Large cardinal operators and elementary embeddings

We introduce and investigate a uniform framework for large cardinal operators. This framework accomodates the Ramsey operator, and we will show that it also accommodates the subtle operator, the ineffability operator, and the pre-Ramsey operator. We use this framework to introduce a new large cardinal operator, and to show that operator to be strictly intermediate between the ineffability operator and the Ramsey operator. This new operator is closely connected to the notion of a Tω-Ramsey cardinal that was recently introduced by Philipp Lücke and the author. As a test application for our framework, we show that a strong form of the key results of James Baumgartner connecting ineffability to subtlety, and Ramseyness to pre-Ramseyness, generalizes to the context of our abstract large cardinal operators.