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addition of the "complety reducible" assumption for the exemple in remark
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Sebastien Palcoux
Sebastien Palcoux
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Let $\mathcal{C}$ be a monoidal category and Mon$_{\mathcal{C}}$ the category of monoids (also called algebra objects) on $\mathcal{C}$.

Questions: are there definitions of rangeimage and kernel for a monoid morphism, and of quotient of monoids? Is there a generalization of the first isomorphism theorem of groups?

Remark: if my question has a negative answer in general, I'm interested in the case where $\mathcal{C}$ is the category of bifinite (dualizable) completely reducible $R$-$R$-bimodules, with $R$ the hyperfinite ${\rm II}_1$ factor.

Let $\mathcal{C}$ be a monoidal category and Mon$_{\mathcal{C}}$ the category of monoids (also called algebra objects) on $\mathcal{C}$.

Questions: are there definitions of range and kernel for a monoid morphism, and of quotient of monoids? Is there a generalization of the first isomorphism theorem of groups?

Remark: if my question has a negative answer in general, I'm interested in the case where $\mathcal{C}$ the category of bifinite (dualizable) $R$-$R$-bimodules, with $R$ the hyperfinite ${\rm II}_1$ factor.

Let $\mathcal{C}$ be a monoidal category and Mon$_{\mathcal{C}}$ the category of monoids (also called algebra objects) on $\mathcal{C}$.

Questions: are there definitions of image and kernel for a monoid morphism, and of quotient of monoids? Is there a generalization of the first isomorphism theorem of groups?

Remark: if my question has a negative answer in general, I'm interested in the case where $\mathcal{C}$ is the category of bifinite (dualizable) completely reducible $R$-$R$-bimodules, with $R$ the hyperfinite ${\rm II}_1$ factor.

Source Link
Sebastien Palcoux
Sebastien Palcoux
  • 27k
  • 5
  • 74
  • 186

Image, kernel, quotient and first isomorphism theorem, in a category of monoid objects

Let $\mathcal{C}$ be a monoidal category and Mon$_{\mathcal{C}}$ the category of monoids (also called algebra objects) on $\mathcal{C}$.

Questions: are there definitions of range and kernel for a monoid morphism, and of quotient of monoids? Is there a generalization of the first isomorphism theorem of groups?

Remark: if my question has a negative answer in general, I'm interested in the case where $\mathcal{C}$ the category of bifinite (dualizable) $R$-$R$-bimodules, with $R$ the hyperfinite ${\rm II}_1$ factor.