The Stacks project

Lemma 4.27.11. Let $\mathcal{C}$ be a category and let $S$ be a right multiplicative system.

  1. The relation on pairs defined above is an equivalence relation.

  2. The composition rule given above is well defined on equivalence classes.

  3. Composition is associative (and the identity morphisms satisfy the identity axioms), and hence $S^{-1}\mathcal{C}$ is a category.

Proof. This lemma is dual to Lemma 4.27.2. It follows formally from that lemma by replacing $\mathcal{C}$ by its opposite category in which $S$ is a left multiplicative system. $\square$

Comments (0)

There are also:

  • 19 comment(s) on Section 4.27: Localization in 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Your email address will not be published.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 04VH. Beware of the difference between the letter 'O' and the digit '0'.