nLab internal direct sum

Idea

In algebra one speaks of internal direct sums for direct sums of subobjects of a given object that are themselves again canonically subobjects of that object.

In contrast to this situation, the ordinary direct sum is sometimes called the external direct sum.

Definition

For modules

Traditionally this is considered for modules $N$ over a ring $R$ and stated as follows (see the references below):

Let $\big\{N_i \subset N\big\}_{i \in I}$ be an indexed set of of submodules $N_i \hookrightarrow N$. Then a submodule

(1)$$\oplus^{int}_{i \in I} N_i \,\xhookrightarrow{\;\; q \;\;}\, N$$

is called their internal direct sum if the following condition holds

• For every $n \in \bigoplus^{int}_{i \in I} N_i$ there is a unique $I$-tuple $(n_i \in N_i)_{i \in I}$ such that their sum in $N$ is $\sum_i n_i \,=\, n$.

Of course, this is equivalent to the condition:

• $\bigoplus^{int}_{i \in I} N_i \,\simeq\, \bigoplus_{i \in I} N_i$ is isomorphic to the abstract (external) direct sum, and $q$ in (1) is the universal morphism induced from the inclusions $N_i \hookrightarrow N$.

In this form the definition clearly generalizes.

Generally

Given an object $B$ and a family of subobjects $A_i$ of $B$ (or more generally a family of morphisms $A_i \to B$, or equivalently a map $\coprod_i A_i \to B$), suppose that the direct sum $\bigoplus_i A_i$ exists. Suppose further that the map $\coprod_i A_i \to B$ factors through the map $\coprod_i A_i \to \bigoplus_i A_i$ (which means that it factors uniquely if $\coprod_i A_i \to \bigoplus_i A_i$ is an epimorphism, as it must be in a regular category). Finally, suppose that the (or a) quotient map $\bigoplus_i A_i \to B$ is an isomorphism. Then we say that $B$ is the internal direct sum of the $A_i$.

In contrast, the abstractly defined direct sum $\bigoplus_i A_i$ may be called an external direct sum. These terms are usually used with concrete categories where the $A_i$ may either be given independently (for an external direct sum) or as subsets of some ambient space (either $B$ or something of which $B$ is a subset) for an internal direct sum. In too abstract a context, there is no difference: on the one hand, any internal direct sum is a fortiori isomorphic to any external direct sum; on the other hand, given an external direct sum, there is a natural map $\coprod_i A_i \to \bigoplus_i A_i$, relative to which the external direct sum is an internal direct sum.

References

On the tradition definition:

• ProofWiki, Internal Direct Sum of Modules

• Lia-Ming Liou, External direct sum and Internal direct sum of vector spaces, lecture notes [pdf]