ScalingStacks

Definition 4.9. Let $K$ be a commutative ring and $\mathcal{C}$ be a (small) $K$-linear category. Then the zeroth Hochschild homology of $\mathcal{C}$, also called the trace of $\mathcal{C}$, is defined as the $K$-module

$$\mathrm{HH}_{0}(\mathcal{C}):=\operatorname{Tr}(\mathcal{C}):=\left(\bigoplus_{x\in\mathrm{Ob}(\mathcal{C})}\operatorname{End}_{\mathcal{C}}(x)\right)\bigg/\mathrm{Span}\{f\circ g-g\circ f\}$$

where the spanning set for the subspace to be divided out is constructed from all pairs of cyclically composable morphisms, i.e. $f\in\operatorname{Hom}_{\mathcal{C}}(x,y)$ and $g\in\operatorname{Hom}_{\mathcal{C}}(y,x)$ for some $x,y\in\mathrm{Ob}(\mathcal{C})$.