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,


1. $2^\kappa$ is regular
2. $2^\kappa$ is a fixed-point of $\aleph$
3. $2^\kappa$ is weakly inaccessible
4. $2^\kappa$ is weakly hyper-inaccessible
5. $2^\kappa$ 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:


1. the map $\kappa \mapsto 2^\kappa$ is injective
2. 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?
> 
> 
> 



Why isn't there more interest in "large powerset axioms"?
I suppose there is some interest in propositions implying large powerset. For example, a classical question at the infancy of set theory and real analysis was whether there is a probability measure on the unit interval that extends Lebesgue measure, in which every set is measurable. This implies that the continuum is weakly Mahlo, Rowbottom, etc. It's also equiconsistent with a two-valued measurable cardinal.


The stronger interest in large cardinals is surely due to the empirical fact that they form a coherent system that measures the relative consistency of everything. It's not clear that large powerset type propositions have a similar property. Some of the things you mention have no strength, like the continuum function is injective. It follows from GCH and it's negation is easily forced.


Related fact: If $2^\omega$ is real-valued measurable, the continuum function is not injective. $2^\kappa = 2^\omega$ for all $\kappa < 2^\omega$. This is due to Prikry, and a proof can be found in chapter 22 of Jech.


