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 t