Equivalence of two definitions of sheaves over a Lawvere-Tierney topology

Question

In an older text Toposes, Triples, and Theories that served as my first introduction to the general theory of (logical as opposed to Grothendieck) topoi, the definition used for a sheaf with respect to a Lawvere-Tierney topology was (paraphrasing):

An object $\mathscr{F}$ is $j$-separated if the diagonal morphism $\Delta : \mathscr{F} \to \mathscr{F} \times \mathscr{F}$ is a closed embedding. $\mathscr{F}$ is a $j$-sheaf if it is separated, and also it is universally closed in separated objects, i.e. whenever $i : \mathscr{F} \to \mathscr{G}$ is a monomorphism with $\mathscr{G}$ separated, $i$ is a closed embedding.

Later, I saw this definition on the ncatlab page (again paraphrasing):

An object $\mathscr{F}$ is a $j$-sheaf if whenever $i : V \to U$ is a dense monomorphism, ${-} \circ i : \operatorname{Hom}(U, \mathscr{F}) \to \operatorname{Hom}(V, \mathscr{F})$ is a bijection. It is $j$-separated if whenever $i : V \to U$ is a dense monomorphism, ${-} \circ i$ is injective.

So, I found myself wondering how it could be proved that these two definitions are equivalent.

First, for the equivalence of the two definitions of separated objects: ($\Rightarrow$) Suppose we have $i : V \to U$ a dense monomorphism, and suppose we have two sections $f, g \in \mathscr{F}(U)$ such that $f\circ i = g\circ i$. Then $(f, g) \in (\mathscr{F} \times \mathscr{F})(U)$ has $i^* (f, g)$ factoring through the diagonal morphism $\Delta_{\mathscr{F}}$. Since $i$ is dense and $\Delta_{\mathscr{F}}$ is a closed embedding, that implies that $(f, g)$ factors through $\Delta_{\mathscr{F}}$, so $f=g$. ($\Leftarrow$) We have that $\pi_1 \circ \operatorname{inc}_{\bar \Delta}$ and $\pi_2 \circ \operatorname{inc}_{\bar \Delta}$ on the closure of the diagonal agree when restricted to the diagonal. Since $\Delta$ is dense in $\bar \Delta$, that implies that $\pi_1 \circ \operatorname{inc}_{\bar \Delta} = \pi_2 \circ \operatorname{inc}_{\bar \Delta}$, so $\bar \Delta$ factors through $\Delta$, so the diagonal is closed.

I also think I have a proof of ($\Leftarrow$) for the definition of sheaves: suppose $\mathscr{F}$ satisfies the second definition of sheaf, and $i : \mathscr{F} \to \mathscr{G}$ is a monomorphism with $\mathscr{G}$ separated. Let $\chi_{\mathscr{F}} : \mathscr{G} \to \Omega$ be the corresponding morphism. Now suppose we have $f \in \operatorname{cl}_{\mathscr{G}}(\mathscr{F}) (U)$ for some object $U$. Then $j(\chi_{\mathscr{F}}(f)) = \operatorname{true}$, so $\chi_{\mathscr{F}}(f)$ corresponds to a dense subobject of $U$. Therefore, there exists (unique) $\tilde f \in \mathscr{F}(U)$ such that $\tilde f |_{\chi_{\mathscr{F}}(f)} = f |_{\chi_{\mathscr{F}}(f)}$. But since $\mathscr{G}$ is separated (by the first definition, and therefore separated by the second definition), that implies that $\tilde f = f$, so $f \in \mathscr{F}(U)$. Since this holds for arbitrary $U$, this shows that $\operatorname{cl}_{\mathscr{G}}(\mathscr{F}) \subseteq \mathscr{F}$, so $i$ is a closed embedding.

