nLab
indecomposable object

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Indecomposable objects

Definition

An object XX of a category CC is indecomposable if it cannot be expressed as a non-trivial coproduct of objects of CC. Formally, XX is indecomposable if given an isomorphism X iU iX \cong \coprod_i U_i, there is a unique index ii such that XU iX \cong U_i and U j0U_j \cong 0 for all jij \neq i, where 00 is an initial object.

The requirement that ii be unique keeps the initial object itself from being indecomposable; this is analogous to being too simple to be simple.

Remarks

If CC is an extensive category, meaning that coproducts in CC are disjoint and pullback-stable, then we have

Proposition

An object XX of CC is indecomposable if and only if it is connected, that is if the hom functor hom(X,)\hom(X,-) preserves coproducts.

Proof

If XX is connected, then a morphism k:X iU ik \colon X \to \coprod_i U_i factors uniquely as ι ik¯:XU i iU i\iota_i \circ \bar k \colon X \to U_i \to \coprod_i U_i, where ι i\iota_i is the coprojection. Suppose kk is invertible – then the composite k 1ι ik¯:XU i iU iXk^{-1} \iota_i \bar k \colon X \to U_i \to \coprod_i U_i \to X is the identity. Consider

U iι i iU ik 1Xk iU i U_i \overset{\iota_i}{\to} \coprod_i U_i \overset{k^{-1}}{\to} X \overset{k}{\to} \coprod_i U_i

Of course kk 1=1k \circ k^{-1} = 1, while k=ι ik¯k = \iota_i \bar k as before. So ι ik¯k 1ι i=ι i\iota_i \circ \bar k k^{-1} \iota_i = \iota_i. But because CC is extensive the coprojections are monic, so k¯k 1ι i=1\bar k k^{-1} \iota_i = 1. Thus k¯\bar k is an isomorphism XU iX \cong U_i, with inverse k 1ι ik^{-1} \iota_i. Because coproducts in CC are disjoint, the pullback of distinct coprojections is 00, and because U iX iU iU_i \cong X \cong \coprod_i U_i, the pullback of ι i\iota_i along ι j\iota_j is an isomorphism, showing that U j0U_j \cong 0 for jij \neq i.

Conversely, assume XX is indecomposable. Given k:X iU ik \colon X \to \coprod_i U_i, we are to produce a unique XU iX \to U_i as above. Because CC is extensive, kk is isomorphic (in the slice category C/ iU iC/\coprod_i U_i) to ik i: iX i iU i\coprod_i k_i \colon \coprod_i X_i \to \coprod_i U_i for some family {k i:X iU i}\{k_i \colon X_i \to U_i\}. But X iX iX \cong \coprod_i X_i gives us by indecomposability of XX an isomorphism XX iX \cong X_i for a unique ii, which composed with k ik_i gives a morphism XX iU iX \cong X_i \to U_i. Because ik i=[ι ik i] i\coprod_i k_i = [\iota_i k_i]_i (where the right-hand side is the copairing of the family {ι ik i} i\{\iota_i k_i\}_i), and because the X j0X_j \cong 0 for jij\neq i, the composite ι ik i:XX iU i iU i\iota_i k_i \colon X \cong X_i \to U_i \to \coprod_i U_i is equal to kk. Hence XX is connected.

If CC is a presheaf category [S op,Set][S^{op}, Set] (thus a Grothendieck topos and so a fortiori (infinitary) extensive), then it is easy to see that the representable functors S(,s)S(-,s) are connected and so indecomposable. Conversely, the objects of [S op,Set][S^{op}, Set] that are indecomposable as well as projective are precisely the objects of the Cauchy completion of SS.

Lambek–Scott indecomposability

Lambek & Scott give a different definition of indecomposability. Generalizing their definition slightly, we may say that an object XX is indecomposable (in the sense of Lambek–Scott) if any jointly epimorphic family {U iX} i\{U_i \to X\}_i of arrows into XX contains at least one epimorphism U iXU_i \twoheadrightarrow X, and moreover the unique arrow 0X0 \to X is not epic (this to ensure that 00 is not indecomposable).

If the epi U iXU_i \twoheadrightarrow X is required to be regular, then in an extensive category the Lambek–Scott definition implies that given above: if k:X iU ik \colon X \cong \coprod_i U_i, then the family {k 1ι i:U i iU iX} i\{k^{-1} \iota_i \colon U_i \to \coprod_i U_i \cong X\}_i is jointly epic, so it contains a regular epi ι ik 1\iota_i k^{-1}. But extensivity implies that ι i\iota_i is a monomorphism, so the regular epi ι ik 1\iota_i k^{-1} is also monic and hence an isomorphism. The converse does not hold in general, but it does hold if XX is projective. See this MathOverflow thread for a discussion.

Indecomposability vs irreducibility

An indecomposable representation is precisely an indecomposable object in an appropriate category RepRep of representations, as one would expect. In contrast, an irreducible representation is precisely a simple object in RepRep. 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 PP is precisely an indecomposable object of PP 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 = 00-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.

Revised on September 20, 2011 00:46:42 by Finn Lawler (86.44.193.15)