What (filtered) (homotopy) (co) limits does $\pi_0:\mathbf{sSets}\to\mathbf{Sets}$ preserve?

Question

Consider the functor $\pi_0:\mathbf{sSets}\to\mathbf{Sets}$.

• $\pi_0$ does not preserve arbitrary limits
• $\pi_0$ does not send homotopy limits to limits
• $\pi_0$ does preserve filtered colimits
• $\pi_0$ does preserve arbitrary colimits

• Does $\pi_0$ send homotopy colimits to colimits? I think, this is true.
• Does $\pi_0$ preserve filtered limits? If not, what is a counterexample?
• Does $\pi_0$ send filtered homotopy limits to filtered limits? If not, what is a counterexample?

Answer 1

$\pi_0 : \mathbf{sSet} \to \mathbf{Set}$ is a left adjoint (exercise), so it preserves all colimits. It's also a left Quillen functor, so it even preserves homotopy colimits. There's no reason to believe anything good happens with (homotopy) limits, filtered or not, but it is true that finite (homotopy) products are preserved.