2025-10-03

This following is a very brief summary of what happened in class on 2025-10-03.

We spent the entire hour looking more deeply into natural transformations. We noted that there are actually two ways one might compose natural transformations, namely a "horizontal composition" and a vertical composition". We noted that we can now talk about natural isomorphisms and the property of two functors being (naturally) isomorphic.

The vertical composition of natural transformations also allowed us to define the notion of "functor category", in which the objects are functors $F:C\to D$ and the arrows are natural transformations between such functors. We denoted that category $D^C$. We'll return to functor categories in the near future.

We then tiptoed towards the concept of a pair of functors being "adjoint." We first talked about the new distinction between two categories being "isomorphic" and being "equivalent", and then started to talk about a new, third possibility.

Next week we'll explore that possibility in detail when we talk about adjoints.

Concepts

• Natural transformations
• Functor categories

References

• Mac Lane, Categories for the Working Mathematician: pp. 16-18, 40-41