
abelian groups) is f.p. iff it appears as the cokernel of a map
between representables.

Postscript:

Where is the abelianess of the target category being used? If it is
abelian then -- using the notation above -- one can prove that

                T(P') --> T(P) --> Cok(Ta) --> Cok(Tb)

is exact. The extension of  T  is defined so that it preserves the
cokernel of  T(P') --> T(P). One invokes abelianess of the target
to know that it therefore preserves the kernel of  Cok(Ta) --> Cok(Tb).