### Homological Algebra Puzzle

#### Posted by John Baez

James Dolan gave me another puzzle today.

This one is a bit more sophisticated. Can you find a really nice solution? I’m also curious to hear how well-known this fact is. (Neither of us know a reference.)

Suppose $A$ is an abelian category and let $Arr(A)$ be the arrow category of $A$. There are obvious functors $ker, coker : Arr(A) \to Arr(A)$. Prove that $coker$ is the left adjoint of $ker$.

Or if that’s too jargon-packed, let me expand it out a bit for you:

Suppose $A$ is an abelian category. Let $Arr(A)$ be the category where an object is a morphism $f : a \to b$ in $A$, and a morphism is a commutative square in $A$. There are functors

$ker, coker : Arr(A) \to Arr(A)$

where $ker$ sends any morphism $f : a \to b$ in $A$ to the obvious morphism

$ker(f) \to a$

while $coker$ sends it to the obvious morphism

$b \to coker(f)$

Show that the functor $coker$ is left adjoint to the functor $ker$.

And here’s a question for the experts: if we have an additive category with kernels and cokernels, and $ker$ is right adjoint to $coker$ in the above sense, is it abelian? If so, is this characterization already known?

## Re: Homological Algebra Puzzle

This adjunction holds in any category enriched in pointed sets with all kernels and cokernels (so it doesn’t imply abelianness). It’s a special case of a more general scheme: in general, quotients of some kind are left adjoint to the corresponding kernels.

This general theory was developed in “Factorizations in bicategories”, by Betti, Schumacher and Street, which was never published, but Betti has a paper in JPAA where it is described: “Adjointness in descent theory”. If I remember correctly, they attribute the general adjunction to Robert Paré. Their proof is, I think, very nice (it takes one line, once kernels and quotients are defined in the appropriate way). You can also look at Section 1.2.2 (and around) of my PhD thesis.