the "walking mono" finitely complete k-algebroid is, by (?some version of) gabriel-ulmer duality, the opposite of the k-algebroid of monos between fd vector spaces over the field k. there are two indecomposable objects 0>->k and k>->k.

now what are the fp op-modules of this k-algebroid? i guess that you can think of them as modules of the endomorphism algebroid of the direct sum of the two indecomposable objects. so what is this endomorphism algebroid like? is it the algebroid of 2-by-2 triangular matrixes?? ...

i guess that that sounds correct... ??so are we claiming that the k-algebroid of fd reps of the a2 quiver is the "walking short exact sequence" abelian k-algebroid? if so then this seems somewhat suggestive... except that i'm not really sure what it's suggesting yet...

let's try working this out a bit more carefully...

the 2 representable modules of the 2-object algebroid should correspond to the quiver reps 1>->1 and 0>->1 ...

so what are the indecomposable a2 reps? do they correspond to just the positive roots? i haven't thought about this stuff in a while...

so is it really true that every object in the "walking short exact sequence" abelian k-algebroid is a finite direct sum of the three indecomposable objects "the subobject", "the total object", and "the quotient object" ??

so if this is on the right track then in what ways can/should it be generailzed??

i'm vaguely wondering whether there's something vaguely like "diaconescu's theorem" lurking here. i don't really know the history of diaconescu's theorem but i always had the vague feeling that it was developed as an analog of something else...