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

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 $J(V)$.

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 $J(U)$.

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

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