By a large powerset axiom, let us mean informally an axiom that says that for some cardinal numbers $\kappa$, we have that $2^\kappa$ is somehow "large" or "difficult to access from below." For example: whenever $\kappa$ is infinite,

- $2^\kappa$ is regular
- $2^\kappa$ is a fixed-point of $\aleph$
- $2^\kappa$ is weakly inaccessible
- $2^\kappa$ is weakly hyper-inaccessible
- $2^\kappa$ is weakly Mahlo

etc. (I do not know if these are all consistent).

Let us also include more ambiguous cases in our definition of "large powerset axioms," like:

- the map $\kappa \mapsto 2^\kappa$ is injective
- for every infinite set $X$, the powerset of $X$ has a subchain of cardinality $2^{|X|}$

Anyway, looking at the literature, there doesn't seem to be a lot of interest in these kinds of axioms, as compared to the amount of attention given to large cardinal axioms.

Question.Is there any particular reason for this lack of interest? For example, are there conservativity results showing that such axioms are "very weak" which might explain the lack of interest?