Covering sieves and the sheaf condition (topologies 3/6)

In the first post of this series, I explained how subobjects of the constant presheaf (resp. constant sheaf) $\mathbf 1_X$ on a small category (resp. small site) with terminal object $X$ correspond to left closed (resp. local) properties on $\mathscr C$ . 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 $\mathscr C$ is a small category with fibre products.

Definition. Given a set of morphisms $\mathscr U = \{f_i \colon U_i \to U\}_{i \in I}$ with the same target $U \in \mathscr C$ , define the sieve $S_{\mathscr U} \subseteq h_U$ generated by $\mathscr U$ as the sieve on $U$ of those morphisms $V \to U$ that factor through some $f_i \colon U_i \to U$ .

It is in a sense the right ideal in $\operatorname{Hom}(-,U)$ generated by the $f_i$ . What does this look like as a subobject of $h_U$ ?

Example. If $I$ has one element, i.e. $\mathscr U = \{V \to U\}$ , then $S_{\mathscr U}$ is the image of the morphism of representable presheaves $h_V \to h_U$ . In the case where $V \to U$ is already a monomorphism (this is always the case when $\mathscr C$ is a poset, such as $\operatorname{Open}(X)$ for some topological space $X$ ), then $h_V \to h_U$ is itself injective (this is the definition of a monomorphism!), so $S_{\mathscr U}$ is just $h_V$ .

In general, $S_{\mathscr U}$ is the image of the map

$\coprod_{i \in I} h_{U_i} \to h_U$

induced by the maps $U_i \to U$ . Indeed, an element of $h_U(V)$ is a morphism $f \colon V \to U$ , and it comes from some $h_{U_i}(V)$ if and only if $f$ factors through $f_i \colon U_i \to U$ .

This shows that, in fact, every sieve $S \subseteq h_X$ is of this form for some set $\{U_i \to U\}_{i \in I}$ : take as index set (the objects of) the slice category $(h \downarrow S)$ , which as in the previous post gives a surjection $\coprod_{(V,\alpha)} h_V \to S$ . This corresponds to generating an ideal by all its elements.

But we can also characterise $S_{\mathscr U}$ without using the word ‘image’ (which somehow computes its first syzygy):

Lemma. Let $\mathscr U = \{U_i \to U\}$ be a set of morphisms with common target, and $S_{\mathscr U}$ the sieve generated by $\mathscr U$ . Then $S_{\mathscr U}$ is the coequaliser of the diagram

$\coprod_{i,j \in I} h_{U_i \underset U\times U_j} \rightrightarrows \coprod_{i \in I} h_{U_i},$

where the maps are induced by the two projections $I^2 \to I$ .

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

$\begin{array}{ccc}\displaystyle\coprod_{i,j \in I} h_{U_i \underset U\times U_j} & \to & \displaystyle\coprod_{i \in I} h_{U_i} \\ \downarrow & & \downarrow \\ \displaystyle\coprod_{j \in I} h_{U_j} & \to & h_U \end{array}$

is a pullback, by the universal property of fibre products $U_i \times_U U_j$ and since fibre products with a fixed set/presheaf of sets commute with coproducts. Then the same goes for the square

since $S_{\mathscr U} \to h_U$ is a monomorphism. But $\coprod_{i \in I} h_{U_i} \to S_{\mathscr U}$ is an epimorphism (objectwise surjection) by definition, so this square is a pushout as well (in $\mathbf{Set}$ , epimorphisms are regular). $\qedsymbol$

