Weak Canonical Bases in NSOP Theories

We study the notion of weak canonical bases in an NSOP theory T with existence. Given where in, the weak canonical base of p is the smallest algebraically closed subset of B over which p does not Kim-fork. With this aim we firstly show that the transitive closure of collinearity of an indiscernible sequence is type-definable. Secondly, we prove that given a total-Morley sequence I in p, the weak canonical base of is, if the hyperimaginary is eliminable to e, a sequence of imaginaries. We also supply a couple of criteria for when the weak canonical base of p exists. In particular the weak canonical base of p is (if exists) the intersection of the weak canonical bases of all total-Morley sequences in p over B. However, while we investigate some examples, we point out that given two weak canonical bases of total-Morley sequences in p need not be interalgebraic, contrary to the case of simple theories. Lastly we suggest an independence relation relying on weak canonical bases, when T has those. The relation, satisfying transitivity and base monotonicity, might be useful in further studies on NSOP theories.