Alternative formulation of Grothendieck topology

category-theorygrothendieck-topologiessheaf-theorytopos-theory

In Mac Lane and Moerdijk's Sheaves in Geometry and Logic there is a reformulation of the Grothendieck topology conditions in terms of arrows, namely a Grothendieck topology on a small category $\mathbb{C}$ is defined as a relation, which is read as "covers", between the sieves of the category and the arrows, such that for any sieve $S$ on some object $C$ the following conditions hold

  1. if $f\in S$, then $S$ covers $f$
  2. if $S$ covers $f\colon D\to C$, then $S$ covers $f\circ h$ for any $h\colon E\to D$ ;
  3. Let $R$ be a sieve on $C$. If $S$ covers $f$ and $R$ covers $g$ for any $g\in S$, then $R$ covers $f$.

Now one can define a set $J(C)$ of sieves as follows
$$
S \in J(C) \iff S \text{ covers } 1_C.
$$
I struggled over the proof of the fact that $\{J(C)\}_{C\in\mathbb{C}}$ is a Grothendieck topology in the previous sense, that is, the following conditions hold

  1. for any $C$ the greatest sieve $\{f\in Mor\ \mathbb{C}\colon cod f = C\}$ belongs to $J(C)$;
  2. if $S\in J(C)$, then $f^*(S)\in J(D)$ for any $f\colon D\to C$;
  3. if $S\in J(C)$ and $R$ is a sieve on $C$, such that $h^*(R)\in J(D)$ for any $h\colon D\to C\in S$, then $R\in J(C)$.

Could you help me with it? Thank you in advance.

Best Answer

I'll use (1), (2), and (3) for the conditions in the MacLane Moerdijk definition and 1., 2., and 3. for the conditions in the usual definition.

It does seem that there's an axiom missing: (4) $S$ covers $f\colon D\to C$ if and only if $f^*S$ covers $1_D$.

What MacLane and Moerdijk write here seems a little unclear to me. Maybe they view (4) as implicitly part of what they mean when they talk about covering arrows?

Anyway, let's check that it's ok to add axiom (4). If we have a Grothendieck topology $J$ satisfying 1., 2., and 3., we get a covering relation between sieves and arrows by declaring that a sieve $S$ on $C$ covers $f\colon D\to C$ if and only if $f^*S\in J(D)$. So $S$ covers $f$ iff $f^* S\in J(D)$ iff $(1_D)^*(f^*S)\in J(D)$ iff $f^*S$ covers $1_D$.

And to see that (4) doesn't follow from (1), (2), and (3), note that these three axioms are purely "object-by-object": they say nothing about the relationship between sieves on different objects. For an explicit counterexample, consider the category $\mathcal{C}$ with just two objects $D$ and $C$ and an arrow $f\colon D\to C$ (and the identity arrows). Consider the covering relation which says that a sieve $S$ on $D$ covers $1_D$ if and only if $1_D\in S$, but every sieve on $C$ covers every arrow with codomain $C$. This satisfies (1), (2), and (3), but it fails (4), because if $S$ is the empty sieve on $C$, then $S$ covers $f$, but $f^*S$, which is the empty sieve on $D$, fails to cover $1_D$.

Ok, now let's prove 1., 2., and 3. using (1), (2), (3), and (4).

  1. Let $S$ be the greatest sieve on $C$. Then $1_C\in S$, so $S$ covers $1_C$ by (1), so $S\in J(C)$.
  2. Suppose $S\in J(C)$ and $f\colon D\to C$ is an arrow. Since $S\in J(C)$, $S$ covers $1_C$, so $S$ covers $f = 1_C\circ f$ by (2), and $f^*S$ covers $1_D$ by (4), so $f^*S\in J(D)$.
  3. Suppose $S\in J(C)$ and $R$ is a sieve on $C$ such that $h^*R\in J(D)$ for all $h\colon D\to C$ in $S$. Then $S$ covers $1_C$ and we would like to show that $R$ covers $1_C$. By (3) it suffices to show that $R$ covers every arrow in $S$. So let $h\colon D\to C$ be in $S$. We have $h^*R\in J(D)$, so $h^*R$ covers $1_D$, so $R$ covers $h$ by (4).

It may seem a little silly to add axiom (4), because it reduces the covering relation on arrows to the covering relation on identity arrows, i.e. on objects. So it's not clear what we gain by reaxiomatizing the notion in terms of the covering relation between sieves and arrows. I don't have a good answer for this, except maybe that axioms (1)-(3) look a little bit simpler than their counterparts 1.-3. (and any mention of pullback sieves is pushed off to axiom (4)).

Related Question