# Morphisms of frames induce morphisms of sites

One can associate a site with an arbitrary frame by defining coverages with suprema. According to Johnstone (Sketches of an Elephant, 2.3.20), "If $$A$$ and $$B$$ are frames, made into sites via their canonical coverages, then a morphism of sites $$A \to B$$ is exactly the same thing as a frame homomorphism, i.e. a function preserving finite meets and arbitrary joins". That definition of a morphism says that a morphism of sites is cartesian and cover-preserving.

A more general definition for non-necessarily cartesian categories is that a morphism of sites is cover-preserving and covering-flat. The definition of covering-flat functors is slightly more complicated and less obvious, so how could one instantiate the covering flatness property for a morphism of frames understood as a morphism of sites?

Another question is also related to morphisms of sites. The Grothendieck topologies on a small category exactly correspond to Lawvere-Tierney topologies on the presheaf topos $${\bf Set}^{{\bf C}^{op}}$$. I believe this correspondence should be extensible to morphisms of sites and presheaf topos functors as well, but I don't quite understand how exactly.

• It is a fact that cover-preserving and covering-flat functors between subcanonical sites that have finite limits must also preserve finite limits. This is surely in the Elephant. Apr 18, 2023 at 22:56
• Thanks! There is not so much that, just a small remark as far as I can see. Anyway, my question was rather about instantiating the definition of covering flatness for the cases of frames in order to get the covering flatness property for the simple case with coverages defined with suprema in a frame. Apr 18, 2023 at 23:01

First, let me give the "morally correct" definition.

Definition. A flat functor $$F : \mathcal{C} \to \textbf{Set}$$ is one such that the weighted colimit functor $${-} \star_\mathcal{C} F : [\mathcal{C}^\textrm{op}, \textbf{Set}] \to \textbf{Set}$$ preserves finite limits. (Recall that $$G \star_\mathcal{C} F$$ is defined so that we have a bijection $$\textrm{Hom} (G \star_\mathcal{C} F, Y) \cong \textrm{Hom} (G, \textrm{Hom} (F, Y))$$ natural in $$G$$, $$F$$, and $$Y$$.)

Of course, we should restrict to the case where $$G \star_\mathcal{C} F$$ does exist for all $$G$$ – but this is not necessarily a condition on $$\mathcal{C}$$. (For example, if $$F (C) = \emptyset$$ for all $$C$$ then $$G \star_\mathcal{C} F = \emptyset$$ for all $$G$$, without any assumption on $$\mathcal{C}$$. However, this $$F$$ is not flat.)

It can be shown that a functor $$F : \mathcal{C} \to \textbf{Set}$$ is flat if and only if the following conditions are satisfied:

• There exists $$C$$ in $$\mathcal{C}$$ such that there exists an element of $$F (C)$$.
• Given $$x_0 \in F (C_0)$$ and $$x_1 \in F (C_1)$$, there exist $$C_2$$, $$f_0 : C_2 \to C_0$$, $$f_1 : C_2 \to C_1$$, and $$x_2 \in F (C_2)$$ such that $$f_0 \cdot x_2 = x_0$$ and $$f_1 \cdot x_2 = x_1$$.
• Given $$f, f' : C_1 \to C_2$$ and $$x_1 \in F (C_1)$$ such that $$f \cdot x_1 = f' \cdot x_1$$, there exist $$C_0$$, $$e : C_0 \to C_1$$, and $$x_0 \in F (C_0)$$ such that $$f \circ e = f' \circ e$$ and $$e \cdot x_0 = x_1$$.

Thus we obtain the usual "elementary" definition of flat functor. The general definition basically comes from replacing $$\textbf{Set}$$ with a topos and interpreting the above conditions in the internal logic. (The equivalence with the "morally correct" definition is constructive, so we could internalise the "morally correct" definition too – but it is much harder to interpret that.)

