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edit approved Aug 7, 2017 at 16:21
jeq
edit approved Aug 7, 2017 at 16:21
jeq
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Hi,

In a strict monoidal category, where the associator, left and right unitor are identity morphisms we have the following relations between (string) diagrams:

http://i46.tinypic.com/ifajb9.png(source)

where $i_x$ and $e_x$ are the unit and counit isomorphisms.

For the first relation for example, the LHS starts by $1\otimes x$ and goes to $x\otimes 1$, but the two are really the same object $x$ since we're in a stirctstrict monidal category, so we go from $x$ to $x$, which is exactly what happens in the RHS.

What I don't understand, is why such equalities are still valid in a weakened monoidal category, where now $x\otimes 1$ and $1\otimes x$ are merely isomorphic.

Hi,

In a strict monoidal category, where the associator, left and right unitor are identity morphisms we have the following relations between (string) diagrams:

http://i46.tinypic.com/ifajb9.png

where $i_x$ and $e_x$ are the unit and counit isomorphisms.

For the first relation for example, the LHS starts by $1\otimes x$ and goes to $x\otimes 1$, but the two are really the same object $x$ since we're in a stirct monidal category, so we go from $x$ to $x$, which is exactly what happens in the RHS.

What I don't understand, is why such equalities are still valid in a weakened monoidal category, where now $x\otimes 1$ and $1\otimes x$ are merely isomorphic.

In a strict monoidal category, where the associator, left and right unitor are identity morphisms we have the following relations between (string) diagrams:

(source)

where $i_x$ and $e_x$ are the unit and counit isomorphisms.

For the first relation for example, the LHS starts by $1\otimes x$ and goes to $x\otimes 1$, but the two are really the same object $x$ since we're in a strict monidal category, so we go from $x$ to $x$, which is exactly what happens in the RHS.

What I don't understand, is why such equalities are still valid in a weakened monoidal category, where now $x\otimes 1$ and $1\otimes x$ are merely isomorphic.

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Pedro
Pedro
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Hi,

In a strict monoidal category, where the associator, left and right unitor are identity morphisms we have the following relations between (string) diagrams:

http://i46.tinypic.com/ifajb9.png

where $i_x$ and $e_x$ are the unit and counit isomorphisms.

For the first relation for example, the LHS starts by $1\otimes x$ and goes to $x\otimes 1$, but the two are really the same object $x$ since we're in a stirct monidal category, so we go from $x$ to $x$, which is exactly what happens in the RHS.

What I don't understand, is why such equalities are still valid in a weakened monoidal category, where now $x\otimes 1$ and $1\otimes x$ are merely isomorphic.

Hi,

In a strict monoidal category, where the associator, left and right unitor are identity morphisms we have the following relations between (string) diagrams:

where $i_x$ and $e_x$ are the unit and counit isomorphisms.

For the first relation for example, the LHS starts by $1\otimes x$ and goes to $x\otimes 1$, but the two are really the same object $x$ since we're in a stirct monidal category, so we go from $x$ to $x$, which is exactly what happens in the RHS.

What I don't understand, is why such equalities are still valid in a weakened monoidal category, where now $x\otimes 1$ and $1\otimes x$ are merely isomorphic.

Hi,

In a strict monoidal category, where the associator, left and right unitor are identity morphisms we have the following relations between (string) diagrams:

http://i46.tinypic.com/ifajb9.png

where $i_x$ and $e_x$ are the unit and counit isomorphisms.

For the first relation for example, the LHS starts by $1\otimes x$ and goes to $x\otimes 1$, but the two are really the same object $x$ since we're in a stirct monidal category, so we go from $x$ to $x$, which is exactly what happens in the RHS.

What I don't understand, is why such equalities are still valid in a weakened monoidal category, where now $x\otimes 1$ and $1\otimes x$ are merely isomorphic.

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Pedro
Pedro
  • 733
  • 4
  • 11

String diagrams for (weak) monoidal categories

Hi,

In a strict monoidal category, where the associator, left and right unitor are identity morphisms we have the following relations between (string) diagrams:

where $i_x$ and $e_x$ are the unit and counit isomorphisms.

For the first relation for example, the LHS starts by $1\otimes x$ and goes to $x\otimes 1$, but the two are really the same object $x$ since we're in a stirct monidal category, so we go from $x$ to $x$, which is exactly what happens in the RHS.

What I don't understand, is why such equalities are still valid in a weakened monoidal category, where now $x\otimes 1$ and $1\otimes x$ are merely isomorphic.