I am reading Lurie's book Higher Algebra. I notice that he mentioned that the functor $X \to \mathbb{Z}Sing(X)$ from topological spaces to Kan complexes is excisive, i.e. mapping the homotopy pushout diagrams in topological spaces to homotopy pullback diagrams in Kan complexes. I don't know why it is true.
I really want to use this example to understand the definition of excisive functors. I guess it's the same as the Mayer-Vietoris theorem. But I don't know how to compute. Is the homotopy pullback diagram of these Kan complexes in fact the homotopy pullback diagram of chain complexes?
The singular simplicial set functor $\def\Sing{{\sf Sing}}\Sing$ is an equivalence of ∞-categories (from topological spaces to simplicial sets). The functor $\def\Z{{\bf Z}}\Z[-]$ is a left adjoint ∞-functor from simplicial sets to connective chain complexes. The inclusion ∞-functor from connective chain complexes to unbounded chain complexes is also a left adjoint functor.
Therefore, the functor $\Z[\Sing(-)]$ is a left adjoint ∞-functor from topological spaces to unbounded chain complexes, so it preserves homotopy pushout squares.
The ∞-category of unbounded chain complexes is a stable ∞-category, so homotopy pushout squares coincide with homotopy pullback squares. Therefore, the ∞-functor $\Z[\Sing(-)]$ from topological spaces to unbounded chain complexes is excisive.
The forgetful ∞-functor from unbounded chain complexes to simplicial sets is the composition of the truncation functor from unbounded chain complexes to nonnegatively graded chain complexes, the Dold–Kan functor from nonnegatively graded chain complexes to simplicial abelian groups, and the forgetful functor from simplicial abelian groups to simplicial sets. All three functors are right adjoint ∞-functors, and, therefore, preserve homotopy limits, in particular, homotopy pullbacks.