Why is the empty theory with a single binary function companionable? I was looking around on the map of the universe and came across this entry. I checked the provided source for more information, but it didn't seem to actually say anything about this theory from a quick source. I then tried to prove that this theory existed myself, but I am struggling to figure out what the existentially closed models are, much less if they form an elementary class.
 A: In Generic expansion and Skolemization in NSOP$_1$ theories by Kruckman and Ramsey (journal DOI, arXiv) there is Theorem 3.8, which is exactly what you are looking for I think. Given any language $L$ it constructs a theory $T^\emptyset_L$ that is the model companion of the empty $L$-theory.
Moreover, given your original motivation, you might be interested in Corollary 3.13: it establishes that the theory is NSOP$_1$ and describes Kim-independence in the theory (over models). Namely: $a$ is Kim-independent from $b$ over $M$ iff $\operatorname{acl}(Ma) \cap \operatorname{acl}(Mb) = \operatorname{acl}(M)$.
Finally, Proposition 3.14 shows that if $L$ contains an $n$-ary function symbol (with $n \geq 2$) then $T^\emptyset_L$ has TP$_2$ and is thus not simple. In particular this establishes the exact position in the map of the theory you asked about.

Part of the initial confusion was due to a wrong link, which should have linked to the abovementioned paper (and it now does). Also, to make sure proper credit is given I would like to refer to the comment of Alex Kruckman below, where he points out that their results are inspired by a preprint of Emil Jerabek.
