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HeinrichD
HeinrichD
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Here is a simple observation: The condition is equivalent to $$\forall a,b \in M. \, [a \cdot [b]]=[a \cdot b]=[[a] \cdot b].$$ Assume that $M$ is a preordered monoid. Then it is natural to assume $a \leq [a]$, and $a \mapsto [a]$ behaves like a "closure operator". The fixed points are the closed elements. There are lots of examples for this. This setting can be generalized and has been studied before:

Assume that $M$ is a monoidal category with underlying category $C$ and $R : C \to C$ is a functor equipped with a natural transformation $\eta : \mathrm{id}_C \to R$ (satisfying $R\eta=\eta R$). Then one may demand that the induced morphisms $$R(a \otimes R(b)) \leftarrow R(a \otimes b) \to R(R(a) \otimes b)$$ are isomorphisms. This situation appears in Day's reflection theorem for closed monoidal categories; here the reflection is called normal. This is used to endow reflective subcategories of $C$ with a monoidal structure. It is also useful for the construction of monoidal localizations, see Day's Note on monoidal localization. I would also consult the papers which cite these.

Here is a simple observation: The condition is equivalent to $$\forall a,b \in M. \, [a \cdot [b]]=[a \cdot b]=[[a] \cdot b].$$ Assume that $M$ is a preordered monoid. Then it is natural to assume $a \leq [a]$, and $a \mapsto [a]$ behaves like a "closure operator". The fixed points are the closed elements. This setting can be generalized and has been studied before:

Assume that $M$ is a monoidal category with underlying category $C$ and $R : C \to C$ is a functor equipped with a natural transformation $\eta : \mathrm{id}_C \to R$. Then one may demand that the induced morphisms $$R(a \otimes R(b)) \leftarrow R(a \otimes b) \to R(R(a) \otimes b)$$ are isomorphisms. This situation appears in Day's reflection theorem for closed monoidal categories; here the reflection is called normal. This is used to endow reflective subcategories of $C$ with a monoidal structure. It is also useful for the construction of monoidal localizations, see Day's Note on monoidal localization. I would also consult the papers which cite these.

Here is a simple observation: The condition is equivalent to $$\forall a,b \in M. \, [a \cdot [b]]=[a \cdot b]=[[a] \cdot b].$$ Assume that $M$ is a preordered monoid. Then it is natural to assume $a \leq [a]$, and $a \mapsto [a]$ behaves like a "closure operator". The fixed points are the closed elements. There are lots of examples for this. This setting can be generalized and has been studied before:

Assume that $M$ is a monoidal category with underlying category $C$ and $R : C \to C$ is a functor equipped with a natural transformation $\eta : \mathrm{id}_C \to R$ (satisfying $R\eta=\eta R$). Then one may demand that the induced morphisms $$R(a \otimes R(b)) \leftarrow R(a \otimes b) \to R(R(a) \otimes b)$$ are isomorphisms. This situation appears in Day's reflection theorem for closed monoidal categories; here the reflection is called normal. This is used to endow reflective subcategories of $C$ with a monoidal structure. It is also useful for the construction of monoidal localizations, see Day's Note on monoidal localization. I would also consult the papers which cite these.

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HeinrichD
HeinrichD
  • 5.5k
  • 22
  • 42

Here is a simple observation: The condition is equivalent to $$\forall a,b \in M. \, [a \cdot [b]]=[a \cdot b]=[[a] \cdot b].$$ Assume that $M$ is a preordered monoid. Then it is natural to assume $a \leq [a]$, and $a \mapsto [a]$ behaves like a "closure operator". The fixed points are the closed elements. This setting can be generalized and has been studied before:

Assume that $M$ is a monoidal category with underlying category $C$ and $R : C \to C$ is a functor equipped with a natural transformation $\eta : \mathrm{id}_C \to R$. Then one may demand that the induced morphisms $$R(a \otimes R(b)) \leftarrow R(a \otimes b) \to R(R(a) \otimes b)$$ are isomorphisms. This situation appears in Day's reflection theorem for closed monoidal categories; here the reflection is called normal. This is used to endow reflective subcategories of $C$ with a monoidal structure. It is also useful for the construction of monoidal localizations, see Day's Note on monoidal localization. I would also consult the papers which cite these.