nLab
direct sum

The direct sum, or weak direct product, is a concept from algebra that actually makes sense in any category $C$ with zero morphisms (that is, any category enriched over the closed monoidal category of pointed sets), as long as the needed (co)limits exist. In many cases, but not all, direct sums and weak direct products are the same, 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 $C$, 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.

Definition

Let $C$ be a category with zero morphisms, and let $(A_i{)}_{i:I}$ be a family of objects of $C$ indexed by a set $I$. Then …

OK, there are two ways to continue the definition, both of which reproduce the usual one for groups (and for such groups with extra structure/property as abelian groups, vector spaces, etc) but which give different results for pointed sets. I'll assign terminology somewhat arbitrarily. If nobody objects to these definitions, then the page should probably be split into two, and then they can wait for somebody to find something useful to do with them. —Toby

Direct sum

… then consider the natural map $r$ from the coproduct ${\coprod }_i{A}_i$ to the product ${\prod }_i{A}_i$ that is defined in terms of the zero morphism as follows:

$$(A_i\to \coprod A\stackrel{r}{\to }\prod A\to A_j)=\{\begin{array}{cc}{\mathrm{Id}}_{A_i}& \mathrm{if}i=j\\ 0_{\mathrm{ij}}& \mathrm{if}i\ne j,\end{array}$$\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 $0_{\mathrm{ij}}$ is the zero morphism from $A_i$ to $A_j$.

Then if $C$ is a regular category or otherwise has a good concept of image, we define the direct sum ${⨁}_i{A}_i$ to be the image of the map $r$.

In constructive mathematics, the definition of $r$ requires that the index set $I$ have decidable equality, which is the case in most applications of interest. An arbitrary index set will still work if $C$ 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 $C$ are not (constructively) so enriched.

Weak direct product

… then consider the finite product?s

$$\prod _{i\in F}{A}_i$$\prod_{i \in F} A_i

as $F$ varies over the finite subsets of the index set $I$. (In constructive mathematics, use the ‘finitely indexed’ or ‘Kuratowski’ notion of finiteness here.)

Mike: Are there categories that have finitely-indexed products? I don’t see how one could construct them even in $\mathrm{Set}$.

Toby: If they don't exist, then that's no big deal; we only want them if $F$ is a subset of $I$, and $F$ will have decidable equality (and therefore be finite in the strictest sense) whenever $I$ does (which it must already have to define direct sums in general). I put in that bit about Kuratowski-finiteness just to be fair in case $I$ is not decidable and may in fact have very few subsets with decidable equality.

However, $\mathrm{Set}$ certainly has finitely-indexed products; it has all (small) products, after all. For example, consider the quotient set $I$ of $2$ given by the truth value $p$ and a constant $I$-indexed family of sets with the value $A$. Then the product of this family is the set of functions from $I$ to $A$. If $p$ is true, then the product is $A$, and if $p$ is false, then the product is $A^2$; in general, the product is a subset of $A^2$:

$$\{a,b:A\mid p\Rightarrow a=b\}.$$\{ a , b : A \;|\; p \;\Rightarrow\; a = b \} .

See?

Mike: Right, I was being silly. The point is rather than having finitely-indexed products is, in general, stronger than having finite products.

Toby: Yes, to which I say: we don't need to have them, since direct sums don't need to exist; but if $I$ has finitely-indexed subsets that may not be finite, then we really should include these (even at the risk that the products might not exist) or the direct sum will be too small.

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

$$\prod _{i\in F}{A}_i\to \prod _{i\in G}{A}_i,$$\prod_{i \in F} A_i \to \prod_{i \in G} A_i ,

where $F\subseteq G$, given by

$$\prod _{i\in F}{A}_i\cong \prod _{i\in F}{A}_i\times \prod _{i\in G\setminus F}1\stackrel{(\mathrm{id},0)}{\to }\prod _{i\in F}{A}_i\times \prod _{i\in G\setminus F}{A}_i\cong \prod _{i\in G}{A}_i.$$\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 .

Then if it exists, the weak direct product ${\prod }_i^{\mathrm{wk}}{A}_i$ is defined to be the directed colimit of this direct system.

Examples

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

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

$$\underset{i:I}{⨁}{A}_i=\{(a_i{)}_i\mid \mathrm{ess}\forall (i:I),a_i=0\},$$\bigoplus_{i: I} A_i = \{ (a_i)_i \;|\; ess \forall (i: I),\; a_i = 0 \} ,

where ‘$\mathrm{ess}\forall$’ 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.

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 $2$ pointed sets, this is different from the weak direct product (which is, as always, the same as the product for finitely many objects).

Discussion

OK, so which of these is correct for pointed sets? In particular, given pointed sets $A$ and $B$ (where in each we call the basepoint $0$), is the direct sum $A\oplus B$ the wedge sum $A\vee B$ (the image of the coproduct in the product) or the product $A\times B$ (since there are only $2$ sets)? I can turn either into a general definition above, but which is the right one?

Or should we use both? There are, after all, two names: ‘direct sum’ and ‘weak direct product’; I'd naturally use these for the image-of-coproduct and almost-zero-product versions, respectively. Is there already a convention in universal algebra? If not, do people like this terminology? Or do you know that one is definitely what is wanted while the other is useless?

—Toby Bartels

Mike: I’ve only ever heard “direct sum” used to mean “coproduct,” or sometimes “finite biproduct,” in an additive category.

Toby: Well, it's definitely also used in the sense of direct sum of groups; the same concept is also called ‘weak direct product’. I thought that I once knew how this worked in general, from a universal-algebra perspective, but when I started writing this, I found that I had forgotten (or never really understood) ….