nLab internal direct sum

Context

Algebra

Limits and colimits

Contents

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 NN over a ring RR and stated as follows (see the references below):

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

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

is called their internal direct sum if the following condition holds

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

Of course, this is equivalent to the condition:

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

In this form the definition clearly generalizes.

Generally

Given an object BB and a family of subobjects A iA_i of BB (or more generally a family of morphisms A iBA_i \to B, or equivalently a map iA iB\coprod_i A_i \to B), suppose that the direct sum iA i\bigoplus_i A_i exists. Suppose further that the map iA iB\coprod_i A_i \to B factors through the map iA i iA i\coprod_i A_i \to \bigoplus_i A_i (which means that it factors uniquely if iA i iA i\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 iA iB\bigoplus_i A_i \to B is an isomorphism. Then we say that BB is the internal direct sum of the A iA_i.

In contrast, the abstractly defined direct sum iA i\bigoplus_i A_i may be called an external direct sum. These terms are usually used with concrete categories where the A iA_i may either be given independently (for an external direct sum) or as subsets of some ambient space (either BB or something of which BB 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 iA i iA i\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:

category: algebra

Last revised on July 5, 2024 at 09:18:46. See the history of this page for a list of all contributions to it.