Proof 2. By the previous post, the presheaf $S_{\mathscr U}$ is the colimit over $(V,\alpha) \in (h \downarrow S_{\mathscr U})$ of $h_V$ (see post for precise statement). Let $D \colon (\bullet \rightrightarrows \bullet) \to \mathbf{Set}$ be the diagram $I^2 \rightrightarrows I$ of the two projections, and let $\mathcal I = \bigcup D = (h \downarrow D)^{\operatorname{op}}$ be the category of elements of $D$ , as in this post. There is a natural functor $F \colon \mathcal I \to (h \downarrow S_{\mathscr U})$ taking $(i,j) \in I^2$ to $(U_i \times_U U_j,h_{U_i \times_U U_j} \to S_{\mathscr U})$ and $i \in I$ to $(U_i,h_{U_i} \to S_{\mathscr U})$ , taking the morphisms $i \leftarrow (i,j) \to j$ in $\mathcal I$ to the projections $U_i \leftarrow U_i \times_U U_j \to U_j$ . We claim that $F$ is cofinal, hence the colimit can be computed over $\mathcal I$ instead (see Tag 04E7).

To verify this, we use the criteria of Tag 04E6. If $(V,\alpha) \in (h \downarrow S_{\mathscr U})$ , then by definition the composition $h_V \stackrel\alpha\to S_{\mathscr U} \hookrightarrow h_U$ is given by a morphism $f \colon V \to U$ that is contained in $S_{\mathscr U}(V)$ . Since $S_{\mathscr U}$ is generated by the $U_i$ , this factors through some $V \to U_i$ over $S_{\mathscr U}$ , giving a map $(V,\alpha) \to F(i)$ .

If $(V,\alpha) \to F(i)$ and $(V,\alpha) \to F(j)$ are two such maps, they factor uniquely through $(V,\alpha) \to F(i,j)$ . The general result for $(V,\alpha) \to F(x)$ and $(V,\alpha) \to F(y)$ for $x,y \in \mathcal I$ (either of the form $i$ or of the form $(i,j)$ ) follows since elements of the form $(i,j)$ always map to the elements $i$ and $j$ , showing that the category $((V,\alpha) \downarrow F)$ is weakly connected. $\qedsymbol$

Corollary. Let $S_{\mathscr U}$ as above, and let $\mathscr F$ be a presheaf on $\mathscr C$ . Then

$\operatorname{Hom}(S_{\mathscr U},\mathscr F) \stackrel\sim\to \operatorname{Eq}\left( \prod_{i \in I} \mathscr F(U_i) \rightrightarrows \prod_{i,j\in I} \mathscr F\Big(U_i \underset U\times U_j\Big) \right).$

Proof. By the lemma above, we compute

$\begin{align*}\operatorname{Hom}(S_{\mathscr U},\mathscr F) &\cong \operatorname{Hom}\left(\operatorname{Coeq}\left(\coprod_{i \in I} h_{U_i \underset U \times U_j} \rightrightarrows \coprod_{i \in I} h_{U_i}\right), \mathscr F\right) \\&\cong \operatorname{Eq}\left(\prod_{i \in I} \operatorname{Hom}(h_{U_i},\mathscr F) \rightrightarrows \operatorname{Hom}\Big(h_{U_i \underset U\times U_j},\mathscr F\Big)\right),\end{align*}$

so the result follows from the Yoneda lemma. $\qedsymbol$

Corollary. Let $\mathscr C$ be a (small) site. Then a presheaf $\mathscr F \colon \mathscr C^{\operatorname{op}} \to \mathbf{Set}$ is a sheaf if and only if for every object $U \in \mathscr C$ and every covering $\{U_i \to U\}_{i \in I}$ in the site, the inclusion $S_{\mathscr U} \to h_U$ induces an isomorphism

$\operatorname{Hom}(h_U,\mathscr F) \stackrel\sim\to \operatorname{Hom}(S_{\mathscr U},\mathscr F).$

Proof. Immediate from the previous corollary. $\qedsymbol$

Thus, the category of sheaves on $\mathscr C$ can be recovered from $[\mathscr C^{\operatorname{op}},\mathbf{Set}]$ if we know at which subobjects $S \subseteq h_U$ 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).

Lovely little lemmas

Or maybe ‘amusing observations’

Covering sieves and the sheaf condition (topologies 3/6)

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