Department of Mathematics

In winter term 2025 I had the pleasure of coaching a student in a self-study course on category theory; which led me to think a lot about my own introduction to the subject in a year-long course taught by Jiří Sichler 50 years ago; and a group of theorems (taught over about 3 months!) that strongly influenced my understanding of the foundations of model theory. The current course left me with the strong desire to revisit this material.

A concrete category is one where the morphisms between two objects can be represented as functions between sets.

How complicated can a concrete category be? How complicated can a category of algebras be?

Of course, foundational questions enter in: if there are not too many measurable cardinals, the category of graphs is already the most complicated concrete category; and any concrete category has a reasonably nice representation in a category of algebras, even in the category of algebras with two unary operations. (!)

If there is a proper class of measurable cardinals, there are monsters lurking out there!

I will give an overview of all of this, free of proofs, of course!

References:
Some of the original references