In Scholze's masterclass on condensed mathematics, there was a discussion whether there is a "good definition of a site". It seems, that the definition of Grothendieck topology using sieves is the most general. If one works with Grothendieck pretopologies one has to worry about existence of certain pullbacks. So I am not interested in pretopologies, instead I want to conceptualize the sieve-definition. This is what I have:
The usual definition of Grothendieck topology based on sieves (see e.g. https://en.wikipedia.org/wiki/Grothendieck_topology, or Alternative formulation of Grothendieck topology) can be rewritten as follows:
Def. A Grothendieck topology on a small category $\mathcal C$ is a functor $J : \mathcal C^{\mathrm{op}} \to \mathrm{Set}$ with the following three properties:
- $J$ is a subfunctor of $\mathrm{Sv}$. Here $\mathrm{Sv}(X)$ is the set of sieves on $X$ and for any morphism $f : Y \to X$ the map $f^* : \mathrm{Sv}(X) \to \mathrm{Sv}(Y)$ is given by pullback of sieves (see https://en.wikipedia.org/wiki/Grothendieck_topology)
- $J_{\mathrm{ind}}$ is a subfunctor of $J$, where $J_{\mathrm{ind}}(X) = \{\mathrm{Hom}_{\mathcal C}(-,X)\}$.
- For all $X \in \mathcal C$ and all $S \in J(X)$, we have $$ J(X) = \{T \in \mathrm{Sv}(X) \mid \forall (f : Y \to X) \in S : f^*T \in J(Y)\}$$
1 and 2 just say, that $J$ is somewhere between the indiscrete and the discrete Grothendieck topology. I am completely happy with that.
The way I read 3 is: Being a $J$-covering sieve is a $J$-local property. This looks like a "circular" sheaf condition.
If $X \in \mathcal C$ and $T$ is a sieve on $X$, then we define a presheaf on $\mathcal C \downarrow X$ by $$ \mathcal F_{T}(f : Y \to X) = \begin{cases} \{*\}, &f^*T \in J(Y), \\ \emptyset, &\text{else.} \end{cases} $$ This is a $J$-sheaf if and only if for all $Y$ and all covering sieves $S \in J(Y)$, the map $$ \mathcal F_T(Y) \to \prod_{(f:V \to Y)\in S} \mathcal F_T(V) $$ is bijective. This for all $X$ and $T$ is equivalent to 3.
So we can take:
- For all $X \in \mathcal C$ and all $T \in \mathrm{Sv}(X)$, the presheaf $\mathcal F_T$ on $\mathcal C \downarrow X$ is a $J$-sheaf.
Grothendieck topologies are often advertised as the "right generalization" of topological spaces for the definition of sheaves. Since 1 and 2 don't say much about $J$, all interesting topological features hide inside of 3. So it might be interesting to ask:
Is there a nice(r) way to express 3? (or the entire definition?)
Creative answers are highly appreciated.
PS: I don't know the details of the discussion in the chat in the masterclass, because I watched it on Youtube (https://www.youtube.com/watch?v=OT65JC3gKPY, at 3:00). Comments about the content of this discussion are also appreciated.
Rather than having $J: \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$, have the codomain of $J$ being the category $\mathbf{DistLat}_1$ of distributive lattices with a maximal element (the notation is ad hoc), since the collection of sieves is trivially closed under unions, and it is a consequence of the ordinary definition of Grothendieck topologies that it is closed under intersections. Moreover, pulling back sieves along morphisms in $\mathcal{C}$ preserves intersections, unions and maximal sieves. The maximal element eliminates the need for your axiom 2, since the maximal sieve is the maximal element of $\mathrm{Sv}(X)$ for every $X$, so being a subfunctor via an inclusion that preserves maximal elements ensures that the maximal sieve is always present.
Your axiom 3 in terms of the sheaf condition isn't as circular as it seems at first glance, since we can define the sheaf condition for any collection of sieves. However, all that's left to add to the axiomatisation is multicomposition (since stability under pullback is implied by functoriality of $J$), and this seems far less complicated to check than either the sheaf condition version of axiom 3 that you wrote. In summary, suppressing the subsidiary definitions of $\mathrm{Sv}$ and $\mathbf{DistLat}_1$, we have:
Def. A Grothendieck topology $J$ on a small category $\mathcal{C}$ is a subfunctor of the functor $\mathrm{Sv}: \mathcal{C}^{\mathrm{op}} \to \mathbf{DistLat}_1$, subject to the condition that for each $S \in J(X)$ and $T \in \mathrm{Sv}(X)$, $$(\forall f:Y \to X \in S) (f^*(T) \in J(Y)) \implies T \in J(X).$$