Excellent metrics are a subclass of good metrics for which the completion map S to S(S)^op is an almost involution, up to idempotent completion.
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Excellent metrics on triangulated categories, and the involutivity of the map taking $\mathcal{S}$ to $\mathfrak{S}({\mathcal{S})^{\mathrm{op}}}$
Excellent metrics are a subclass of good metrics for which the completion map S to S(S)^op is an almost involution, up to idempotent completion.