2024-10-07

This following is a very brief summary of what happened in class on 2024-10-07.

We introduced the notion of a natural transformation between two functors. Intuitively, a natural transformation between two functors $F,G:C\to D$ is a map from the image of one functor to the image of the other. More precisely, it is the data of a family of arrows ${\tau }_c:F(c)\to G(c)$ in $D$, one for each object $c\in C$, that satisfies the expected naturality condition.

We took a peek at our first few examples of natural transformations, namely:

• the determinant
• the "insertion of generators" in the free vector space construction
• the idea/definition of maps between diagrams of the same shape

We also defined functor categories. For given categories $J$ and $C$, we let $C^J$ denote the category with:

• objects: all functors $F:J\to C$
• arrows: natural transformations between such functors

We also quickly looked at few small examples, noting that the functor category $C^{\mathbf{\text{0}}}$ "looks like" the category $\mathbf{\text{1}}$, while the functor category $C^{\mathbf{\text{1}}}$ "looks like" the category $C$. (We will soon formalize the notion of "looks like" in the definition of an "equivalence of categories.")

Concepts

• Natural transformations
• Functor categories

References

• Mac Lane: pages 16-18