### Flat Functors and Morphisms of Sites

#### Posted by Mike Shulman

There’s something about the notion of *flat functor* that confused me vaguely in the background for a long time. Eventually I tracked it to its source: the same term is used with two slightly different meanings! As a preview of part of my CT2011 talk, today I’ll explain what those meanings are, and how a mutual generalization of them gives us a better notion of “morphism of sites”.

The starting point is that a *Set-valued* functor $F\colon C \to Set$ is **flat** if its category of elements is cofiltered. If $C$ has finite limits, then this is equivalent to $F$ preserving them; thus a flat functor “would preserve all finite limits if they existed.” This is also a reasonable thing to say because $F\colon C\to Set$ is flat if and only if the induced functor $[C^{op},Set] \to Set$ preserves finite limits (which $[C^{op},Set]$ always has, unlike $C$). In topos-theoretic terms, $F\colon C\to Set$ is flat if and only if the induced adjunction $Set \leftrightarrows [C^{op},Set]$ is a geometric morphism; thus flat functors $F\colon C\to Set$ are equivalent to points of the presheaf topos $[C^{op},Set]$.

Now let’s generalize this to codomain categories other than $Set$. Starting from the first definition, a natural line of thought might go like this. The category of elements of $F\colon C\to Set$ is the same as the comma category $(1/F)$, where $1$ is the 1-element set. In particular, it only knows about maps from $1$ into the images of $F$—but this is enough to characterize finite limits in $Set$. With a codomain other than $Set$, therefore, we may expect to need maps out of all objects rather than just out of $1$.

Thus, it seems reasonable to define a functor $F\colon C\to D$ to be **representably flat** if for each $d\in D$, the comma category $(d/F)$ is cofiltered—or equivalently, if the functor $Hom_D(d,F(-))\colon C \to Set$ is flat. (This is often called just “flat”, but I’m adding an adjective for clarity later on.) If $C$ has finite limits, then $F$ is representably flat if and only if it preserves them.

This notion of flatness has some nice properties. For instance, if $C$ has finite limits, then $F\colon C\to D$ is representably flat if and only if it preserves them. Moreover, if $C$ and $D$ are small, then $F\colon C \to D$ is representably flat if and only if the induced functor $Lan_F\colon [C^{op},Set] \to [D^{op},Set]$ preserves finite limits. Therefore, if $C$ and $D$ are small sites and $F\colon C \to D$ is representably flat, then the composite of $Lan_F$ with sheafification provides a left-exact left adjoint to $F^\ast\colon Sh(D) \to [C^{op},Set]$. If $F$ also preserves covers, then $F^\ast$ lands inside $Sh(C)$, so in this case we have an induced geometric morphism $Sh(D) \leftrightarrows Sh(C)$. Thus, one often defines a **morphism of sites** to be a representably flat functor which preserves covers (cf. for instance C2.3.7 in *Sketches of an Elephant*).

On the other hand, however, the fact that points of $[C^{op},Set]$ are equivalent to flat functors $F\colon C\to Set$ suggests that $[C^{op},Set]$ should be the classifying topos for flat functors defined on $C$. That is, for any (Grothendieck) topos $E$, geometric morphisms $E \to [C^{op},Set]$ should be equivalent to “flat functors defined on $C$” internal to $E$. This is true as long as we define “flat functors defined on $C$ internal to $E$” correctly. The straightforward approach is to write down a geometric theory whose models in $Set$ are flat Set-valued functors.

If you like geometric logic, that’s a nice exercise, but it just comes out to a functor $F\colon C \to E$ with the following property. For any finite diagram $G\colon I\to C$, consider the family of all cones $\Delta(x)\to G$ over $G$ in $C$. Each of these cones induces a cone $\Delta(F x) \to F G$ in $E$, which therefore factors through the limit $\lim F G$ (which, of course, exists in $E$). We then ask that the family of all these factorizations is jointly epimorphic onto $\lim F G$.

This is just the notion of flatness for a Set-valued functor, rephrased in terms of diagrams. Let’s call such a functor **internally flat**; then geometric morphisms $E \to [C^{op},Set]$ are equivalent to internally flat functors $C\to E$. More precisely, $F\colon C\to E$ is internally flat if and only if the induced functor $[C^{op},Set] \to E$ preserves finite limits, which is a natural generalization of the characterization of flat Set-valued functors we started from. See for instance VII.7-8 of *Sheaves in Geometry and Logic*, and B3.2.3 of *Sketches of an Elephant*.

Now, representable flatness and internal flatness are *not* the same! In fact, even for Set-valued functors, internal-flatness is equivalent to flatness in the original sense, but representable flatness is a good deal stronger (although of course they agree when the domain $C$ has finite limits).

**Exercise:** Find a functor $F\colon C\to Set$ which is internally, but not representably, flat.

I don’t know about you, but I find this a bit bothersome, especially since people don’t usually put adjectives like “representably” and “internally” in front of “flat” to clarify which notion they mean. However, I got much happier about it when I realized that representable and internal flatness are actually two special cases of a single general notion.

