Today we can define homology before I head up to the Baltimore/DC area for the weekend. Anyone near DC who wants to hear about anafunctors can show up at George Washington University’s topology seminar on Friday.
As a preliminary, we need to know what quotients in an abelian category are. In we think of an abelian group and a subgroup and consider two elements of to be equivalent if they differ by an element of . This causes problems for us because we don’t have any elements to work with.
Instead, remember that comes with an “inclusion” arrow , and that the quotient has a projection arrow . The inclusion arrow is monic, the projection is epic, and an element of the quotient is zero if and only if it comes from an element of that is actually in . That is, we have a short exact sequence . But we know in any abelian category that this short exact sequence means that the projection is the cokernel of the inclusion. So in general if we have a monic we define .
Now we define a chain complex in an abelian category to be a sequence with arrows so that . In particular, an exact sequence is a chain, since the composition of two arrows in the sequence is the zero homomorphism. But a chain complex is not in general exact. Homology will be the tool to measure exactly how the chain complex fails to be exact.
So let’s consider the following diagram
where . We can factor as for an epic and a monic . We can also construct the kernel of . Now , so because is epic. This means that factors through , and the arrow must be monic.
Now, if the sequence were exact then would be the same as , and the arrow we just constructed would be an isomorphism. But in general it’s just a monic, and so we can construct the quotient . When the sequence is exact this quotient is just the trivial object , so the failure of exactness is measured by this quotient.
In the case of a chain complex we consider the above situation with and , so they connect through . We define and , which are both subobjects of . Then the “homology object” is the quotient . We can string these together to form a new chain complex where all the arrows are zero. This makes sense because if we think of the case of abelian groups, consists of equivalence classes of elements of , and when we hit any element of by we get . Thus the residual arrows when we pass from the original chain complex to its homology are all zero morphisms.