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.