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David Carchedi
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Yuri Sulyma
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Localization of symmetric monoidal category

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?