If you drop the condition that your category is symmetric monoidal, and content yourself with a category that's merely monoidal, then the tensor category of $R$-$R$-bimodules for $R$ a factor (factor = von Neumann algebra with trivial center) is an example .

The "quantum dimension" or "statistical dimension" of an $R$-$R$-bimodule is an $\mathbb R_{\ge 0}$-valued invariant with all the properties that you want.

See, e.g., my paper: https://arxiv.org/abs/1110.5671, specifically Proposition 5.2.

If you drop the condition that your category is symmetric monoidal, and content yourself with a category that's merely monoidal, then the tensor category of $R$-$R$-bimodules for $R$ a factor (factor = von Neumann algebra with trivial center) is an example .

The "quantum dimension" or "statistical dimension" of an $R$-$R$-bimodule is an $\mathbb R_{\ge 0}\cup\{\infty\}$-valued invariant with all the properties that you want.

See, e.g., my paper: https://arxiv.org/abs/1110.5671, specifically Proposition 5.2.

If you drop the condition that your category is symmetric monoidal, and content yourself with a category that's merely monoidal, then the tensor category of $R$-$R$-bimodules for $R$ a factor (factor = von Neumann algebra with trivial center) is an example .

The "quntum dimension" or "statistical dimension" of an $R$-$R$-bimodule is an $\mathbb R_{\ge 0}$-valued invariant with all the properties that you want.

See, e.g., my paper: https://arxiv.org/abs/1110.5671, specifically Proposition 5.2.

If you drop the condition that your category is symmetric monoidal, and content yourself with a category that's merely monoidal, then the tensor category of $R$-$R$-bimodules for $R$ a factor (factor = von Neumann algebra with trivial center) is an example .

The "quantum dimension" or "statistical dimension" of an $R$-$R$-bimodule is an $\mathbb R_{\ge 0}$-valued invariant with all the properties that you want.

See, e.g., my paper: https://arxiv.org/abs/1110.5671, specifically Proposition 5.2.