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    • CommentRowNumber1.
    • CommentAuthorZhen Lin
    • CommentTimeMar 11th 2012
    • (edited Mar 11th 2012)

    Let C\mathbf{C} be a small category and let E\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:CEA : \mathbf{C} \to \mathbf{E} such that the induced functor () CA:C^E(-) \otimes_\mathbf{C} A : \widehat{\mathbf{C}} \to \mathbf{E} is left exact. Now, when E=Set\mathbf{E} = \mathbf{Set}, I know this is equivalent to the definition given in the nLab article flat functor: the comma category (1A)(1 \downarrow A) is cofiltered if and only if AA is flat in the above sense (and I guess this extends without problems to the case (EA)(E \downarrow A) for a general set EE). This is Theorem 3 in [Sheaves and Geometry and Logic, Ch. VII, §6]. The article also asserts that, when E\mathbf{E} is a topos, if AA is representably flat, then AA is flat in the above sense. This seems dubious, since we should at least take E\mathbf{E} to be a locally small and cocomplete topos (e.g. a Grothendieck topos). Regardless, assuming E\mathbf{E} is a sufficiently nice topos, is the converse true? That is, if AA is flat, is AA representably flat? For some reason Mac Lane and Moerdijk phrase everything internally in terms of E\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 C\mathbf{C} is a finitely complete category and E\mathbf{E} is a locally small cocomplete topos, are

    1. AA being a flat functor
    2. AA being a representably flat functor
    3. AA being left exact

    all equivalent? For the case E=Set\mathbf{E} = \mathbf{Set}, Mac Lane and Moerdijk prove (1) ⇒ (3) ⇒ (2) ⇔ (1), and I can see that (1) ⇒ (3) ⇒ (2) hold even when E\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…

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeMar 12th 2012

    You may be interested in this blog post. Does that answer some or all of your questions?

    • CommentRowNumber3.
    • CommentAuthorZhen Lin
    • CommentTimeMar 12th 2012
    • (edited Mar 12th 2012)

    Interesting – thanks! If I understand correctly, all the definitions coincide when C\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)?

    • CommentRowNumber4.
    • CommentAuthoreparejatobes
    • CommentTimeMar 12th 2012

    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 yFLan_y F are what is called there postulated, then

    Lan yFLan_y F preserves finite limits iff FF is flat

    where FF 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 yFLan_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 EE is small with subcanonical topology, a colimit is postulated if it is preserved by the Yoneda embedding of EE into the topos E^\widehat E of sheaves on EE

    You may also find interesting

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeMar 12th 2012

    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.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMar 12th 2012

    I have reorganized and added material to the page flat functor, so that hopefully it can answer this question by itself in the future.

    • CommentRowNumber7.
    • CommentAuthorZhen Lin
    • CommentTimeMar 13th 2012

    That is much clearer, thanks! Still, there’s one little thing I have to be sure of: when C\mathbf{C} is small, and E\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 C\mathbf{C} can be lifted to an internal category in E\mathbf{E}).

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeMar 13th 2012

    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. (-:

    • CommentRowNumber9.
    • CommentAuthorZhen Lin
    • CommentTimeMay 17th 2012

    I’ve been thinking about flat functors and I have some further questions.

    1. 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 opSet\mathbb{C}^{op} \to \mathbf{Set} as a “right \mathbb{C}-module” and write the Yoneda extension of a “left \mathbb{C}-module” as F{-} \otimes_{\mathbb{C}} F. But the definition in terms of elements seems to give something very different.

    2. 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?

    3. Let GG be an internal group in an elementary topos \mathcal{E}, and let XX be an internal left GG-torsor. Then, since X1X \to 1 is epic, there is a (generalised) element xx of XX, and we can define an internal morphism f:GXf : G \to X by ggxg \mapsto g \cdot x. This is an internal epimorphism/monomorphism by transitivity/freeness (resp.). This means f:GXf : G \to X is an internal isomorphism. (Right…?) This seems disturbing, since it looks as if \mathcal{E} believes that any two left GG-torsors are isomorphic. What’s really going on?

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeMay 18th 2012

    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 GXG\to X” is internally true.

    Anyway, why does it bother you that \mathcal{E} believes any two GG-torsors are isomorphic? That doesn’t make them externally isomorphic.

    • CommentRowNumber11.
    • CommentAuthorZhen Lin
    • CommentTimeMay 18th 2012

    Why do you say that?

    Well, by “definition in terms of elements” I mean something like this. Let RR be a ring. A left RR-torsor is a left RR-module MM with the following properties:

    1. There is a non-zero element of MM.

    2. Given mMm \in M and nMn \in M, there exist elements r,sr, s of RR and an element pp of MM such that rp=mr \cdot p = m and sp=ns \cdot p = n.

    3. Given rRr \in R, sRs \in R, and mMm \in M such that rm=smr \cdot m = s \cdot m, there is an element tt of RR and an element pp of MM such that tr=tst r = t s and tp=mt \cdot p = m.

    In the case that RR is a field, we see that a left RR-torsor must be a one-dimensional vector space over RR. I think if RR is an integral domain then it can be any faithful RR-submodule of its fraction field. But this is far from a complete classification of flat RR-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 GG-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 GG-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 “XX is isomorphic to GG as GG-torsors” but which is not automatically valid for all GG-torsors XX?

    • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 18th 2012

    Then, since X1X \to 1 is epic, there is a (generalised) element xx of XX, and we can define an internal morphism f:GXf : G \to X by ggxg \mapsto g \cdot x.

    Actually what you would get is a morphism G×UG×XXG\times U \to G\times X \to X, where x:UXx:U\to Xis the generalised element, and hence that XX is isomorphic to GG as a GG-object in the internal language (witnessed by the morphism G×UX×UG\times U \to X\times U).

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeMay 18th 2012

    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 “XX is isomorphic to GG as GG-torsors” but which is not automatically valid for all GG-torsors XX?

    No, I don’t think so. Just like in sets, any two isomorphic GG-torsors have all the same properties, the same is true internally.

    • CommentRowNumber14.
    • CommentAuthorZhen Lin
    • CommentTimeMay 18th 2012

    Externally, a GG-torsor XX is trivial if and only if there is a global element of XX, 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 1X1 \to X,” does not work, since it really means “XX is inhabited.” That’s quite disappointing, because it seems to be saying that cohomology is invisible to the internal logic…

    • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 18th 2012

    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.

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeMay 18th 2012

    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 SetSet is trivial.

    • CommentRowNumber17.
    • CommentAuthorZhen Lin
    • CommentTimeMay 18th 2012

    I haven’t seen that point of view before. It makes a lot of sense. Thanks!