Each algebraic theory has some free algebras. Which algebraic theories have the property that all their algebras are free? It turns out that it is not hard to state precisely which theories have this property; proving that they are the only ones is a much more difficult task. Proofs were given in the 1970s in the language of logic by Steven Givant[1], and in the 2010s in the language of classical universal algebra by Keith A. Kearnes, Emil W. Kiss, and Ágnes Szendrei[2]. Tom Leinster and I investigated this result from a category theoretic point of view[3]: I will talk about the theories in question, highlighting some of their interesting properties.

I will also spend some time asking what it might mean for us to live in a world where everyone is free, and in particular, how we might get there.

[1] Steven Givant. Universal Horn classes categorical or free in power. Annals of Mathematical Logic, 15(1):1–53, 1978.
[2] Keith A. Kearnes, Emil W. Kiss, and Ágnes Szendrei. Varieties whose finitely generated members are free. Algebra Universalis, 79, 2018.
[3] Maia Woolf. Algebraic theories all of whose algebras are free. MScR Thesis, University of Edinburgh, 2025.