Dear 1-topos theorists, Anders Kock has a nice paper from the last millennium (1990), about Algebras for the partial map classifier monad https://link.springer.com/chapter/10.1007/BFb0084225 https://tildeweb.au.dk/au76680/jonna5.pdf As he remarks and is well known, and also trivial, in a boolean topos, all algebras are free. Then he goes on to say many interesting things that hold in all 1-toposes. Jon Sterling recently conjectured that, in an arbitrary topos, not all algebras are free. I came up with an example. My question is whether this example is well known, and, moreover, whether more examples are known. First of all, the subobject classifier Ω is a free algebra on one generator. If you think of Ω as the powerset of the terminal object 𝟙, then the structure map of Ω as a free algebra is *union*. (†) But, you can check, also *intersection* exhibits Ω = 𝓟 𝟙 as an algebra. I have proved that this algebra is free if and only if the principle of excluded middle holds, that is, the topos is boolean. Is this known? Then I wanted to find more counter-examples to "every algebra is a free algebra". I tried, first, exponential powers of Ω. But they are free in all toposes. Then I tried, more generally, arbitrary products of free algebras. But, again, they are free in all toposes. Does anybody know a source of more counter-examples? At the moment, the only counter-example I know is (†). It is embarrasing to know only one counter-example. Best wishes, Martin PS. I have written my proofs (on paper and) in a proof assistant, namely Agda (and this is publicly available and advertised in various forums). So this gives some confidence regarding the above claims. I still have to write a human-readable version for public consumption, but here I am more interested in knowning what people already know about this. You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/groups/groupsubscription?source=EscalatedMessage&action=files&smtp=categories%40mq.edu.au&bO=true&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Leave group<https://outlook.office365.com/groups/groupsubscription?source=EscalatedMessage&action=leave&smtp=categories%40mq.edu.au&bO=true&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g>