Suppose that $\mathcal{C}$ is a *finitary* category, so for any two objects $A$ and $B$ we have that $|\mathrm{Ext}^i(A,B)| < \infty$ for $i\geq 0$, suppose $\mathcal{C}$ has *finite global dimension*, so $\mathrm{Ext}^i(A,B) = \{0\}$ for all sufficiently large $i$^{*}. In such a category $\mathcal{C}$ we can define the *Euler form* of two objects
$$
\langle A,B \rangle_m = \left( \prod_{i=0}^{\infty} \Big|\mathrm{Ext}^i(A,B)\Big|^{(-1)^i}\right)^{\frac{1}{2}}\,.
$$

This is a definition I've encountered a few times while studying quiver representations and Hall algebras. It's a constant we need to introduce when giving a Hall algebra an associative algebra structure (see these lecture notes by Olivier Schiffmann for details). I've yet to get a good grasp on what exactly this is counting in terms of the objects and morphisms in the category though. Is this related to the good ol' Euler characteristic of a topological space in some non-superficial way? How should I be thinking about this form?

***** Typically I've only seen this in the context of *hereditary* categories, where the $\mathrm{Ext}^i$ groups vanish for $i>1$, and so we only need to consider $\mathrm{Hom}(A,B)$ and $\mathrm{Ext}^1(A,B)$ for any objects $A$ and $B$.