Let $\mathcal A$ be an additive 1-category, equipped with some class of weak equivalences $\mathcal W$. Let $\mathcal A[\mathcal W^{-1}]$ be the localization of $\mathcal A$ at $\mathcal W$ (so $\mathcal A[\mathcal W^{-1}]$ is an $\infty$-category). Let's assume that $\mathcal W$ is stable under finite direct sums, so that $\mathcal A[\mathcal W^{-1}]$ is an [additive $\infty$-category](https://ncatlab.org/nlab/show/additive+%28infinity%2C1%29-category), and hence admits a (unique) enrichment in the $\infty$-category $Sp\_{\geq 0}$ of connective spectra.


**Question:** Under what conditions does the $Sp\_{\geq 0}$-enrichment of $\mathcal A[\mathcal W^{-1}]$ lift to an enrichment in connective $H\mathbb Z$-modules (= topological abelian groups = connective chain complexes)?


I want to say the answer is "always", but probably there is some technical condition which must be assumed, since I don't have a great handle on how to think of $H\mathbb Z$-modules in a "model-independent" way... The idea should be that the strictness of the additive structure on $\mathcal A$ itself translates to strictness on the additive structure of $\mathcal A[\mathcal W^{-1}]$.


I'd be happy to understand things under more restrictive hypotheses, e.g. hypotheses implying that $\mathcal A[\mathcal W^{-1}]$ is not just additive but in fact stable.



When is an $\infty$-categorical localization of an additive 1-category enriched in topological abelian groups?
The answer is indeed always.


The fact that $\mathcal A$ is an additive 1-category makes it canonically a module over $Proj\_\mathbb Z$, the 1-category of finitely generated projective $\mathbb Z$-modules.


Your assumption on $W$ makes this action compatible with $W$, and because localization is a product-preserving procedure, $\mathcal A[W^{-1}]$ acquires the structure of a module over $Proj\_\mathbb Z$.


Now for an additive $\infty$-category $\mathcal B$, a module structure over $Proj\_\mathbb Z$ induces a lift to $D\_{\geq 0}(\mathbb Z)$ of the mapping connective spectra of $\mathcal B$ (in fact, an actual enrichment but this is more complicated to construct). This follows from the equivalence $D\_{\geq 0}(\mathbb Z) \simeq Fun^\times((Proj\_\mathbb Z)^{op}, Spaces)$ induced by the restricted Yoneda embedding.


