I am currently confused about the empty set in terms of its path components and how this fits into the Quillen adjunction between topological spaces and simplicial sets. Probably, one of my definitions are not precisely correct:
The category of simplicial sets is a cofibrantly generated model category with generating acyclic cofibrations the inclusions of all horns into standard simplicial sets. The map $\emptyset \to \Delta^0$ is such a horn inclusion, so it should be an acyclic cofibration.
However, an acyclic cofibration is a weak equivalence, so it should induce isomorphisms on all homotopy groups after realization. The realization of this map is the map $\emptyset \to \ast$ of topological spaces and this should be a weak equivalence, i.e. induce isomorphisms on all homotopy groups ($k \geq 0$) for all basepoints: $\pi_k (\emptyset,x) \to \pi_k(\ast,y)$
However, $\emptyset$ does not have any basepoints, such that the condition is empty, so that $\emptyset \to \ast$ is a weak equivalence. But this feels terribly wrong to me, since it implies that the emptyset is weakly homotopy equivalent to any space. A solution can be found by making a basepoint-independent definition of $\pi_0$ and set it to be equivalence classes of points. Then $\pi_0 (\emptyset) = \emptyset$ and $\pi_0(\ast) = \ast$, so $\emptyset \to \ast$ is not a weak equivalence.
But this would in turn imply that $\emptyset \to \Delta^0$ is not an acyclic cofibration of simplicial sets, because it is not a weak equivalence after realization.
So I do not find the right definition of $\pi_0$ to make it compatible with the Quillen adjunction, but I think it is just a mistake in my reasoning somewhere.
Any help would be greatly appreciated.