I am not a computer scientist and have no knowledge of programming. However, I wondered continuations occur as natural and interesting mathematical structures, perhaps as algebraic or type theoretic structures of some kind, and quite apart from computer science considerations.
I understand that continuations are special kinds of monad, and so are mathematically monads. I am aware that they are used in computer science and also in linguistics.
However, I wondered whether the particular type of monad that continuations exemplify occurs in mathematical structures that mathematicians study.
The following link has since come to my attention, that discusses the relation between continuations and the Yoneda embedding:
However, I would be particularly interested in examples of mathematical structures that act like continuations in fields such as algebra, etc (i.e, outside of category theory).
Barker in https://www.nyu.edu/projects/barker/barker-cw.pdf also discusses how we can understand continuations as ways of mapping entities to the principal ultrafilter containing them, so perhaps there are structures involving ultrafilters that are examples of continuations in pure mathematics.