When does "splits" imply "cosplits"?

Question

In the category of groups, there are lots of "exact sequences", e.g. 4 → H → 2, that neither split nor cosplit, where H is the eight-element group of quaternions, and lots of sequences like 4 → D → 2 that split but do not cosplit, where D is the eight-element dihedral group. By "2" and "4" I mean the cyclic groups of those orders. By "exact sequence" A → B → C, I mean that A is the kernel of the quotient B → C (equivalently C is the cokernel of the subobject A → B). A sequence A → B → C "splits" if there is a map C → B so that the compotision C → B → C is the identity; cosplitting is on the other side.

So in groups, a split exact sequence does not necessarily cosplit. (In fact, I have a hard time thinking of any cosplit sequences.) On the other hand, my friends who do ring theory state definitions like "A ring is semisimple if any short exact sequence of modules splits". Why don't they ask for the sequence to cosplit? Does that come for free? (Or am I misremembering the definition?)

More generally, what conditions does one have to place on a category so that "splits" implies "cosplits"?

Answer 1

Yes, it comes for free. A short exact sequence of abelian groups, or R-modules (R not necessarily commutative), splits iff it cosplits iff the middle term is the sum of the other two terms. The key here is that maps of modules and be added/subtracted:

Let the middle maps in 0→A→B→C→0 be f:A→B and g:B→C. Then if there's a splitting q:C→B, then (1_B-qg):B→B is a projection onto A as a submodule of B, i.e. a cosplitting. Together the splitting and cosplitting exhibit B as the direct sum of A and C.

A dual trick shows that a cosplit sequences are split: if p:B→A is a cosplitting, then (1_B-fp):B→B is a projection onto a submodule of itself which is isomorphic to C via g, i.e. a cosplitting, so again B is the sum of A and C via these maps.

More generally, this same trick works in any abelian category. One way to recognize this instantly is via Freyd's Exact Embedding theorem, which roughly implies that you can pretend a diagram in an abelian category is a diagram of R-modules for some R.

... whoah... are you guys related?

Answer 2

So the fact that you had a hard time thinking of cosplit sequences of groups and the last question got me thinking (along the lines of Joel's comment actually)... what I came up with is probably standard (to people who well know it).

Suppose the sequence A-> B-> C is an exact sequence of groups. Then if it splits B is a quotient of the free product A*C (the coproduct in Grp) and if it cosplits B is a subgroup of the product AxC.
The idea (in the splitting case - this is enough since they are dual) is to use the universal property + the splitting to get a map from the coproduct. Then use the factorizations + exactness + element chase to check it is an epi in terms of right cancellation. There might be a slicker way to do this, but a way to get it for free from universal properties didn't occur to me.

This gives a philosophical explanation of why split sequences are easy to find, they are given by a presentation for B in terms of A and C (e.g. your dihedral group example). I think cosplitting seems a bit weirder since one is defining a group as a subgroup of a product - is this less natural to people who actually do group theory?

I haven't checked but I suspect that one cannot have a split and cosplit sequence in Grp - it seems like the fact that there is no biproduct should be an obstruction to this based on the above but I am not sure. The largest class of categories which springs to mind where Andrew's trick works would be quasi-abelian categories for strict exact sequences (so pretty much exact categories).

Finally I thought I'd point out some whackier examples where one has splitting iff cosplitting behaviour. In a triangulated category one has that every monomorphism is split and every epimorphism is split - in particular a triangle "splits" iff it "cosplits". The same is actually true for "exact sequences of triangulated categories". A fully faithful exact functor S -> T admits a right adjoint (cosplitting) iff T -> T/S admits a right adjoint (splitting).