I am reading MacPherson's paper "Chern Classes for singular varieties".
Proposition1 : There is a unique covariant functor from compact complex algebraic variety to abelian groups whose value on a variety is the group of constructible functions from that variety to the integers and whose value $f_*$ on a map $f$ satisifies $$f_*(1_W)(p)=\chi(f^{-1}(p)\cap W)$$, where $1_W$ is the function takes 1 on $W$ and zero otherwise, and $\chi$ denotes the toplogical Euler characteristic.
In the proof, he mentioned stratification theory but I am not very familiar with that. I don't see why $f_*(1_w)$ is constructible function on $Y$. Is there anyone who can help me explain it to me? Thank you so much.
Here is an attempt. So if you start with a morphism $f:X\to Y$ and a subvariety $W\subset X$, as you said you define the function $$f_\ast \mathbb 1_W:Y\to \mathbb Z, \qquad p\mapsto \chi(f^{-1}(p)\cap W).$$ By definition, in order to check that $f_\ast\mathbb 1_W$ is constructible, you want to write it as $$f_\ast\mathbb 1_W=\sum_{n\in\mathbb Z}n\cdot \mathbb 1_{Y_n}, \qquad\qquad(\star)$$ where $Y_n\subset Y$ are subvarieties. Now define $$Y_n=\{y\in Y\,|\,\chi(f^{-1}(y)\cap W)=n\}.$$ You then have a stratification $Y=\coprod_n Y_n$ and $(\star)$ follows. Hope this helps.