It's a proof for ($\Rightarrow$) that is giving me some trouble. Given $i : V \to U$, and a section $f \in \mathscr{F}(V)$, my basic idea would be to try forming the pushout of $f$ and $i$. Then, you might need to take the separated quotient of that pushout (in general, the closure of the diagonal is an equivalence relation, so you can take the quotient of that equivalence relation) and let $\mathscr{G}$ be the separated quotient. I suspect that the morphism $\mathscr{F} \to \mathscr{G}$ should be a monomorphism with dense image; however, I get bogged down in details when I try to prove this fact. If I could prove that, then it would follow that that morphism is an isomorphism, and composing its inverse with the morphism $U \to \mathscr{G}$ should give the desired section of $\mathscr{F}(U)$ lifting $f$.

Answer 1

After coming back to this question to ask it here, I found a somewhat different approach to that part: Note that the situation we have here, of a monomorphism $V \to U$ and a section $V \to \mathscr{F}$, is precisely that of a partial morphism $U \dashrightarrow \mathscr{F}$. On the other hand, in a topos, we have partial morphism classifiers $\bar{\mathscr{F}}$, for example as the interpretation of $\{ S : P(\mathscr{F}) \mid \forall x, y : \mathscr{F}, (x \in S \land y \in S) \rightarrow x = y \}$. Therefore, we have a section of $\bar{\mathscr{F}}(U)$, and we want to show it comes from a section of $\mathscr{F}(U)$.

However, $\bar{\mathscr{F}}$ is not separated in general, so to use the first sheaf condition on $\mathscr{F}$, we will need to form the separated quotient $\bar{\mathscr{F}}^{\operatorname{sep}}$. Here, the following lemma will be useful:

Lemma: Suppose $\mathscr{F}$ is separated and $i : \mathscr{F} \to \mathscr{G}$ is a monomorphism. Then the composition $\pi \circ i : \mathscr{F} \to \mathscr{G}^{\operatorname{sep}}$ is also a monomorphism.

Proof: This is probably easiest in terms of the internal language. So, suppose we have $x, y : \mathscr{F}$ with $\pi(i(x)) = \pi(i(y))$. Then by the construction of the separated quotient, we must have $j(i(x) = i(y))$ true. But since $i$ is monic, $i(x) = i(y) \rightarrow x = y$ is also true, so $j(i(x) = i(y)) \rightarrow j(x = y)$. So, we conclude $j(x = y)$ is true; but since $\mathscr{F}$ is separated, this implies $x = y$. $\square$

Now, using this lemma, we see that the canonical inclusion $\mathscr{F} \to \bar{\mathscr{F}}$ induces an inclusion $\mathscr{F} \to \bar{\mathscr{F}}^{\operatorname{sep}}$. For the given section of $\bar{\mathscr{F}}(U)$, the image in $\bar{\mathscr{F}}^{\operatorname{sep}}(U)$ is in the closure of $\mathscr{F}$, since the truth value of membership in $\mathscr{F}$, as a section of $\Omega(U)$, contains $V$.
By the first sheaf condition applied to $\mathscr{F}$, however, $\mathscr{F}$ is closed in $\bar{\mathscr{F}}^{\operatorname{sep}}$, so that implies that the given section of $\bar{\mathscr{F}}^{\operatorname{sep}}(U)$ comes from a section of $\mathscr{F}(U)$.

Answer 2

A key property of dense and closed monomorphisns of a Lawvere-Tierney topology is that closed monomorphisms are the right-orthogonal monomorphisms of the dense monomorphisms. This means that a monomorphism $m\colon G\to G'$ is closed if and only if for every dense monomorphism $d\colon U\rightarrowtail F$ and morphisms $f\colon U\to G$ and $h\colon F\to G'$ so that $m\circ f=h\circ d$, there exists a (necessarily unique) morphism $t\colon F\to G$ so that $f=t\circ d$ and $h=m\circ t$. $$\require{AMScd}\begin{CD}U@>f>>G\\@VdVV\overset t\nearrow@VVmV\\F@>h>>G' \end{CD}$$ In other words, a monomorphism $m\colon G\rightarrowtail G'$ is closed if and only if a densely-defined partial morphism $(m,f)$ has a total extension $t$ when the densely-defined partial morphism $m\circ(f,d)=(m\circ f,d)$ has a total extension $h\colon F\to G'$, in which casee $m\circ t=h$.

