Some Remarks on Kim-dividing in NATP Theories

In this note, we prove that Kim-dividing over models is always witnessed by a coheir Morley sequence in NATP theories. Following the strategy of Chernikov and Kaplan [8], we obtain some corollaries which hold in NATP theories. Namely, (i) if a formula Kim-forks over a model, then it quasi-divides over the same model, (ii) for any tuple of parameters $b$ and a model $M$, there exists a global coheir $p$ containing $\text{tp}(b/M)$ such that $B \ind^K_M b'$ for all $b'\models p|_{MB}$. We also show that for coheirs in NATP theories, condition (ii) above is a necessary condition for being a witness of Kim-dividing, assuming that a witness of Kim-dividing exists (see Definition 4.1 in this note). That is, if we assume that a witness of Kim-dividing always exists over any given model, then a coheir $p\supseteq \text{tp}(a/M)$ must satisfy (ii) whenever it is a witness of Kim-dividing of $a$ over a model $M$. We also give a sufficient condition for the existence of a witness of Kim-dividing in terms of pre-independence relations. At the end of the paper, we leave a short remark on Mutchnik's recent work [16]. We point out that the class of $\omega$-NDCTP$_2$ theories, a subclass of the class of NATP theories, contains all NTP$_2$ theories and NSOP$_1$ theories. We also note that Kim-forking and Kim-dividing are equivalent over models in $\omega$-NDCTP$_2$ theories, where Kim-dividing is defined with respect to invariant Morley sequences, instead of coheir Morley sequences as in [16].