Definition. Let $$\mathcal{C}$$ and $$\mathcal{D}$$ be categories and let $$K$$ be a Grothendieck topology on $$\mathcal{D}$$. A $$K$$-flat functor $$F : \mathcal{C} \to \mathcal{D}$$ is one satisfying the following conditions:

• Let $$\mathcal{D}'$$ be the full subcategory of $$\mathcal{D}$$ spanned by those $$D$$ in $$\mathcal{D}$$ for which there exist $$C$$ in $$\mathcal{C}$$ and $$D \to F (C)$$ in $$\mathcal{D}$$. Then, for each $$D$$ in $$\mathcal{D}$$, $$\mathcal{D}'_{/ D}$$ is a $$K$$-covering sieve on $$D$$.

• Suppose given $$C_0, C_1$$ in $$\mathcal{C}$$ and $$D$$, $$x_0 : D \to F (C_0)$$, and $$x_1 : D \to F (C_1)$$ in $$\mathcal{D}$$. Then there exists a $$K$$-covering sieve $$\mathcal{U} \subseteq \mathcal{D}_{/ D}$$ such that, for every $$(D', g)$$ in $$\mathcal{U}$$, there exist $$C_2$$, $$f_0 : C_2 \to C_0$$, and $$f_1 : C_2 \to C_1$$ in $$\mathcal{C}$$ and $$x_2 : D' \to F (C_2)$$ such that $$F f_0 \circ x_2 = x_0 \circ g$$ and $$F f_1 \circ x_2 = x_1 \circ g$$.

• Suppose given $$f, f' : C_1 \to C_2$$ in $$\mathcal{C}$$ and $$D$$ and $$x_1 : D \to F (C_1)$$ such that $$F f \circ x_1 = F f' \circ x_1$$. Then there exists a $$K$$-covering sieve $$\mathcal{V} \subseteq \mathcal{D}_{/ D}$$ such that, for every $$(D', g)$$ in $$\mathcal{V}$$, there exist $$C_0$$ and $$e : C_0 \to C_1$$ in $$\mathcal{C}$$ and $$x_0 : D' \to F (C_0)$$ in $$\mathcal{D}$$ such that $$f \circ e = f' \circ e$$ and $$F e \circ x_0 = x_1$$.

This is genuinely a generalisation of the earlier definition: take $$\mathcal{D} = \textbf{Set}$$ and $$K$$ to be the canonical topology. If we instead specialise to the case where $$\mathcal{C}$$ and $$\mathcal{D}$$ are preorders, many of the clauses in the above definition become trivial (because any parallel pair of morphisms is trivial). In particular, the third condition becomes trivial. Let me change notation for this specialisation.

Definition. Let $$C$$ and $$D$$ be preordered sets and let $$K$$ be a Grothendieck topology on $$D$$. A $$K$$-flat monotone map $$f : C \to D$$ is one satisfying the following conditions:

• For each $$d \in D$$, $$\bigcup_{c \in C} \{ d' \in D : d' \le d \text{ and } d' \le f (c) \}$$ is a $$K$$-covering sieve on $$d$$.

• Suppose given $$c_0, c_1$$ in $$C$$ and $$d \in D$$ such that $$d \le f (c_0)$$ and $$d \le f (c_1)$$. Then $$\bigcup_{c_2 \le c_0, c_2 \le c_1} \{ d \in D : d' \le d \text{ and } d' \le f (c_2) \}$$ is a $$K$$-covering sieve on $$d$$.

Finally, we specialise to the case where $$D$$ is a frame and $$K$$ is the canonical topology on $$D$$.

Definition. Let $$C$$ be a preordered set and let $$D$$ be a frame. A flat monotone map $$f : C \to D$$ is one satisfying the following conditions:

• $$\sup {\{ f (c) : c \in C \}} = \top$$.

• For all $$c_0$$ and $$c_1$$ in $$C$$, $$\sup {\{ f (c_2) : c_2 \le c_0 \text{ and } c_2 \le c_1 \}} = f (c_0) \land f (c_1)$$.

In particular, if $$C$$ is a meet semilattice, we deduce that a monotone map $$C \to D$$ is flat if and only if $$f (\top) = \top$$ and $$f (c_0 \land c_1) = f (c_0) \land f (c_1)$$. So a flat monotone map between frames is exactly the same thing as a meet-preserving map between them.