2025-10-07

This following is a very brief summary of what happened in class on 2025-10-07.

We began by recapping the new notion of adjoint functors. We also noted that the natural bijection $\tau$ of the adjunction can be used to construct natural transformations $\eta :I_C\Rightarrow GF$ and $\epsilon :FG\Rightarrow I_D$, called the unit and counit of the adjunction, respectively.

We then returned to the example of the abelianization functor as a left adjoint to the forgetful functor $U:\mathbf{A}\mathbf{b}\to \mathbf{G}\mathbf{r}\mathbf{p}$.

For a new example (which we'll generalize next class) we quickly reviewed the example of the free vector space construction, which provides a left adjoint to the forgetful functor $U:{\mathbf{V}\mathbf{e}\mathbf{c}}_k\to \mathbf{S}\mathbf{e}\mathbf{t}$.

Finally, we started to see how the equalizer (and coequalizer) constructions can be reframed as provided right (resp., left) adjoints of a "constant" (or "diagonal") functor $\mathrm{\Delta }:\mathbf{S}\mathbf{e}\mathbf{t}\to {\mathbf{S}\mathbf{e}\mathbf{t}}^J$, where $J$ is the category with exactly two objects and two distinct parallel arrows between them.

Next time we'll finally tackle free module construction, as a functor left adjoint to the forgetful functor $U:R\text{-}\mathbf{\text{Mod}}\to \mathbf{S}\mathbf{e}\mathbf{t}$.

Concepts

• Adjoints
• Examples of adjoints

References

• Mac Lane, Categories for the Working Mathematician: pp. 79-89