Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2012
   • (edited Apr 16th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to _[[flat functor]]_ right after the very first definition ("$C \to Set$ is flat if its category of elements is cofiltered") a remark which spells out explicitly what this means in components. Just for convenience of the reader.

   I have added to flat functor right after the very first definition (“$C \to Set$ is flat if its category of elements is cofiltered”) a remark which spells out explicitly what this means in components. Just for convenience of the reader.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2012
   • (edited Apr 16th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added in the section _[Topos-valued functors](http://ncatlab.org/nlab/show/flat+functor#ToposValuedFlatFunctors)_ right after the definition the remark that in a topos with enough points, internal flatness is stalkwise $Set$-flatness.

   I have added in the section Topos-valued functors right after the definition the remark that in a topos with enough points, internal flatness is stalkwise $Set$-flatness.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 16th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexI noticed that any mention of _[[Diaconescu's theorem]]_ was missing from the entry _[[flat functor]]_, so I added [a section](http://ncatlab.org/nlab/show/flat+functor#DiaconescuTheorem).

   I noticed that any mention of Diaconescu’s theorem was missing from the entry flat functor, so I added a section.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeOct 18th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexHere is a question, probably to Mike, on the section _[Site-valued functors](http://ncatlab.org/nlab/show/flat+functor#SiteValuedFunctors)_: I may be too tired, but I have trouble parsing this here: > $ \{ h\colon v\to u | T h \;\text{ factors through the }\; F\text{-image of some cone over }\; D \} $ What "$T h$"? Maybe I am mixed up.

   Here is a question, probably to Mike, on the section Site-valued functors:

   I may be too tired, but I have trouble parsing this here:

   $\{ h\colon v\to u | T h \;\text{ factors through the }\; F\text{-image of some cone over }\; D \}$

   What “$T h$”? Maybe I am mixed up.

5. • CommentRowNumber5.
   • CommentAuthorZhen Lin
   • CommentTimeOct 18th 2012
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexIt says "$T$ is a cone over $F \circ D$ with vertex $u$", so $T h$ must be a cone over $F \circ D$ with vertex $v$.

   It says “$T$ is a cone over $F \circ D$ with vertex $u$”, so $T h$ must be a cone over $F \circ D$ with vertex $v$.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeOct 18th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexMaybe we could write $h^* T$ or the like.

   Maybe we could write $h^* T$ or the like.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeOct 18th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI think $h^* T$ would be confusing; that looks to me like $h$ and $T$ have the same *codomain* and we are pulling $T$ back along $h$. Here the codomain of $h$ is the domain of $T$ and we are just composing every morphism in $T$ with $h$.

   I think $h^* T$ would be confusing; that looks to me like $h$ and $T$ have the same codomain and we are pulling $T$ back along $h$. Here the codomain of $h$ is the domain of $T$ and we are just composing every morphism in $T$ with $h$.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeOct 31st 2012
   • (edited Oct 31st 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThere is really just one sensible way to interpret the notation, so it doesn't really matter. I was a bit over-tired when I asked the above question. But nevertheless, let me remark: the notation "$h^*$" has also the common interpretation of pullback in the sense of pullback of functions, functors, etc by _precomposition_ with $h$. And this is what we do here.

   There is really just one sensible way to interpret the notation, so it doesn’t really matter. I was a bit over-tired when I asked the above question.

   But nevertheless, let me remark: the notation “$h^*$” has also the common interpretation of pullback in the sense of pullback of functions, functors, etc by precomposition with $h$. And this is what we do here.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeOct 31st 2012
   • (edited Oct 31st 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSomething different: somebody please give me a sanity check, it's so easy to get mixed up about variances in this business: let $$ \Delta_0 \to \Delta $$ be the functor into the simplex category out of the non-full subcategory of finite linear non-empty graphs (hence regard each $\Delta[n]$ as a sequence of $n$ elementary edges and morphisms in $\Delta_0$ have to send elementary edges to elementary edges). This is a (representably) flat functor, right?

   Something different:

   somebody please give me a sanity check, it’s so easy to get mixed up about variances in this business:

   let

   $$\Delta_0 \to \Delta$$

   be the functor into the simplex category out of the non-full subcategory of finite linear non-empty graphs (hence regard each $\Delta[n]$ as a sequence of $n$ elementary edges and morphisms in $\Delta_0$ have to send elementary edges to elementary edges).

   This is a (representably) flat functor, right?

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeNov 3rd 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexDoes that mean every morphism in $\Delta_0$ is an injection? If so, it doesn't seem like it could possibly be flat. Consider the identity $[1]\to [1]$ and the projection $[1]\to[0]$. If those were to factor through a span $[1] \leftarrow [n] \to [0]$ in $\Delta_0$, then $n$ would have to be $0$ by injectivity, but then the composite $[1]\to[0]\to [1]$ couldn't be the identity.

    Does that mean every morphism in $\Delta_0$ is an injection? If so, it doesn’t seem like it could possibly be flat. Consider the identity $[1]\to [1]$ and the projection $[1]\to[0]$. If those were to factor through a span $[1] \leftarrow [n] \to [0]$ in $\Delta_0$, then $n$ would have to be $0$ by injectivity, but then the composite $[1]\to[0]\to [1]$ couldn’t be the identity.