Applying this to the case of the diagonal morphism $m=\Delta_G\colon G\rightarrowtail G\times G$ shows immediately that the diagonal morphism is closed if and only if every densely-defined partial morphisms to $G$ has at most one total extension. This is the equivalence of the definitions of separated objects.

Note that since partial morphisms and their extensions with codomain a subobject are in bijection with partial morphisms and their extension in the object that factor throught the subobject, subobjects of separated objects are separated.

A sheaf object is an object $F$ that is right-orthogonal to dense monomorphisms, i.e. for which any densely-defined partial morphism with codomain $F$ has exactly one extension. In particular, sheaf objects are separated, as are their subobjects.

If $G'$ in the above diagram is a sheaf object, then applying right-orthogonality of closed morphisms shows that a monomorphism $m\colon G\rightarrowtail G'$ is closed if and only if $G$ is a sheaf object. In other words, a subobject of a sheaf object is itself a sheaf if and only if the subobject is a closed subobject.

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Another pair of properties is that dense monomorphisms are closed under composition and that every monomorphism factors uniquely as a dense monomorphism followed by a closed monomorphism. In particular, the isomorphisms are precisely the monomorphisms that are both dense and closed.

These suffice to show that every monomorphism from a sheaf object $F$ to suboject $G$ of a sheaf $J$ is closed. To see this, let $F\rightarrowtail G\rightarrowtail J$ be the corresponding monomorphisms.

Factor $F\rightarrowtail G$ as a dense monomorphism $F\rightarrowtail H$ followed by a closed monomorphism $H\rightarrowtail G$. We want to show that $F\rightarrowtail H$ is an isomorphism.

To that end, factor the composite $H\rightarrow G\rightarrowtail J$ as a dense monomorphism $H\rightarrow K$ followed by a closed monomorphism $K\rightarrowtail J$. Note that $K$ is a closed subobject of the sheaf $J$, whence a sheaf. But now the composite of dense monomorphisms $F\rightarrowtail H\rightarrowtail K$ is not only a dense monomorphism, but because $F$ and $K$ are sheaves, also closed, and hence an isomorphism. In particular, the monomorphism $F\rightarrowtail H$ has a monomorphic retraction $H\rightarrowtail F$, so is an isomorphism as desired.

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The proof of equivalence of the definitions of sheaf objects can then be completed by showing that every separated object is the subobject of a sheaf object. This can be done if the category is cartesian closed and has pullbacks and a subobject classifier $\Omega$ for strong monomorphisms. Here's a summary of the argument in Chapter 4 of Oswald Wyler's Topoi and Quasitopoi (he assumes the category is a quasitopos for the chapter, but partial morphism classifiers are not needed for the following).

In that context one can check that there is a sheaf $\tilde\Omega$ that classifies closed strong subobjects, with admitting a morphism $\tilde\Omega\to\Omega$. One can also check that if $F$ is a sheaf, then the exponential object $[G,F]$ is also a sheaf. In particular, $[G,\tilde\Omega]$ is a sheaf, and in fact a classifier for closed relations to $G$. The induced morphism $[G,\tilde\Omega]\to[G,\Omega]$ corrresponds to closed strong relations being relations.

Now the (strong) diagonal embedding of an object $G$ corresponds to a (strong) relation from $G$ to itself, hence to a monomorphism $G\rightarrowtail[G,\Omega]$. Then $G$ is separated if and only if the diagonal embedding is closed, i.e. if and only if the monomorphism $G\rightarrowtail[G,\Omega]$ factors through (necessarily a) monomorphism $G\rightarrowtail[G,\tilde\Omega]$.