Let $\mathcal M$ be a symmetric monoidal category, $S\subset \mathcal M$ a collection of objects and morphisms. I would like to construct the localization $\mathcal M \mathop{\longrightarrow}^T \mathcal M[S^{-1}]$, defined by

  1. for each $x\in S$, there is $y\in\mathcal M[S^{-1}]$ such that $Tx\otimes y \cong I$;

  2. for each $x \mathop{\longrightarrow}^f y \in S$, $Tf$ is an isomorphism in $\mathcal M[S^{-1}]$;

  3. maybe there are some coherence conditions on the above;

  4. $T$ is the universal symmetric monoidal functor with these properties.

Is there a reference which covers this construction and its properties?


Localising with respect to morphisms is fairly standard. Assuming that the tensor product of two morphisms in $S$ is again in $S$, the localised category should inherit a symmetric monoidal structure, just by the universal property.

Localising with respect to objects is Quillen's $\mathcal{A}^{-1}\mathcal{A}$ construction. This is explained in Grayson's paper "Higher algebraic K-theory. II (after Daniel Quillen)", for example. To avoid making mistakes with this, you should also read Thomason's paper "Beware the phony multiplication on Quillen's $\mathcal{A}^{-1}\mathcal{A}$".


A reference regarding the monoidal structure on $M[S^{-1}]$ is Brian Day's Note on monoidal localisations. It also talks about the enriched setting, and about monoidal completion.


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