In the first post of this series, I explained how subobjects of the constant presheaf (resp. constant sheaf) on a small category (resp. small site) with terminal object correspond to left closed (resp. local) properties on . In this post, I will explain the main examples that intervene in setting up topoi, and show how to define the sheaf condition using sieves (instead of coverings).
For simplicity, assume is a small category with fibre products.
Definition. Given a set of morphisms with the same target , define the sieve generated by as the sieve on of those morphisms that factor through some .
It is in a sense the right ideal in generated by the . What does this look like as a subobject of ?
Example. If has one element, i.e. , then is the image of the morphism of representable presheaves . In the case where is already a monomorphism (this is always the case when is a poset, such as for some topological space ), then is itself injective (this is the definition of a monomorphism!), so is just .
In general, is the image of the map
induced by the maps . Indeed, an element of is a morphism , and it comes from some if and only if factors through .
This shows that, in fact, every sieve is of this form for some set : take as index set (the objects of) the slice category , which as in the previous post gives a surjection . This corresponds to generating an ideal by all its elements.
But we can also characterise without using the word ‘image’ (which somehow computes its first syzygy):
Lemma. Let be a set of morphisms with common target, and the sieve generated by . Then is the coequaliser of the diagram
We will give two proofs, one using the description of coequalisers of sets, and the other using that presheaves are colimits of representable presheaves, as discussed in the previous post.
Proof 1. The diagram
is a pullback, by the universal property of fibre products and since fibre products with a fixed set/presheaf of sets commute with coproducts. Then the same goes for the square
since is a monomorphism. But is an epimorphism (objectwise surjection) by definition, so this square is a pushout as well (in , epimorphisms are regular).
Proof 2. By the previous post, the presheaf is the colimit over of (see post for precise statement). Let be the diagram of the two projections, and let be the category of elements of , as in this post. There is a natural functor taking to and to , taking the morphisms in to the projections . We claim that is cofinal, hence the colimit can be computed over instead (see Tag 04E7).
To verify this, we use the criteria of Tag 04E6. If , then by definition the composition is given by a morphism that is contained in . Since is generated by the , this factors through some over , giving a map .
If and are two such maps, they factor uniquely through . The general result for and for (either of the form or of the form ) follows since elements of the form always map to the elements and , showing that the category is weakly connected.
Corollary. Let as above, and let be a presheaf on . Then
Proof. By the lemma above, we compute
so the result follows from the Yoneda lemma.
Corollary. Let be a (small) site. Then a presheaf is a sheaf if and only if for every object and every covering in the site, the inclusion induces an isomorphism
Proof. Immediate from the previous corollary.
Thus, the category of sheaves on can be recovered from if we know at which subobjects we should localise (make the inclusion invertible). Next week, we will use this to give a definition of a Grothendieck topology, abstracting and generalising the notion of a site (i.e. Grothendieck pretopology).