2025-09-30

This following is a very brief summary of what happened in class on 2025-09-30.

We continued our discussion from last class, of four basic constructions on modules:

  1. Direct products of modules
  2. Direct sums of modules
  3. Intersections of submodules of a given module
  4. Sums of submodules of a given module

After first reviewing the constructions in the case of a pair of (sub)modules, we then proceeded to analyze the first two constructions in the more general case of a finite collection of modules, and then finally in the most general case of an arbitrary family of modules.

This led us to finally come to terms with the vague notions of "ordered tuples" and "formal sums", replacing them (at least in the infinite case) with set maps from the indexing set.

Next class we will finish this discussion, by quickly summarizing the general case, why the infinite direct sum is (usually) not isomorphic to the infinite direct product, and how all four constructions (even in the general infinite case) fit into a simply categorical framework.

And then we'll finally introduce the last player in the foundations of category theory: natural transformations!

Concepts

References