ScalingStacks

Remark 3.16. The behavior of a homology theory with coefficients in $\mathcal{V}$ depends critically on the symmetric monoidal structure chosen on $\mathcal{V}$. For instance, for the symmetric monoidal $\infty$-category $(\m_{k},\oplus)$ of $k$-modules over a fixed field $k$ with direct sum, a homology theory is forced to be ordinary homology with coefficients in $k$; while for $(\m_{k},\otimes)$, a homology theories is typically not a homotopy invariant of manifolds.