On $\Sigma _1^1$-completeness of quasi-orders on $\kappa ^\kappa $
We prove under $V=L$ that the inclusion modulo the non-stationary ideal is a $\Sigma_1^1 $-complete quasi-order in the generalized Borel-reducibility hierarchy ($\kappa \gt \omega $). This improvement to known results in $L$ has many new consequences concerning the $\Sigma_1^1 $-completeness of quasi-orders and equivalence relations such as the embeddability of dense linear orders as well as the equivalence modulo various versions of the non-stationary ideal. This serves as a partial or complete answer to several open problems stated in the literature. Additionally the theorem is applied to prove a dichotomy in $L$: If the isomorphism of a countable first-order theory (not necessarily complete) is not $\Delta_1^1 $, then it is $\Sigma_1^1 $-complete.
We also study the case $V\ne L$ and prove $\Sigma_1^1 $-completeness results for weakly ineffable and weakly compact $\kappa $.