Hi Richard, Thanks! This is very interesting and nice. Although I didn't say this explicitly in my original message, I am looking for algebras that can be defined in any (elementary) topos (by diagrams or in the internal language) that are not free in all toposes. Do you also have examples of those? (Other than the one I gave in the opening message of this thread.) I am sure there should be a plentiful supply of algebras that one can define in all toposes, but which are not free in all toposes. I guess that such an exploration would lead to a better understanding of the algebras of this monad in an arbitrary topos. In particular, we may ask whether Anders Kock's characterization of the algebras as posets that are complete in a certain way may help to shed light on this question. Best, Martin On 07/01/2026 01:07, Richard Garner wrote:
Hi Martin,
Maybe one can look at the Sierpinski topos. In there, if my calculations are correct, the partial map classifier takes A ---> B to 1+A+B ---> 1+B, and an algebra for this monad is a map p: A-->B equipped with a section s together with points in A and B which are preserved by p and s.
In this case, Omega seems to have exactly two non-isomorphic algebra structures, which can be realised by union and intersection.
But for a general p: A-->B, I think there are either:
- no algebra structures (if B is empty or p is non-surjective), or else - exactly one non-isomorphic algebra structure for each distinct cardinality possessed by one of the fibres of p
Indeed, in the second case, pick an element of B whose fibre has the given cardinality. This is your B-point *. Then use AC to pick a section s of p, and take the point in A to be s(*). If you chose a different section, you get an isomorphic algebra. If you chose a different basepoint in B of the same cardinality, you will again get an isomorphic algebra.
In particular, Omega(Omega(1)) has one fibre of cardinality 1, one of cardinality 2 and one of cardinality 3, and so should have three distinct algebra structures.
All the best,
Richard
Martin Escardo <escardo.martin@gmail.com> writes:
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>