2025-10-02

This following is a very brief summary of what happened in class on 2025-10-02.

We began by outline the notions of direct product and direct sum for an arbitrary family ${M_{s}}_{s \in S}$ of $R$ -modules indexed by a set $S$ . We noted that the direct product $\prod_{s \in S} M_{s}$ can be constructed as the set whose elements are functions $f$ from $S$ (to either the universal $U$ , or any set in the universe containing all of the elements of all of the modules in the family) satisfying the condition that $f (s) \in M_{s}$ for every $s \in S$ . We briefly outlined the binary operation on that set of functions ("add outputs") as well as the $R$ -action ("act on outputs"), as well as the "projection" maps $π_{t} : \prod_{s \in S} M_{s} \to M_{t}$ for each $t \in S$ . We then sketched out how one can verify this module (together with those morphisms) satisfy the "expected universal property" of a product.

Following that, we briefly explained how the same module also has "injections" $i_{t} : M_{t} \to \prod_{s \in S} M_{s}$ , but these cannot satisfy the analogous universal property of a direct sum (i.e., coproduct). The fix, however, was to restrict to the submodule of the direct product consisting of those functions $f$ for which $f (s) = 0_{M_{s}}$ for all but finitely many $s$ .

We then moved on to introducing the notion of a natural transformation between functors. We covered the definition and a few first examples, but we still have a lot more to see.

Concepts

References

Dummit & Foote, Abstract Algebra: Section 10.3
Mac Lane, Categories for the Working Mathematician: pp. 16-18