A presheaf $P$ on $X$ is a sheaf iff for every covering sieve $S$ on an open set $U$ of $X$ one has $PU=\varprojlim_{V\in S}PV.$

Question

This is Exercise II.2 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]".

The Details:

Adapted from Adámek et al.'s, "Abstract and Concrete Categories: The Joy of Cats", p. 48 . . .

Definition 1: A full subcategory $\mathbf{A}$ of a category $\mathbf{B}$ if $\mathbf{A}$ is a subcategory of $\mathbf{B}$ such that for all $A, A'\in{\rm Ob}(\mathbf{A})$, we have $${\rm Hom}_{\mathbf{A}}(A, A')={\rm Hom}_{\mathbf{B}}(A, A').$$

From Mac Lane and Moerdijk, p. 37

Definition 2: Given an object $C$ in the category $\mathbf{C}$, a sieve on $C$ [. . .] is a set $S$ of arrows with codomain $C$ such that

$f \in S$ and the composite $fh$ is defined implies $fh \in S$.

Let $X$ be a topological space with $\mathcal{O}(X)$ its set of open sets.

Adapted from p. 25, ibid. . . .

Definition 3: Let $\mathbf{C}$ be a category. Then $\hat{\mathbf{C}}=\mathbf{Sets}^{\mathbf{C}^{{\rm op}}}$ is the category of presheaves of $\mathbf{C}$.

On p. 66, ibid. . . .

Definition 4: A sheaf of sets $F$ on a topological space $X$ is a functor $F:\mathcal{O}(X)^{{\rm op}}\to\mathbf{Sets}$ such that each open covering $U=\bigcup_iU_i, i\in I$, of open subsets of $U$ of $X$ yields an equaliser diagram

$$ FU\stackrel{e}{\dashrightarrow}\prod_{i\in I}FU_i\overset{p}{\underset{q}{\rightrightarrows}}\prod_{i,j\in I}(U_i\cap U_j),$$

where for $t\in FU,$ $e(t)=\{ t\rvert_{U_i}\mid i\in I\}$ and for a family $t_i\in FU_i$,

$$p\{ t_i\}=\{t_i\rvert_{(U_i\cap U_j)}\}\quad\text{ and }\quad q\{ t_i\}=\{t_j\rvert_{(U_i\cap U_j)}\}.$$

From p. 70 ibid. . . .

A sieve $S$ on $U$ is said to be a covering sieve for $U$ when $U$ is the union of all the open sets $V$ in $S$.

The definition of a limit can be found on page 21 ibid. It is standard and quite lengthy, so I will omit it here.

The Question:

Exercise II.2: A sieve $S$ on $U$ in $\mathcal{O}(X)$ may be regarded as a full subcategory of $\mathcal{O}(X)$. Prove that a presheaf $P$ on $X$ is a sheaf iff for every covering sieve $S$ on an open set $U$ of $X$ one has $$PU = {\lim_{\longleftarrow}}_{V\in S}PV.\tag{1}$$

Thoughts:

Let $P:\mathcal{O}(X)^{{\rm op}}\to\mathbf{Sets}$ be a presheaf.

$(\Rightarrow)$ Suppose $P$ is a sheaf. Let $S$ be a covering sieve of an open set $U$ of $X$.

I'm not sure what to do.

I think one has to make use of the statement about the full subcategory of $\mathcal{O}(X)$ in the sense that the uniqueness of the limit on the RHS of $(1)$ is taken to be that of the LHS by consideration of the ${\rm Hom}$-sets of $S$ and $\mathcal{O}(X)$. This idea is not fully fledged in my mind though. Does it even make sense?

$(\Leftarrow)$ I'm at a loss here.

Further Context:

To give you a rough idea of my abilities, consider the following questions of mine:

• A sieve $S$ on $U$ in the category $\mathcal{O}(X)$ is principal iff the corresponding subfunctor $S\subset 1_U\cong{\rm Hom}(-,U)$ is a sheaf.

• What does $S^z$ mean for each $z\in\mathbb{C}$?

• Shouldn't "cone" be "cocone" in the definition of a colimit?

I am studying topos theory recreationally.

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Please help :)

Answer 1

Here are some hints. First, for a preliminary observation: given that we already have functions $P(U) \to P(V)$ for each $V \in S$, and this forms a cone from $P(U)$ to the diagram $P(S)$, I would expect that the statement $$PU = \varprojlim_{V \in S} PV$$ implicitly means that this cone is a limit.

($\Rightarrow$) Given any other cone $f : X \to P(S)$, for each $x \in X$, consider that $S$ is a cover of $U$, and we also have $(f_V(x)) \in \prod_{V\in S} P(V)$.

($\Leftarrow$) Given a covering set $\{ V_i \mid i \in I \}$, the set of $V$ such that $V \subseteq V_i$ for some $i\in I$ will form a sieve; and in particular, for each $i, j \in I$ we have morphisms $V_i \cap V_j \to V_i$, $V_i \cap V_j \to V_j$ which are both in that sieve.