2024-10-10

This following is a very brief summary of what happened in class on 2024-10-11.

We began by recapping the definition of an adjunction between two categories. We then spent the remainder of class considering the situation of a left adjoint to the forgetful functor $U:R\mathbf{-}\mathbf{Mod}\to \mathbf{Set}$. First, we described the properties such an adjoint would need to have. We then carried out an analysis of what properties $F(X)$ would need to possess for some particular sets. We quickly deduced that we must have:

• $F(\mathrm{\varnothing })\simeq 0$, the zero module

• $F(\{\bullet \})\simeq R$, for any singleton set $\{\bullet \}$

• Using the fact that left adjoints commute with colimits (together with the facts that the colimit of a discrete collection of objects in $\mathbf{Set}$ is their disjoint union, while in $R\mathbf{-}\mathbf{Mod}$ is their direct sum), we deduced that for any set $X$ we must have

  $$F(X)\simeq F(\underset{x\in X}{⨆}\{x\})\simeq \underset{x\in X}{⨁}F(\{x\})\simeq \underset{x\in X}{⨁}R$$

This inspired us to realize that only possible definition of $F(X)$ must be $\underset{x\in X}{⨁}R$, i.e., that the set of elements of $F(X)$ is the set of all finite, formal $R$-linear combinations of elements of $X$.

Concepts

• Adjoints
• Properties of adjoints
• Free modules

References

• Dummit & Foote: Section 10.3