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Neuromath
Neuromath
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Selinger ("A survery of graphical languages for monoidal categories", 2009) defines a "spacial monoidal category" as one that satisifies the following string-diagram axiom

enter image description here

Or textually as sayingThis diagram says that for every $h : I \to I$ and object $A$, $$ \rho_A \circ (\text{id}_A \otimes h) \circ \rho^{-1}_A = \lambda_A \circ (h \otimes \text{id}_A) \circ \lambda_A^{-1}$$$$ \rho_A \circ (\text{id}_A \otimes h) \circ \rho^{-1}_A = \lambda_A \circ (h \otimes \text{id}_A) \circ \lambda_A^{-1},$$ where $\lambda_A$ and $\rho_A$ are the left and right unitors.

What is an example of a monoidal category that is not spacial by this definition?

Selinger ("A survery of graphical languages for monoidal categories", 2009) defines a "spacial monoidal category" as one that satisifies the following string-diagram axiom

enter image description here

Or textually as saying for every $h : I \to I$ and object $A$, $$ \rho_A \circ (\text{id}_A \otimes h) \circ \rho^{-1}_A = \lambda_A \circ (h \otimes \text{id}_A) \circ \lambda_A^{-1}$$

What is an example of a monoidal category that is not spacial by this definition?

Selinger ("A survery of graphical languages for monoidal categories", 2009) defines a "spacial monoidal category" as one that satisifies the following string-diagram axiom

enter image description here

This diagram says that for every $h : I \to I$ and object $A$, $$ \rho_A \circ (\text{id}_A \otimes h) \circ \rho^{-1}_A = \lambda_A \circ (h \otimes \text{id}_A) \circ \lambda_A^{-1},$$ where $\lambda_A$ and $\rho_A$ are the left and right unitors.

What is an example of a monoidal category that is not spacial by this definition?

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edit approved Nov 7, 2019 at 3:04
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edit approved Nov 7, 2019 at 3:04
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Selinger ("A survery of graphical languages for monoidal categories", 2009) defines a "spacial monoidal category" as one that satisifies the following string-diagram axiom

enter image description here

Or textually as saying for every $h : I \to I$ and object $A$, $$ \rho_A \circ (\text{id}_A \otimes h) \circ \rho^{-1}_A = \lambda_A \circ (h \otimes \text{id}_A) \circ \lambda_A^{-1}$$

What is an example of a monoidal category that is not spacial by this definition?

Selinger ("A survery of graphical languages for monoidal categories", 2009) defines a "spacial monoidal category" as one that satisifies the following string-diagram axiom

enter image description here

What is an example of a monoidal category that is not spacial by this definition?

Selinger ("A survery of graphical languages for monoidal categories", 2009) defines a "spacial monoidal category" as one that satisifies the following string-diagram axiom

enter image description here

Or textually as saying for every $h : I \to I$ and object $A$, $$ \rho_A \circ (\text{id}_A \otimes h) \circ \rho^{-1}_A = \lambda_A \circ (h \otimes \text{id}_A) \circ \lambda_A^{-1}$$

What is an example of a monoidal category that is not spacial by this definition?

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Neuromath
Neuromath
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Monoidal category that is not spacial

Selinger ("A survery of graphical languages for monoidal categories", 2009) defines a "spacial monoidal category" as one that satisifies the following string-diagram axiom

enter image description here

What is an example of a monoidal category that is not spacial by this definition?