To wit, suppose that $C$ is a category and $D$ is a site, and define $F\colon C\to D$ to be **covering flat** if, for any finite diagram $G\colon I\to C$, and any cone $T\colon \Delta(u) \to F G$ over $F G$ in $D$, the sieve

$\{ h\colon v \to u \;|\; \text{there exists a cone }\; S\colon \Delta(w) \to G \;\text{ such that }\; T h \;\text{ factors through }\; F(S) \}$

is a covering sieve of $u$ in $D$.

Let’s see how this reproduces representable flatness and internal flatness. First, suppose $D$ is a Grothendieck topos with its canonical topology, whose covering sieves are the jointly epimorphic ones. Then for any $G\colon I \to C$, the limiting cone $\Delta(\lim F G) \to F G$ is a particular cone $T$ (as in the definition of covering flatness). In that case, the sieve in question is generated by the family of all factorizations through $\lim F G$ of cones over $G$ in $C$. Thus, covering flatness implies that this family is jointly epic, which is internal flatness. Covering flatness appears to say even more than this, since it refers to *any* cone over $F G$; but when $F G$ has a limit, it suffices to consider the limiting cone, since covering sieves are stable under pullback. So when the codomain is a topos (such as $Set$) with its canonical topology, covering flatness reduces to internal flatness.

Second, suppose $D$ has the trivial topology, in which a sieve is covering just when it contains a split epimorphism. Then covering flatness asserts that for any $G\colon I \to C$ and any cone $T\colon \Delta(u) \to F G$, there exists a cone $S\colon \Delta(w) \to G$ such that $T$ factors through $F(S)$ (since there is some $h$ in the above sieve which is split epic). But $G$ and $T$ together are precisely a finite diagram in $(u/F)$, and $S$ such that $T$ factors through $F(S)$ is precisely a cone over this diagram in $(u/F)$. Thus, when the codomain has a trivial topology, covering flatness reduces to representable flatness.

So that’s nice, but can we do anything else with covering flatness? Well, recall that one of the uses of representable flatness was to define morphisms of sites. If $F\colon C \to D$ is representably flat, then $Lan_F\colon [C^{op},Set] \to [D^{op},Set]$ preserves finite limits; hence if $D$ is a site then so does the composite
$[C^{op},Set] \xrightarrow{Lan_F} [D^{op},Set] \xrightarrow{sheafify} Sh(D)$
since sheafification always preserves finite limits. But at least *a priori*, representable flatness is more than we need for this, since we don’t actually need $Lan_F$ itself to preserve finite limits, only its composite with sheafification. The weaker notion of covering flatness is exactly right!

**Exercise:** Prove that if $C$ is a small category and $D$ is a small site, then a functor $F\colon C \to D$ is covering flat if and only if the composite $[C^{op},Set] \xrightarrow{Lan_F} [D^{op},Set] \xrightarrow{sheafify} Sh(D)$ preserves finite limits.

Therefore, if we were to define a **morphism of sites** $F\colon C \to D$ to be a functor which is (1) covering flat and (2) cover-preserving, this would be sufficient to induce a geometric morphism $Sh(D) \to Sh(C)$. I don’t know whether there are any especially interesting functors which are morphisms of sites in this sense but not the classical one, but the extra generality is aesthetically pleasing and formally convenient. For instance, one nice thing is that the inclusion of any dense sub-site is always a morphism of sites in the new sense, though not necessarily in the classical one.

More importantly, this notion of “morphism of sites” now matches exactly the corresponding theory for classifying topoi. Recall that when $E$ is a topos, geometric morphisms $E\to [C^{op},Set]$ are equivalent to internally flat, or equivalently covering flat, functors $C\to E$. If $C$ is moreover itself a small site, then geometric morphisms $E\to Sh(C)$ are classically known to correspond to internally/covering flat functors which are also cover-preserving—in other words, morphisms of sites in the new sense, where $E$ is equipped with its canonical topology. That means that modulo size issues (which I’ll address in my CT talk), the functor

$Sh(-) \colon Site \to Topos^{op}$

(where $Site$ is defined using the new notion of morphism of sites) is *left adjoint* to the forgetful functor

$U\colon Topos^{op} \to Site$

which sends a topos to its underlying category equipped with its canonical topology, and a geometric morphism to its inverse image functor. But things are actually even better than that, because $U$ is *fully faithful*: a functor between Grothendieck topoi is the inverse image functor of a geometric morphism precisely when it is a morphism of sites for the canonical topologies. In other words, the (2-)category of topoi is a reflective subcategory of the (2-)category of sites.

I’ll finish with a puzzle for the reader, connecting the new notion of covering flatness back to the original intuition for flat functors.

**Puzzle:** Under what conditions on $C$ and $D$ can we say that $F\colon C\to D$ is covering flat if and only if it preserves finite limits? *(Hint: we need some conditions on $D$ in addition to $C$.)*

## Re: Flat Functors and Morphisms of Sites

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That’s nice. It gives a refined precise sense of how $Sh(-) : Site^{op} \to Topos$ is a completion operation.

Have you thought about possible higher analogs of this definition of covering-flat functors?