Confusion with flat functors

category-theory

Let $F: C \to E$ be a functor, where $C$ is small and $E$ is complete and cocomplete. Recall that $F$ is called representably flat if one (hence any) of the following equivalent conditions hold:

Fact: The following are equivalent:

  1. For every $e \in E$, the category $(e \downarrow F)^{op}$ is filtered.

  2. For every $e \in E$, the functor $Hom(e,F): C \to Set$ is flat (in the sense that $Elts(Hom(e,F): C \to Set)^{op}$ is filtered.

  3. For every $e \in E$, the left Kan extension of $Hom(e,F): C \to Set$ along the Yoneda embedding is a functor $Psh(C) \to Set$ which preserves finite limits.

  4. The left Kan extension of $F: C \to E$ along the Yoneda embedding is a functor $Psh(C) \to E$ which preserves finite limits.

I am in the following quandary: I believe the proofs I've read showing that the above conditions are equivalent. And yet it seems there is a counterexample.

"Counterexample?"

Take $C = \{0,1\}$ to be the discrete category with two objects, $E = Psh(\{0,1\})$ to be the corresponding presheaf category, and take $F: \{0,1\} \to Psh(\{0,1\})$ to be the Yoneda embedding. Then

  • The left Kan extension along Yoneda of $F$ is the identity functor $Psh(\{0,1\}) \to Psh(\{0,1\})$, which preserves finite limits, so that (4) is satisfied.

But

  • In (1)/(2), take $e = \emptyset$ to be the initial object (i.e the empty presheaf). The functor $Hom(\emptyset,F): \{0,1\} \to Set$ is the constant functor at 1. (The opposite of) its category of elements is equivalent to $\{0,1\}$ again, which is not filtered, so (1)/(2) is not satisfied.

So unfortunately, this is one of those questions where my question is:

Question: Where am I screwing up here?

Best Answer

You have shown that (4) is not equivalent to (1)-(2)-(3). Indeed, in the nLab article you cite, the closest analogue of (4) is instead about the left Kan extension functor $\textbf{Psh} (\mathcal{C}) \to \textbf{Psh} (\mathcal{E})$, and $\mathcal{E}$ is required to be small, not complete.

For the purposes of understanding geometric morphisms $\mathcal{D} \to \mathcal{C}$, where $\mathcal{C}$ and $\mathcal{D}$ are sites, the correct notion of "flat" is "covering-flat", and a functor $\mathcal{C} \to \mathcal{D}$ is covering-flat if and only if the left Kan extension $\textbf{Psh} (\mathcal{C}) \to \textbf{Sh} (\mathcal{D})$ preserves finite limits. (To get a functor $\textbf{Sh} (\mathcal{C}) \to \textbf{Sh} (\mathcal{D})$, we require $\mathcal{C} \to \mathcal{D}$ to be cover-preserving.)

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