(By an $n$-category, I mean an $(n,n)$-category.)
This is potentially a dumb question, but its been on my mind for a couple of years now, so I'm throwing in the towel and asking it. If it is a dumb question, I figure its better to find out as soon as possible and not let it fester any longer.
I'd be interested to know whether or not there's a natural answer to the question: where should the numbering for $n$-categories start? E.g. should what are currently called $2$-categories really be called $2$-categories? Or should they perhaps be thought of as $3$-somethings or $4$-somethings instead? With this goal in mind, I ask:
Question. Does there exist a fixed "fudge factor" $k \in \mathbb{Z}$ such that there exists a natural and useful binary operation $f$ on $\infty$-categories that has the property that if $\mathbf{B}$ is a $b$-category and $\mathbf{A}$ is an $a$-category, then $f(\mathbf{B},\mathbf{A})$ is most naturally viewed $(b+a+k)$-category?
If so, do different choices of natural and useful operations $f$ yields different values for $k$?