The Stacks project

Bhargav Bhatt, private communication.

Lemma 4.24.3. Let $u$ be a left adjoint to $v$ as in Definition 4.24.1.

  1. If $v \circ u$ is fully faithful, then $u$ is fully faithful.

  2. If $u \circ v$ is fully faithful, then $v$ is fully faithful.

Proof. Proof of (2). Assume $u \circ v$ is fully faithful. Say we have $X$, $Y$ in $\mathcal{D}$. Then the natural composite map

\[ \mathop{\mathrm{Mor}}\nolimits (X,Y) \to \mathop{\mathrm{Mor}}\nolimits (v(X),v(Y)) \to \mathop{\mathrm{Mor}}\nolimits (u(v(X)), u(v(Y))) \]

is a bijection, so $v$ is at least faithful. To show full faithfulness, we must show that the second map above is injective. But the adjunction between $u$ and $v$ says that

\[ \mathop{\mathrm{Mor}}\nolimits (v(X), v(Y)) \to \mathop{\mathrm{Mor}}\nolimits (u(v(X)), u(v(Y))) \to \mathop{\mathrm{Mor}}\nolimits (u(v(X)), Y) \]

is a bijection, where the first map is natural one and the second map comes from the counit $u(v(Y)) \to Y$ of the adjunction. So this says that $\mathop{\mathrm{Mor}}\nolimits (v(X), v(Y)) \to \mathop{\mathrm{Mor}}\nolimits (u(v(X)), u(v(Y)))$ is also injective, as wanted. The proof of (1) is dual to this. $\square$

Comments (0)

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Your email address will not be published.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 0FWV. Beware of the difference between the letter 'O' and the digit '0'.