2024-10-04

This following is a very brief summary of what happened in class on 2024-10-04.

We spent our second day working our way towards a formal definition of "universal property." In particular, we:

• Formalized the definition of a commutative diagram, using functors from "shape" categories
• Formalized the definition of a cone to/from a commutative diagram, by first defining the "diagonal functor" corresponding to a "constant diagram", and then introducing the notion of a natural transformation between functors
• Semi-formalized the definition of limits (and colimits), as universal cones to (or from) a given diagram.

We'll return once more to universal properties, after we've officially seen natural transformations and Yoneda's lemma. When we do, we'll be able to clean up and make rigorous all of our ideas so far.

Concepts

• Universal Properties II - Commutative diagrams, cones and limits

References

• Mac Lane: Chapter III (select content)