The Stacks project

Definition 22.26.3. Let $R$ be a ring. Let $\mathcal{A}$ be a differential graded category over $R$. Then we let

  1. the category of complexes of $\mathcal{A}$1 be the category $\text{Comp}(\mathcal{A})$ whose objects are the same as the objects of $\mathcal{A}$ and with

    \[ \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}(\mathcal{A})}(x, y) = \mathop{\mathrm{Ker}}(d : \mathop{\mathrm{Hom}}\nolimits ^0_\mathcal {A}(x, y) \to \mathop{\mathrm{Hom}}\nolimits ^1_\mathcal {A}(x, y)) \]

  2. the homotopy category of $\mathcal{A}$ be the category $K(\mathcal{A})$ whose objects are the same as the objects of $\mathcal{A}$ and with

    \[ \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(x, y) = H^0(\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y)) \]

[1] This may be nonstandard terminology.

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Your email address will not be published.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 09L7. Beware of the difference between the letter 'O' and the digit '0'.



View Definition 22.26.3 as pdf