Marcello Herreshoff

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From discussions I had with Sam, Scott, and Jack:

To solve the problem, it would suffice to find a reflexive domain with a retract onto .

This is because if you have a reflexive domain , that is, an with a continuous surjective map , and is a retract of , then there's also a continuous surjective map .

Proof: If is a retract of then we have a retraction and a section with . Construct . To show that is a surjection consider an arbitrary . Thus, . Since is a surjection there must be some with . It follows that . Since was arbitrary, is also a surjection.

"Self-Reference and Fixed Points: A Discussion and an Extension of Lawvere's Theorem" by Jorge Soto-Andrade and Francisco J. Varela seems like a potentially relevant result. In particular, they prove a converse Lawvere result in the category of posets (though they mention doing this for in an unsolved problem.) I'm currently reading through this and related papers with an eye to adapting their construction to (I think you can't just use it straight-forwardly because even though you can build a reflexive domain with a retract to an arbitrary poset, the paper uses a different notion of continuity for posets.)