Generalized cardinal invariants for an inaccessible $κ$ with compactness at $κ^{++}$
We show that if the existence of a supercompact cardinal $κ$ with a weakly compact cardinal $λ$ above $κ$ is consistent, then the following are consistent as well (where $\mathfrak{t}(κ)$ and $\mathfrak{u}(κ)$ are the tower number and the ultrafilter number, respectively): (i) There is an inaccessible cardinal $κ$ such that $κ^+ < \mathfrak{t}(κ)= \mathfrak{u}(κ)< 2^κ$ and $SR(κ^{++})$ hold, and (ii) There is an inaccessible cardinal $κ$ such that $κ^+ = \mathfrak{t}(κ) < \mathfrak{u}(κ)< 2^κ$ and $SR(κ^{++}), TP(κ^{++})$ and $\neg wKH(κ^+)$ hold. The cardinals $\mathfrak{u}(κ)$ and $2^κ$ can have any reasonable values in these models. We obtain these results by combining the forcing construction from Brooke-Taylor, Fischer, Friedman and Montoya with the Mitchell forcing and with (new and old) indestructibility results for compactness principles. Apart from $\mathfrak{u}(κ)$ and $\mathfrak{t}(κ)$ we also compute the values of $\mathfrak{b}(κ)$, $\mathfrak{d}(κ)$, $\mathfrak{s}(κ)$, $\mathfrak{r}(κ)$, $\mathfrak{a}(κ)$, $\mathrm{cov}(M_κ)$, $\mathrm{add}(M_κ)$, $\mathrm{non}(M_κ)$, $\mathrm{cof}(M_κ)$ which will all be equal to $\mathfrak{u}(κ)$. In (ii), we compute $\mathfrak{p}(κ) = \mathfrak{t}(κ) = κ^+$ by observing that the $κ^+$-distributive quotient of the Mitchell forcing adds a tower of size $κ^+$. Finally, we observe that (i) and (ii) hold also for the traditional invariants on $κ= ω$, using Mitchell forcing up to a weakly compact cardinal; in this case we also obtain the disjoint stationary sequence property $DSS(ω_2)$, which implies the negation of the approachability property $\neg AP(ω_2)$.