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Let $\mathbf{C}$ be a small category and let $\mathbf{E}$ be a locally small and cocomplete category. For the purposes of this discussion, I am defining a flat functor to be a functor $A : \mathbf{C} \to \mathbf{E}$ such that the induced functor $(-) \otimes_\mathbf{C} A : \widehat{\mathbf{C}} \to \mathbf{E}$ is left exact. Now, when $\mathbf{E} = \mathbf{Set}$, I know this is equivalent to the definition given in the nLab article flat functor: the comma category $(1 \downarrow A)$ is cofiltered if and only if $A$ is flat in the above sense (and I guess this extends without problems to the case $(E \downarrow A)$ for a general set $E$). This is Theorem 3 in [Sheaves and Geometry and Logic, Ch. VII, §6]. The article also asserts that, when $\mathbf{E}$ is a topos, if $A$ is representably flat, then $A$ is flat in the above sense. This seems dubious, since we should at least take $\mathbf{E}$ to be a locally small and cocomplete topos (e.g. a Grothendieck topos). Regardless, assuming $\mathbf{E}$ is a sufficiently nice topos, is the converse true? That is, if $A$ is flat, is $A$ representably flat? For some reason Mac Lane and Moerdijk phrase everything internally in terms of $\mathbf{E}$ in [Ch. VII, §8], and the claim that comes closest is Lemma 4.
Ultimately though, what I want to know is this: when $\mathbf{C}$ is a finitely complete category and $\mathbf{E}$ is a locally small cocomplete topos, are
all equivalent? For the case $\mathbf{E} = \mathbf{Set}$, Mac Lane and Moerdijk prove (1) ⇒ (3) ⇒ (2) ⇔ (1), and I can see that (1) ⇒ (3) ⇒ (2) hold even when $\mathbf{E}$ is only assumed to be a locally small and cocomplete category. The troublesome step is (2) ⇒ (1), which the nLab article asserts is true but gives no reference for…
You may be interested in this blog post. Does that answer some or all of your questions?
Interesting – thanks! If I understand correctly, all the definitions coincide when $\mathbf{C}$ is finitely complete, which is probably good enough for most purposes. But the post doesn’t discuss flatness in terms of the Yoneda extension being left exact… how is this definition related to the others, in the general case (of toposes)?
You may find info about this in Postulated colimits and left exactness of Kan extensions :: A Kock. This is a retyped version of an old (1989) preprint.
If (and that’s a big if) I’m reading what it says there right:
if all the colimits used to calculate $Lan_y F$ are what is called there postulated, then
$Lan_y F$ preserves finite limits iff $F$ is flat
where $F$ flat is in the internal-logic sense that Mike is referring to in the post he linked to above. Now, in a general Grothendieck topos all small colimits are postulated (Proposition 2.1 there), thus we get the characterization in terms of $Lan_y F$.
Concerning the technical-looking definition of postulated colimit, a nice, more conceptual way of looking at it is also in that paper:
If $E$ is small with subcanonical topology, a colimit is postulated if it is preserved by the Yoneda embedding of $E$ into the topos $\widehat E$ of sheaves on $E$
You may also find interesting
all the definitions coincide when C is finitely complete
Yes, at least when the codomain category is a topos, or more generally a site with finite limits and extremal-epimorphic covering families. If the codomain is a general site, then covering-flat doesn’t imply representably-flat even if both domain and codomain are finitely complete.
flatness in terms of the Yoneda extension being left exact… how is this definition related to the others
It is also equivalent (when the codomain is a topos). I think this is VII.9.1 in Sheaves in Geometry and Logic.
I have reorganized and added material to the page flat functor, so that hopefully it can answer this question by itself in the future.
That is much clearer, thanks! Still, there’s one little thing I have to be sure of: when $\mathbf{C}$ is small, and $\mathbf{E}$ is a locally small and cocomplete topos, does being representably flat imply being internally flat? This would provide the (2) ⇒ (1) step in my original post. Also, one wonders if the business about being internally flat should be rephrased in terms of internal diagrams or somesuch (since being cocomplete implies that any small category $\mathbf{C}$ can be lifted to an internal category in $\mathbf{E}$).
does being representably flat imply being internally flat?
If you read carefully, yes. Representable flatness is covering-flatness relative to the trivial topology; internal flatness is covering-flatness relative to the canonical topology. Since the canonical topology contains the trivial topology, the one implies the other.
Also, one wonders if the business about being internally flat should be rephrased in terms of internal diagrams or somesuch
It certainly could be. I don’t know that it should be. (-:
I’ve been thinking about flat functors and I have some further questions.
The definition in terms of the Yoneda extension is strongly analogous to the definition of “flat module” in commutative algebra – it works word-for-word and symbol-for-symbol if I think of a presheaf $\mathbb{C}^{op} \to \mathbf{Set}$ as a “right $\mathbb{C}$-module” and write the Yoneda extension of a “left $\mathbb{C}$-module” as ${-} \otimes_{\mathbb{C}} F$. But the definition in terms of elements seems to give something very different.
Every representable copresheaf is projective and is flat. This is even true for internal copresheaves on internal categories in an elementary topos. But are projective copresheaves flat in general, or have I been thinking about commutative algebra too much?
