One day I read a generating function in a paper. For any "sufficietly nice topological space", $C$:
$$ \sum_{l \geq 0 } q^{2l}\chi(\mathrm{Sym}^l[C]) = (1 - q^2)^{-\chi(C)} = \sum_{l \geq 0} \binom{l + \chi(C)-1}{l} q^{2l} $$
First of all, I'm not sure what "sufficiently nice" means here. I'm guessing any CW complex will do. *I wonder what's an example of a space where this formula doesn't work*.

This formula suggests $$ \chi(\mathrm{Sym}^l[C]) = \binom{l -1+ \chi(C)}{l} = \chi \binom{l-1 + C }{l} $$ where the right hand side is some "categorification".

Removing the $\chi$'s, is there some sense in which $$ \mathrm{Sym}^l[C] = \binom{l -1+ C }{l} $$ is rigorous?