so let's consider for example the functor f : _ab gp_ -> _ab gp_ given by x |-> ext(z/2,x). we want to express this as a syntactic construction that makes sense in any abelian category; i guess that this is "cokernel of the endomorphism 2 on x".

for example how is f realized on the opposite of the category of abelian groups?? hmm, it seems to be "kernel of 2", which is just "homming from z/2" ....???....


is there an analog of the "extension" interpretation of f in the general case?

for another example let's consider g given by x |-> tor(x,z/2) (or whatever that notation should be...).

back to f... let's consider f as a module of the ringoid of fp abelian groups... or something like that... try to give a nice presentation of it... ??so take one generator g at the object z, and one relator "2g" ... ??is that correct??