```4 Oct 2004
2:30 - 4:00   M Barr

Absolute homology

Abstract:
What does it mean to for a functor to preserve homology?  There is no
universal mapping property that homology has, but mere isomorphism is not
enough.  Let A be an abelian category and DO(A) denote the category of
differential objects over A: pairs (C,d) s.t. dd = 0.  Any additive
functor F: A --> B induces DO(F): DO(A) ---> DO(B) given by DO(F)(C,d) =
(FC,Fd).  (Actually, you don't need F additive to give this definition; it
need only preserve 0, but additivity is required to say anything useful
about it.)  If H is the homology functor, we thus get a square
DO(F)
DO(A) ------> DO(B)
|             |
|             |
H |             |H
|             |
v     F       v
A ----------> B
It does not commute, nor is there even a 2 cell in either direction.
What I will show we have is illustrated below:
DO(A)                               DO(A)
/|\                                 /|\
/ | \                               / | \
/  |  \                             /  |  \
DO(F)/   |   \H                         H/   |   \DO(F)
/   Phi   \                         /   Phi'  \
/     |     \                       /     |     \
v      |      v                     v      |      v
DO(B) <== | ==>  A                     A  ==> | <== DO(B)
\     |     /                       \     |     /
\    |    /                         \    |    /
H\   |   /F                         F\   |   /H
\  |  /                             \  |  /
\ | /                               \ | /
vvv                                 vvv
B                                   B

and, moreover,
Phi ------> HF
|          |
|          |
|          |
|          |
|          |
v          v
FH -----> Phi'
commutes.  The two diagrams are dual to each other so what I am about to
say applies to either one.  We will say that F preserves the homology of
(C,d) when both 2 cells are ismomorphisms, when evaluated there.  I will
show that the left 2 cell is an isomorphism for all (C,d) iff F is left
exact and the right one is an isomorphism iff F is right exact.  The same
is true of the right hand diagram (but note that that diagram is mirror
image reversed from the first).

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