Definition 7.47.6. Let $\mathcal{C}$ be a category. A topology on $\mathcal{C}$ is given by a rule which assigns to every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ a subset $J(U)$ of the set of all sieves on $U$ satisfying the following conditions

1. For every morphism $f : V \to U$ in $\mathcal{C}$, and every element $S \in J(U)$ the pullback $S \times _ U V$ is an element of $J(V)$.

2. If $S$ and $S'$ are sieves on $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, if $S \in J(U)$, and if for all $f \in S(V)$ the pullback $S' \times _ U V$ belongs to $J(V)$, then $S'$ belongs to $J(U)$.

3. For every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the maximal sieve $S = h_ U$ belongs to $J(U)$.

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.

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 00Z4. Beware of the difference between the letter 'O' and the digit '0'.