The Stacks project

Different from the definition in [Exposé V, Definition 5.1, SGA1]. Compare with [Definition 7.2.1, BS].

Definition 58.3.6. Let $\mathcal{C}$ be a category and let $F : \mathcal{C} \to \textit{Sets}$ be a functor. The pair $(\mathcal{C}, F)$ is a Galois category if

  1. $\mathcal{C}$ has finite limits and finite colimits,

  2. every object of $\mathcal{C}$ is a finite (possibly empty) coproduct of connected objects,

  3. $F(X)$ is finite for all $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and

  4. $F$ reflects isomorphisms1 and is exact2.

Here we say $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ is connected if it is not initial and for any monomorphism $Y \to X$ either $Y$ is initial or $Y \to X$ is an isomorphism.

[1] Namely, given a morphism $f$ of $\mathcal{C}$ if $F(f)$ is an isomorphism, then $f$ is an isomorphism.
[2] This means that $F$ commutes with finite limits and colimits, see Categories, Section 4.23.

Comments (0)

There are also:

  • 7 comment(s) on Section 58.3: Galois 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Your email address will not be published.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 0BMY. Beware of the difference between the letter 'O' and the digit '0'.