The Stacks project

Lemma 4.43.6. Let $\mathcal{C}$ be a monoidal category. If $Y$ is a left dual to $X$, then

\[ \mathop{\mathrm{Mor}}\nolimits (Z' \otimes X, Z) = \mathop{\mathrm{Mor}}\nolimits (Z', Z \otimes Y) \quad \text{and}\quad \mathop{\mathrm{Mor}}\nolimits (Y \otimes Z', Z) = \mathop{\mathrm{Mor}}\nolimits (Z', X \otimes Z) \]

functorially in $Z$ and $Z'$.

Proof. Consider the maps

\[ \mathop{\mathrm{Mor}}\nolimits (Z' \otimes X, Z) \to \mathop{\mathrm{Mor}}\nolimits (Z' \otimes X \otimes Y, Z \otimes Y) \to \mathop{\mathrm{Mor}}\nolimits (Z', Z \otimes Y) \]

where we use $\eta $ in the second arrow and the sequence of maps

\[ \mathop{\mathrm{Mor}}\nolimits (Z', Z \otimes Y) \to \mathop{\mathrm{Mor}}\nolimits (Z' \otimes X, Z \otimes Y \otimes X) \to \mathop{\mathrm{Mor}}\nolimits (Z' \otimes X, Z) \]

where we use $\epsilon $ in the second arrow. A straightforward calculation using the properties of $\eta $ and $\epsilon $ shows that the compositions of these are mutually inverse. Similarly for the other equality. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 4.43: Monoidal categories

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FFQ. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 4.43.6 as pdf