Here's a question from a non-set-theorist, but a sometime-user of large cardinals.
The name Cantor's attic is pretty evocative for the collection of large cardinal axioms: looking through the pages there really feels like I'm in an old, neglected attic, full of boxes which were labeled with names at various times and then stored haphazardly here and there. In order to understand the relationships between different large cardinal axioms, it seems there's no alternative to opening each and every box and comparing their contents yourself. (This sense of disorder is ironic, given that one of the chief features of the attic is that it is well-ordered by consistency strength!)
But there are patterns to be found.
For instance, from a large cardinal axiom $\phi$ we can generate more large cardinal axioms by asking for a proper class of $\phi$-cardinals in $V_\kappa$, or by asking for the class of $\phi$-cardinals to be stationary in $V_\kappa$. Call the latter operation Mahlofication.
There seem to also be "operations" of superfication, Woodinification, and Shelahfication, but I don't know much about them. One complication is that it seems one often needs to modify a large cardinal by introducing various parameters into its definition in order to apply some of these operations.
Question 1: What does it mean to Mahlofy, superfy, Woodenify, or Shelahfy a large cardinal notion? Is it correct to say that these operations all ask for cardinal $\kappa$ such that $V_\kappa$ has "a lot" of the original type of large cardinals? I think these operations generally increase consistency strength -- is it by "a lot" or "a little"? And are these operations "idempotent", or can they be "iterated"?
Question 2: Are there other important "operations" on large cardinal notions?
Qustion 3: Is the list of large cardinal notions at Cantor's attic, say, generated by a substantially smaller list under some collection of "operations"?
Question 4: What about relations between "operations" -- for instance, are there distinct large cardinal notions which become identified under Mahlofication or something? If you restrict attention to large cardinal notions fixed by Mahlofication or something, does the attic look a bit neater?
Obviously these questions are all broad and ill-defined and possibly nonsensical. But if there's anything here which is not nonsense and can be addressed in a precise or at least meaningful manner, I'd be grateful to hear about it!