Description of generated Grothendieck topology

Question

Let $C$ be a small category, and let $\tau$ be a set of sieves in $C$. Assume that $\tau$ contains all the maximal sieves, and is stable under pullbacks. How to describe the Grothendieck topology $\tau'$ generated by $\tau$ explicitly? I guess that $$\tau'(X) = \{T \text{ sieve on } X : \exists S \in \tau(X) \forall (a : Y \to X)\in S \,(a^* (T) \in \tau(Y))\}$$ works, but I am not sure. Perhaps we have to iterate this.

Answer 1

Here is a one-step construction. Say a $\tau$-tree on an object $X$ in $\mathcal{C}$ is a set $\Phi$ that satisfies the following conditions:

• Every element of $\Phi$ is a composable sequence of morphisms in $\mathcal{C}$, say $(f_1, \ldots, f_n)$, such that ($f_1 \circ \cdots \circ f_n$ is defined and) $\operatorname{codom} f_1 = X$.
• The empty sequence is in $\Phi$.
• If $(f_1, \ldots, f_n, f_{n+1}) \in \Phi$ then $(f_1, \ldots, f_n) \in \Phi$.
• For every $(f_1, \ldots, f_n) \in \Phi$, $$\{ u : (f_1, \ldots, f_n, u) \in \Phi \}$$ is either empty or a $\tau$-sieve. (If this set is empty, we say $(f_1, \ldots, f_n)$ is a leaf of $\Phi$.)
• Every element of $\Phi$ occurs as a prefix of some leaf of $\Phi$.

Now, say a $\tau$-covering sieve on $X$ is a sieve on $X$ that contains $$\{ f_1 \circ \cdots \circ f_n : (f_1, \cdots, f_n) \text{ is a leaf of } \Phi \}$$ for some $\tau$-tree $\Phi$ on $X$. I leave it to you to verify that this defines the smallest Grothendieck topology on $\mathcal{C}$ that contains $\tau$.

Answer 2

Yeah, it looks like you have to iterate. Just take a finite chain in which every morphism between adjacent objects generates a covering sieve, as do the identities. Then this is stable under pullback because every nontrivial cover pulls back to a trivial cover, but if there are at least three elements then the $\tau'$ you suggest misses the single-element sieve by which the initial element covers the final.

EDIT I see no obstacle to choosing an arbitrary well ordered set here, and starting with the biggest nontrivial sieves, so that a limit ordinal is covered by maps out of all strictly smaller ordinals. So even transfinitely many iterations may be required.