Let $G$ be an internal group in an elementary topos $\mathcal{E}$, and let $X$ be an internal left $G$-torsor. Then, since $X \to 1$ is epic, there is a (generalised) element $x$ of $X$, and we can define an internal morphism $f : G \to X$ by $g \mapsto g \cdot x$. This is an internal epimorphism/monomorphism by transitivity/freeness (resp.). This means $f : G \to X$ is an internal isomorphism. (Right…?) This seems disturbing, since it looks as if $\mathcal{E}$ believes that any two left $G$-torsors are isomorphic. What’s really going on?
But the definition in terms of elements seems to give something very different.
Why do you say that?
are projective copresheaves flat in general
No. Consider the coproduct of two representables; this is projective, but it doesn’t preserve terminal objects.
…we can define an internal morphism…
I think I know what you’re getting at, but words like “internal morphism” aren’t usually the way people talk. I think it would be more common to say that “there exists a morphism $G\to X$” is internally true.
Anyway, why does it bother you that $\mathcal{E}$ believes any two $G$-torsors are isomorphic? That doesn’t make them externally isomorphic.
Why do you say that?
Well, by “definition in terms of elements” I mean something like this. Let $R$ be a ring. A left $R$-torsor is a left $R$-module $M$ with the following properties:
There is a non-zero element of $M$.
Given $m \in M$ and $n \in M$, there exist elements $r, s$ of $R$ and an element $p$ of $M$ such that $r \cdot p = m$ and $s \cdot p = n$.
Given $r \in R$, $s \in R$, and $m \in M$ such that $r \cdot m = s \cdot m$, there is an element $t$ of $R$ and an element $p$ of $M$ such that $t r = t s$ and $t \cdot p = m$.
In the case that $R$ is a field, we see that a left $R$-torsor must be a one-dimensional vector space over $R$. I think if $R$ is an integral domain then it can be any faithful $R$-submodule of its fraction field. But this is far from a complete classification of flat $R$-modules, no?
No. Consider the coproduct of two representables; this is projective, but it doesn’t preserve terminal objects.
Ah, thanks. I wondered if I needed a connectedness hypothesis of some kind.
Anyway, why does it bother you that $\mathcal{E}$ believes any two $G$-torsors are isomorphic? That doesn’t make them externally isomorphic.
Well, I was expecting that the internal logic would be able to distinguish between non-isomorphic $G$-torsors. But I suppose the question is rather delicate. Is there some logical formula in the internal language of $\mathcal{E}$ which may be roughly interpreted as “$X$ is isomorphic to $G$ as $G$-torsors” but which is not automatically valid for all $G$-torsors $X$?
Then, since $X \to 1$ is epic, there is a (generalised) element $x$ of $X$, and we can define an internal morphism $f : G \to X$ by $g \mapsto g \cdot x$.
Actually what you would get is a morphism $G\times U \to G\times X \to X$, where $x:U\to X$is the generalised element, and hence that $X$ is isomorphic to $G$ as a $G$-object in the internal language (witnessed by the morphism $G\times U \to X\times U$).
by “definition in terms of elements” I mean something like this.
I think you’ve been thinking about commutative algebra too much. (-: Elementwise definitions of concepts in Set-based category theory don’t generally carry over to enriched category theory.
Is there some logical formula in the internal language of $\mathcal{E}$ which may be roughly interpreted as “$X$ is isomorphic to $G$ as $G$-torsors” but which is not automatically valid for all $G$-torsors $X$?
No, I don’t think so. Just like in sets, any two isomorphic $G$-torsors have all the same properties, the same is true internally.
Externally, a $G$-torsor $X$ is trivial if and only if there is a global element of $X$, so I guess it boils down to whether the internal logic can tell whether something has a global element or not. The most obvious formulation, “There exists a morphism $1 \to X$,” does not work, since it really means “$X$ is inhabited.” That’s quite disappointing, because it seems to be saying that cohomology is invisible to the internal logic…
But isn’t what you’re thinking about cohomology defined in the external logic? H^1(1,G) is a quotient of the hom-groupoid in the bicategory of internal groupoids and anafunctors, and this is an external construction.
Yes, cohomology is essentially by definition about the difference between internal and external. So it’s to be expected that “internal cohomology” is trivial, just like the cohomology of $Set$ is trivial.
I haven’t seen that point of view before. It makes a lot of sense. Thanks!
Re #14, “That’s quite disappointing, because it seems to be saying that cohomology is invisible to the internal logic…”: Yes, that’s true, and to be expected, as Mike clarified. But let me add two remarks:
I was quite surprised a few days ago to find out that a sheaf (of sets or of modules, doesn’t matter) on a space $X$ is injective in the usual sense if and only if it is injective from the point of view of the stack semantics of $Sh(X)$. Therefore you can internally define Ext functors (as right derived functors of Hom) and obtain in that way the sheaf-$\mathcal{E}xt$ functors. With a bit more work, you can even internally define Tor as left derived functors of the ordinary tensor product to obtain sheaf-$\mathcal{T}or$ (even though internal projectivity isn’t equivalent to external projectivity at all). What you can’t decribe internally are the non-sheafy global Ext functors.
Let $f : X \to Y$ be a morphism of topological spaces (or locales). Then there is an internal mirror image of $X$ in $Sh(Y)$, the internal locale with $f_* \Omega_{Sh(X)}$ as frame of opens. So from the internal point of view of $Sh(Y)$, it’s as we’re given a map $f : X \to pt$. Thus, internally, it makes sense to calculate sheaf cohomology of $X$. From the external point of view, we’ll actually be computing the higher derived images of $f$.
Thus the internal language still has something to say about cohomology.
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