Grothendieck Topologies As Specifications of Universal Epimorphisms

GROTHENDIECK TOPOLOGIES AS SPECIFICATIONS OF UNIVERSAL EPIMORPHISMS

DAVID I. SPIVAK

1. An alternate definition of Grothendieck topology Deﬁnition 1.1. Let C be a category. A sieve over C is a monomorphism σ ,→ P in the category Pre(C) of presheaves on C. This monomorphism is also called a sieve on P (over C).

Deﬁnition 1.2. Choose a set CovC of monomorphisms in Pre(C), which we will call the covering sieves. We will say that CovC is a localizing system if it satisﬁes the property that a sieve P,→ Q is a covering sieve if and only if, for all maps R → Q, where R ∈ Pre(C) is a representable presheaf, the pullback P ×Q R,→ R is a covering sieve.

Lemma 1.3. Let CovC be a localizing system. Then it is closed under pullback. Proof. Let P 0 ,→ P be a covering sieve and Z → P a morphism. Then for any morphism R → Z from a representable presheaf R, we form the all Cartesian diagram R0 / R p Z0 / Z p P 0 / P and notice that R0 → R is a covering sieve. Thus Z0 → Z is as well. Fix a category C. One way to look at Grothendieck topologies is that one wishes to express which monomorphisms in Pre(C) should also be epimorphisms. One hears echoes of this idea in the name “covering sieve”. That is, one decides which sieves should cover, or “surject onto”, a given presheaf. Once this is decided, one proceeds to localize that category Pre(C) in order to force the chosen sieves to be epimorphisms. But notice that in any category D, the set of epimorphisms form a subcategory (i.e. it is closed under composition), say E ⊂ D. Moreover the subcategory E has two special properties: (1) E contains every identity morphism in D, and (2) if the composite f g A −→ B −→ C is in E, then g ∈ E. 1 2 DAVID I. SPIVAK

This motivates the following deﬁnition.

Deﬁnition 1.4. Let C be a category and χ a set of morphisms in C. Suppose that (1) Every identity morphism in C is in χ, (2) if A → B and B → C are each in χ then their composition A → C is also in χ, and f g (3) for every pair of composable morphisms A −→ B −→ C in C, if the composition gf is in χ, then g is also in χ. In this case we say that B is an epitype subcategory of C. We say that a localizing system CovC is a Grothendieck topology on C if it is an epitype subcategory of C.

The following proposition shows that our deﬁnition of Grothendieck topology is equivalent to the usual one.

Proposition 1.5. Let C be a category and let CovC be a set of monomorphisms in Pre(C). Then CovC is a Grothendieck topology on C if and only if the following conditions hold.

(1) For each C ∈ C, the identity sieve C → C is in CovC. (2) If σ ,→ C is in CovC and f : D → C is a morphism in C, then the pullback −1 f σ ,→ D is in CovC. (3) Let C ∈ C be an object, τ ,→ C a sieve on C, and σ ,→ C a covering sieve. −1 If for each composition f : D → σ → C, the pullback f τ ,→ D is in CovC, then τ ,→ C is in CovC. Proof. By deﬁnition of localizing system, axiom 3 is equivalent to the assertion 3’: Let C ∈ C be an object, τ ,→ C a sieve on C and σ ,→ C a covering sieve. If the pullback τ ×C σ ,→ σ is a covering sieve of σ, then τ is a covering sieve of C.

First, suppose that CovC is a Grothendieck topology in our sense. We prove each of the three axioms. 1: By deﬁnition of epitype. 2: By deﬁnition of localizing system. 3’: Consider the diagram

τ ×C σ / σ p τ / C. We assume that the top map is a covering sieve and that σ is a covering sieve, and we wish to show that τ ,→ C is a covering sieve. Since covering sieves form a category, the composition τ ×C σ → σ → C is a covering sieve. Finally, since CovC is an epitype subcategory, τ → C must also be a covering sieve.

Now suppose that the three axioms hold. We need to show that CovC is an epitype subcategory of C. By axiom 1, C contains all identity morphisms in Pre(C). Let us next show that the set of covering sieves is closed under composition. GROTHENDIECK TOPOLOGIES AS SPECIFICATIONS OF UNIVERSAL EPIMORPHISMS 3

If f : A → B and g : B → C are covering sieves, then form the pullback square

f id (1) A / B / B

p p g A / B / C. f g (The right hand square is a pullback because g is a monomorphism.) Since the top composition is a covering sieve, the bottom one is too by axiom 3. Finally, to show that it is an epitype subcategory, suppose that the composition A → B → C is a covering sieve. Again consider diagram 1. Since the bottom composition is a covering sieve, axiom 3’ (ﬂipped on its side) implies that B → C is a covering sieve if A → A is.