Matthew wrote:

> Do you think we can recover all of the "yanking conditions" of a [symmetric compact closed monoidal category](https://en.wikipedia.org/wiki/Compact_closed_category#Symmetric_compact_closed_category)?

> \[

A\xrightarrow{\cong} A\otimes I\xrightarrow{A\otimes\eta} A\otimes (A^\ast\otimes A) \xrightarrow{\cong} (A\otimes A^\ast) \otimes A \xrightarrow{\varepsilon\otimes A} I\otimes A\xrightarrow{\cong} A \\\\

A^\ast\xrightarrow{\cong} I\otimes A^\ast\xrightarrow{\eta\otimes A^\ast}(A^\ast\otimes A)\otimes A^\ast\xrightarrow{\cong} A^\ast\otimes (A\otimes A^\ast)\xrightarrow{A^\ast \otimes\varepsilon} A^\ast\otimes I\xrightarrow{\cong} A^\ast

\]

> Here "\\(\ast\\)" would be "\\(\mathrm{op}\\)", "\\(\otimes\\)" would be "\\(\times\\)", and "\\(\to\\)" would be "\\(\nrightarrow\\)

Yes! And \\( \eta\\) is the cap, and \\(\varepsilon\\) is the cup.

This will be the subject of my next lecture, and you folks have done most of the work for me!

I call the yanking identities the **zig-zag equations.** If we draw a cap like a cap, a cup like a cup, and an identity like a pipe, they say this:

> Do you think we can recover all of the "yanking conditions" of a [symmetric compact closed monoidal category](https://en.wikipedia.org/wiki/Compact_closed_category#Symmetric_compact_closed_category)?

> \[

A\xrightarrow{\cong} A\otimes I\xrightarrow{A\otimes\eta} A\otimes (A^\ast\otimes A) \xrightarrow{\cong} (A\otimes A^\ast) \otimes A \xrightarrow{\varepsilon\otimes A} I\otimes A\xrightarrow{\cong} A \\\\

A^\ast\xrightarrow{\cong} I\otimes A^\ast\xrightarrow{\eta\otimes A^\ast}(A^\ast\otimes A)\otimes A^\ast\xrightarrow{\cong} A^\ast\otimes (A\otimes A^\ast)\xrightarrow{A^\ast \otimes\varepsilon} A^\ast\otimes I\xrightarrow{\cong} A^\ast

\]

> Here "\\(\ast\\)" would be "\\(\mathrm{op}\\)", "\\(\otimes\\)" would be "\\(\times\\)", and "\\(\to\\)" would be "\\(\nrightarrow\\)

Yes! And \\( \eta\\) is the cap, and \\(\varepsilon\\) is the cup.

This will be the subject of my next lecture, and you folks have done most of the work for me!

I call the yanking identities the **zig-zag equations.** If we draw a cap like a cap, a cup like a cup, and an identity like a pipe, they say this: