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direct sum

Direct sums and weak direct products

Idea

The notion of direct sum, or weak direct product, is a concept from algebra that actually makes sense in any category CC with zero morphisms (that is, any category enriched over the closed monoidal category of pointed sets), as long as the needed (co)limits exist.

As basic and familiar example is the direct sum V 1V 2V_1 \oplus V_2 of two vector spaces V 1V_1 and V 2V_2 over some field, or, more generally of two modules over some ring. Generally, for II a set and {V i} iI\{V_i\}_{i \in I} an II-indexed family of vector space or modules, their direct sum iIV i\oplus_{i \in I} V_i is the collection of formal linear combinations of elements in each of the V iV_i. This may in part motivate the terminology: an element in a direct sum is a sum of elements, at least in these cases.

This generalises in two distinct ways, which we call direct sums and weak direct products. In many cases (as in the example above), these coincide, but not always. Also in many cases, direct sums will be the same as coproducts. In any case, finitary weak direct products are the same as products but the infinitary versions are (almost always) different.

Terminology

The name ‘weak direct product’ comes from the concept of direct product in algebra for a product in a concrete category that is created by the forgetful functor; the weak direct product will be a subobject of the direct product (and the entire direct product in finitary cases). But here we will not restrict ourselves to the context of such a concrete category.

The term ‘direct sum’ comes from the finitary biproduct (simultaneously product and coproduct) in additive categories. The additive character of these biproducts extends in the infinitary case (where biproducts generally no longer appear) to the coproduct rather than to the product. Even when the direct sum is not the same as the coproduct, it still retains some of this flavour.

In the classical examples of CC, the direct sum and weak direct product are the same. However, the general definitions below distinguish them in some cases, and we use the terms ‘direct sum’ and ‘weak direct product’ to best evoke the ‘like a coproduct’ and ‘part of a product’ senses.

Definitions

Let 𝒞\mathcal{C} be a category with products and coproducts, as well as zero morphisms. Let II be a set, and let (A i) iI(A_i)_{i \in I} be an II-indexed family of of objects in 𝒞\mathcal{C}, hence a function A:IObj(𝒞)A : I \to Obj(\mathcal{C}).

We now define both the direct sum and weak direct product of this family. The A iA_i will be called the direct summands or (weak) direct factors.

Direct sum

Here we must assume moreover that 𝒞\mathcal{C} is a regular category (or otherwise has a good concept of image).

Definition

Let rr be the morphism from the coproduct iA i\coprod_i A_i to the product iA i\prod_i A_i characterized by having the following components

(A iArAA j)={Id A i ifi=j 0 ij ifij, \left( A_i \to \coprod A \stackrel{r}{\to} \prod A \to A_j \right) = \left\{ \array{ Id_{A_i} & if i = j \\ 0_{ij} & if i \neq j ,} \right. \,

where 0 ij0_{ij} is the zero morphism from A iA_i to A jA_j.

The direct sum over the family {A i}\{A_i\} is the image

iIA I iA i \oplus_{i \in I} A_I \hookrightarrow \prod_i A_i

of the morphism rr.

Remark

In constructive mathematics, the definition of rr requires that the index set II have decidable equality, which is the case in most applications of interest. An arbitrary index set will still work if 𝒞\mathcal{C} is enriched over the category of sets and partial functions; this may be embedded as a full subcategory of the category of pointed sets, and the embedding is an equivalence of categories if and only if the law of excluded middle holds. But the usual examples of 𝒞\mathcal{C} are not (constructively) so enriched. Fortunately, the usual examples of II have decidable equality.

Weak direct product

Here we consider the finitary products

iFA i \prod_{i \in F} A_i

as FF varies over the finite subsets of the index set II. (In constructive mathematics, use ‘finitely indexed’ or ‘Kuratowski finite’ here … although if II has decidable equality, as is the case in the usual examples, then every finitely indexed subset of II is actually finite in the strictest sense.)

These finite products form a direct system indexed by the directed set 𝒫 finI\mathcal{P}_{fin}I of finite subsets of II (ordered by inclusion) with the map

iFA i iGA i, \prod_{i \in F} A_i \to \prod_{i \in G} A_i ,

where FGF \subseteq G, given by

iFA i iFA i× iGF1(id,0) iFA i× iGFA i iGA i. \prod_{i \in F} A_i \cong \prod_{i \in F} A_i \times \prod_{i \in G \setminus F} 1 \stackrel{(id, 0)}{\to} \prod_{i \in F} A_i \times \prod_{i \in G \setminus F} A_i \cong \prod_{i \in G} A_i .
Definition

If it exists, the weak direct product i wkA i\prod^wk_i A_i is defined to be the directed colimit of this direct system.

Examples

Example

In the categories Grp or Ab of (abelian) groups, the direct sum and weak direct product agree. For finitely many objects, it is the same as the direct product, which is the product in both categories.

Remark

In Ab, where finite products are also finite coproducts, the direct sum continues to be the coproduct, while in Grp, it lies between the coproduct (the free product) and the product.

So in Ab the direct sum is the object equipped with a collection of morphisms

A j A k ι j ι k iIA i \array{ A_j &&\cdots && A_k \\ & {}_{\mathllap{\iota_j} }\searrow &\cdots& \swarrow_{\mathrlap{\iota_{k}}} \\ && \oplus_{i \in I} A_i }

which is characterized up to unique isomorphism by the following universal property: for every other abelian group KK equipped with maps

A j A k f j f k K \array{ A_j &&\cdots && A_k \\ & {}_{\mathllap{f_j} }\searrow &\cdots& \swarrow_{\mathrlap{f_{k}}} \\ && K }

there is a unique homomorphism ϕ: iIA iK\phi : \oplus_{i \in I} A_i \to K such that f i=ϕι i f_i = \phi \circ \iota_i for all iIi \in I.

Proposition

In these examples, the direct sum can also be described in more elementary terms as a subgroup of the direct product:

i:IA i={(a i) i:I|ess(i:I),a i=0}, \bigoplus_{i: I} A_i = \left\{ (a_i)_{i : I} \;|\; ess \forall (i: I),\; a_i = 0 \right\} \,,

where ‘essess \forall’ means ‘for all but finitely many’. This makes it clear that the direct sum equals the direct product when there are only finitely many objects involved.

For 𝒞=\mathcal{C} = Ab, RRMod this is the group of formal linear combinations of elements in the summands.

Example

For RR a ring, the direct sums in the category RRMod or modules over RR are given by those on the underlying abelian groups.

Example

In the category of pointed sets, the direct sum and weak direct product are different. The weak direct product is still given as a pointed subset of the direct product as above. The direct sum, on the other hand, is the same as the wedge sum, which is the same as the coproduct in this category. Even for 22 pointed sets, this is different from the weak direct product (which is, as always, the same as the product for finitely many objects).

Example

In the category of Banach spaces (with short linear maps), the direct sum is the l 1l^1 direct sum, while the weak direct product is the l l^\infty direct sum. (There is in fact a range of l pl^p direct sums for 1p1 \leq p \leq \infty, although I don't know what if any universal properties they all satisfy.) In this case, the direct sum is the same as the coproduct, while the weak direct product is the same as the product even for infinitely many objects. See direct sum of Banach spaces.

Revised on November 12, 2012 16:57:12 by Urs Schreiber (89.204.155.124)