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeNov 3rd 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexI added to the "representable flatness" section of [[flat functor]] an explicit description of what this means in terms of objects and morphisms. By the way, where did the words "transitivity" and "freeness" come from in Remark 1? I've never heard them used to describe those conditions before.

    I added to the “representable flatness” section of flat functor an explicit description of what this means in terms of objects and morphisms.

    By the way, where did the words “transitivity” and “freeness” come from in Remark 1? I’ve never heard them used to describe those conditions before.

12. • CommentRowNumber12.
    • CommentAuthorZhen Lin
    • CommentTimeNov 3rd 2012
    • PermaLink
    Author: Zhen Lin
    Format: MarkdownItexIn the case where $C$ is the delooping of a group $G$, "transitivity" and "freeness" reduce to exactly the usual "transitivity" and "freeness" axioms for a $G$-torsor.

    In the case where $C$ is the delooping of a group $G$, “transitivity” and “freeness” reduce to exactly the usual “transitivity” and “freeness” axioms for a $G$-torsor.

13. • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeNov 4th 2012
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexAh. Generally I think I am against importing words from group torsors to describe flat functors when they lose their original intuition thereby. Group torsors are *such* a special case of flat functors. I would be more inclined to call those properties something like "product cones" and "equalizer cones".

    Ah.

    Generally I think I am against importing words from group torsors to describe flat functors when they lose their original intuition thereby. Group torsors are such a special case of flat functors. I would be more inclined to call those properties something like “product cones” and “equalizer cones”.

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeNov 5th 2012
    • PermaLink
    Author: Urs
    Format: MarkdownItexre #10, thanks, of course. What was I thinking?

    re #10,

    thanks, of course. What was I thinking?

15. • CommentRowNumber15.
    • CommentAuthorTim Campion
    • CommentTimeFeb 24th 2019
    • PermaLink
    Author: Tim Campion
    Format: MarkdownItexThe page claimed that a functor $C \to Set$ is flat if and only if it preserves finite limits. I added the hypothesis that $C$ must have finite limits. <a href="https://ncatlab.org/nlab/revision/diff/flat+functor/32">diff</a>, <a href="https://ncatlab.org/nlab/revision/flat+functor/32">v32</a>, <a href="https://ncatlab.org/nlab/show/flat+functor">current</a>

    The page claimed that a functor $C \to Set$ is flat if and only if it preserves finite limits. I added the hypothesis that $C$ must have finite limits.

    diff, v32, current

16. • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 25th 2019
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexActually I think what it claimed is that such a functor is flat if and only if its [[Yoneda extension]] $[C^{op},Set] \to Set$ preserves finite limits, and that doesn't require $C$ to have finite limits.

    Actually I think what it claimed is that such a functor is flat if and only if its Yoneda extension $[C^{op},Set] \to Set$ preserves finite limits, and that doesn’t require $C$ to have finite limits.

17. • CommentRowNumber17.
    • CommentAuthornLab edit announcer
    • CommentTimeApr 15th 2020
    • PermaLink
    Author: nLab edit announcer
    Format: MarkdownItexCorrect the reference to Borceux's Handbook of Categorical Algebra. Jens Hemelaer <a href="https://ncatlab.org/nlab/revision/diff/flat+functor/34">diff</a>, <a href="https://ncatlab.org/nlab/revision/flat+functor/34">v34</a>, <a href="https://ncatlab.org/nlab/show/flat+functor">current</a>

    Correct the reference to Borceux’s Handbook of Categorical Algebra.

    Jens Hemelaer

    diff, v34, current

18. • CommentRowNumber18.
    • CommentAuthorzskoda
    • CommentTimeSep 25th 2022
    • PermaLink
    Author: zskoda
    Format: MarkdownItex* M.E. Descotte, E.J. Dubuc, M. Szyld, _On the notion of flat 2-functors_, arXiv:[1610.09429](https://arxiv.org/abs/1610.09429) updated to * M.E. Descotte, [[E.J. Dubuc]], M. Szyld, _Sigma limits in 2-categories and flat pseudofunctors_, (v1: On the notion of flat 2-functors) [arXiv:1610.09429](https://arxiv.org/abs/1610.09429) Adv. Math. __333__ (2018) 266--313 <a href="https://ncatlab.org/nlab/revision/diff/flat+functor/38">diff</a>, <a href="https://ncatlab.org/nlab/revision/flat+functor/38">v38</a>, <a href="https://ncatlab.org/nlab/show/flat+functor">current</a>

    • M.E. Descotte, E.J. Dubuc, M. Szyld, On the notion of flat 2-functors, arXiv:1610.09429

    updated to

    • M.E. Descotte, E.J. Dubuc, M. Szyld, Sigma limits in 2-categories and flat pseudofunctors, (v1: On the notion of flat 2-functors) arXiv:1610.09429 Adv. Math. 333 (2018) 266–313

    diff, v38, current