According to Wikipedia, a Weil cohomology theory is a functor from the category of smooth projective varieties over a field $k$, to graded algebras over a field $K$ of characteristic zero, together with a bunch on properties, such as Poincaré duality, Künneth formula, Lefschetz axiom, etc.
Standard examples of such things are Betti cohomology for varieties defined over subfields of $\mathbb{C}$, de Rham cohomology for varieties over fields of characteristic $0$, $\ell$-adic étale cohomology over fields of characteristic $\neq\ell$, and crystalline/rigid cohomology over perfect fields of characteristic $p$.
In all these examples, we actually have more. For example, we can define $H^*(-)$ for any variety (not necessarily smooth and proper), there are versions with supports, there is a generalised Poincaré duality and Künneth formula which the proper (resp. smooth and proper) assumptions can be relaxed, and there is an excision sequence. Here is my question.
Is there a standard term for such an 'extended' Weil cohomology theory?
I guess one potential answer might be 'a formalism of the 6 operations', axiomatised in the introduction to this paper, (and touched upon in this MO question). However I'm really interested in the more restricted case, where we only have absolute cohomology (with and without supports) with constant coefficients. For example, until recent work by Daniel Caro, rigid cohomology had the status of such an 'extended' Weil cohomology theory, but without the full 6 operations formalism.
