A $4$-dimensional compact manifold with free $\pi_1$ has L-S category $2$.

I don't really understand the misunderstanding: this question seems perfectly sensible to me. The OP recognizes that the expression $S^0 =$ terminal sum terminal is a categorical notion, so perhaps a sensible $S^0$ can be defined in other categories. There is no problem provided initial is not terminal; so they ask the idea can be saved in categories where inital $=$ terminal.

Yes, it needs the $p$-local hypothesis

@NeilStrickland I looked at MathSciNet, and I only saw stuff about nearly rational spaces---rational spaces with fundamental group.