Marcello Herreshoff

10

"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.)

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

To solve the problem, it would suffice to find a reflexive domain X with a retract onto [0,1].

This is because if you have a reflexive domain X, that is, an X with a continuous surjective map f::X→XX, and A is a retract of X, then there's also a continuous surjective map g::X→AX.

Proof: If A is a retract of X then we have a retraction r::X→A and a section s::A→X with r∘s=1A. Construct g(x):=r∘f(x). To show that g is a surjection consider an arbitrary q∈AX. Thus, s∘q::X→X. Since f is a surjection there must be some x with f(x)=s∘q. It follows that g(x)=r∘f(x)=r∘s∘q=q. Since q was arbitrary, g is also a surjection.