Yeah, it’s a bit weird to call this particular problem the converse Lawvere problem. The name suggests that there is a converse Lawvere problem for any concrete category and any base object $Y$--namely, to find an object $X$ and a map $f: X \to Y^X$ such that every morphism $X \to Y$ has the form $f(x)$ for some $x \in X$. But, among a small group of researchers, we ended up using the name for the case where $Y$ is the unit interval and the category is that of topological spaces, or alternatively a reasonable topological-like category. In this post, I play with that problem a bit, since I pass to considering functions with computability properties, rather than a topological definition. (I also don’t exactly define a category here...) I agree that what’s going on here is a UTM property.
Yeah, it’s a bit weird to call this particular problem the converse Lawvere problem. The name suggests that there is a converse Lawvere problem for any concrete category and any base object $Y$--namely, to find an object $X$ and a map $f: X \to Y^X$ such that every morphism $X \to Y$ has the form $f(x)$ for some $x \in X$. But, among a small group of researchers, we ended up using the name for the case where $Y$ is the unit interval and the category is that of topological spaces, or alternatively a reasonable topological-like category. In this post, I play with that problem a bit, since I pass to considering functions with computability properties, rather than a topological definition. (I also don’t exactly define a category here...) I agree that what’s going on here is a UTM property.
I think this context of considering different categories is interesting. There’s an interplay between diagonal-lemma-type fixed-point theorems and topological fixed-point theorems going on in work like probabilistic truth predicates, reflective oracles, and logical induction. For more on analogies between fixed-point theorems, you can see e.g. A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points and Scott Garrabrant’s fixed point sequence.