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).