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Let $\mathcal{C}$ be a symmetric monoidal category. In the 1973 paper "Note on monoidal localisation" by Brian Day, the multiplicative system of morphism in $\mathcal{C}$ has been discussed. See also this mathoverflow question by Martin Brandenburg.

My question is: can we consider a multiplicative system consists of both objects and morphisms in $\mathcal{C}$? This means that we have a collection of objects $x_i$ and a collection of morphisms $f_i$ such that $x_i \otimes x_j$ is still in the collection $x_i$ and $x_i$ and $f_j$ satisfies some "compatible condition". And can we define a localization along this more general multiplicative system?

Notice that in this viewpoint the case in the first paragraph can be considered as the multplicative system with only one object $1$ (and a system of morphisms).

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    $\begingroup$ This is a bit above my categorical pay grade, so I will leave a hopefully helpful comment rather than an answer. A general strategy to working with objects in any category is to encode them via their identity morphisms. Is it enough in your case to use the theory of monoidal localization but with some identity morphisms in the mix? $\endgroup$
    – Theo Johnson-Freyd
    Commented Sep 27, 2012 at 14:02
  • $\begingroup$ @Theo: Yes I need some morphism in the mix. But still I'm interested in the case your mentioned: we consider a multiplicative system of objects and the identity morphisms of each object. Then what should be the requirement on the collection of objects to make them a multiplicative system? $\endgroup$
    – Zhaoting Wei
    Commented Sep 27, 2012 at 15:31

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