An object of a category is indecomposable if it cannot be expressed as a non-trivial coproduct of objects of . Formally, is indecomposable if given an isomorphism , there is a unique index such that and for all , where is an initial object.
The requirement that be unique keeps the initial object itself from being indecomposable; this is analogous to being too simple to be simple.
An object of is indecomposable if and only if it is connected, that is if the hom functor preserves coproducts.
If is connected, then a morphism factors uniquely as , where is the coprojection. Suppose is invertible – then the composite is the identity. Consider
Of course , while as before. So . But because is extensive the coprojections are monic, so . Thus is an isomorphism , with inverse . Because coproducts in are disjoint, the pullback of distinct coprojections is , and because , the pullback of along is an isomorphism, showing that for .
Conversely, assume is indecomposable. Given , we are to produce a unique as above. Because is extensive, is isomorphic (in the slice category ) to for some family . But gives us by indecomposability of an isomorphism for a unique , which composed with gives a morphism . Because (where the right-hand side is the copairing of the family ), and because the for , the composite is equal to . Hence is connected.
If is a presheaf category (thus a Grothendieck topos and so a fortiori (infinitary) extensive), then it is easy to see that the representable functors are connected and so indecomposable. Conversely, the objects of that are indecomposable as well as projective are precisely the objects of the Cauchy completion of .
Lambek & Scott give a different definition of indecomposability. Generalizing their definition slightly, we may say that an object is indecomposable (in the sense of Lambek–Scott) if any jointly epimorphic family of arrows into contains at least one epimorphism , and moreover the unique arrow is not epic (this to ensure that is not indecomposable).
If the epi is required to be regular, then in an extensive category the Lambek–Scott definition implies that given above: if , then the family is jointly epic, so it contains a regular epi . But extensivity implies that is a monomorphism, so the regular epi is also monic and hence an isomorphism. The converse does not hold in general, but it does hold if is projective. See this MathOverflow thread for a discussion.
An indecomposable representation is precisely an indecomposable object in an appropriate category of representations, as one would expect. In contrast, an irreducible representation is precisely a simple object in . Every irreducible representation is indecomposable, but the converse holds only in special situations (such as the category of finite-dimensional linear representations of a real semisimple Lie group).
However, one level decategorified, an irreducible element of a poset is precisely an indecomposable object of when thought of as a thin category. In contrast, a simple object is analogous to an atomic element, although they are not the same thing. (One might say that atomic = -simple.) Again, every atomic element is irreducible, but the converse holds only in special situations (such as the power set of any set).
The bottom line is that ‘irreducible’ and ‘indecomposable’ sometimes mean the same thing but sometimes don't, and ‘irreducible’ doesn't even mean the same thing across different fields.