Thursday, 27 March 2008

ct.category theory - Indecomposable objects in a category


Briefly: there's a simple difference in how they treat 0. That fixed, still neither implies the other in general. In a regular extensive category, a slight modification of the LS definition implies the Elephant one. I suspect they're not fully equivalent in anything short of a topos. As Mike Shulman points out, even in a topos they are not equivalent.




The simple difference: 0 is always indecomposable by Lambek and Scott's definition (since any map into 0 is epi), but never by the Elephant's (since the uniqueness condition won't hold; or by considering when the coproduct decomposition is empty). So, let's temporarily change one of the definitions to fix this. I'd suggest we add “…and the map 0toX is not epi.” to Lambek and Scott's definition. (As you noted, their binary condition generalises to a k-ary one; this is just the case k=0.)



In eg Top, however, we can see that the Elephant def still doesn't imply the LS def. [0,1] satisfies the former (it's not decomposable by an iso), but not the latter (it is decomposable by an epi). Even more, it’s decomposable by a regular epi (more on this distinction below).



Conversely, the LS definition doesn't imply the Elephant one either; it fails in eg mathbfSetmathrmop, since in mathbfSet, 0 is co-decomposable by iso (0congAtimes0) but not co-decomposable by monos (for any map (f,g)colon0toAtimesB, not just one but both of f and g are mono).



When do they imply each other? If we upgrade the LS definition to involve regular epis, then in a regular lextensive category, it implies the Elephant definition, if I'm not mistaken. For this, suppose X is “indecomposable by reg epis”, and suppose XcongA+B — WLOG X=A+B. The coproduct inclusions are then jointly reg epi, so one of them is reg epi. But it's also mono (in a lextensive category, every coproduct inclusion is a pullback of 1to1+1, so is mono); so it's iso. There's a little more fiddly stuff to check involving messing around with 0, but it's all the same sort of thing.



Edit from Mike Shulman's comments: if moreover we're in a pretopos, all epis are regular, so there the original LS definition will imply the Elephant definition. On the other hand, the Elephant definition doesn't imply the LS even in a topos: the terminal object of mathbfSh([0,1]) is a counterexample, essentially for the same reasons that [0,1] was a counterexample in mathbfTop.



However, the two definitions are equivalent for projective objects… and I guess that's how this situation has arisen, since a common use of indecomposable objects in topos theory is the theorem that the indecomposable projectives in a presheaf category are exactly the retracts of representables. (This is useful because it lets us recover the idempotent-completion of mathbfC, which is very close to mathbfC itself, from [mathbfCmathrmop,mathbfSet].)

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