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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2012
    • (edited Apr 16th 2012)

    I have added to flat functor right after the very first definition (“CSetC \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.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2012
    • (edited Apr 16th 2012)

    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 SetSet-flatness.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2012

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

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2012

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

    What “ThT h”? Maybe I am mixed up.

    • CommentRowNumber5.
    • CommentAuthorZhen Lin
    • CommentTimeOct 18th 2012

    It says “TT is a cone over FDF \circ D with vertex uu”, so ThT h must be a cone over FDF \circ D with vertex vv.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2012

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

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeOct 18th 2012

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

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeOct 31st 2012
    • (edited Oct 31st 2012)

    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 *h^*” has also the common interpretation of pullback in the sense of pullback of functions, functors, etc by precomposition with hh. And this is what we do here.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeOct 31st 2012
    • (edited Oct 31st 2012)

    Something different:

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


    Δ 0Δ \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 Δ[n]\Delta[n] as a sequence of nn elementary edges and morphisms in Δ 0\Delta_0 have to send elementary edges to elementary edges).

    This is a (representably) flat functor, right?

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeNov 3rd 2012

    Does that mean every morphism in Δ 0\Delta_0 is an injection? If so, it doesn’t seem like it could possibly be flat. Consider the identity [1][1][1]\to [1] and the projection [1][0][1]\to[0]. If those were to factor through a span [1][n][0][1] \leftarrow [n] \to [0] in Δ 0\Delta_0, then nn would have to be 00 by injectivity, but then the composite [1][0][1][1]\to[0]\to [1] couldn’t be the identity.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeNov 3rd 2012

    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.

    • CommentRowNumber12.
    • CommentAuthorZhen Lin
    • CommentTimeNov 3rd 2012

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

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeNov 4th 2012


    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”.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeNov 5th 2012

    re #10,

    thanks, of course. What was I thinking?

    • CommentRowNumber15.
    • CommentAuthorTim Campion
    • CommentTimeFeb 24th 2019

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

    diff, v32, current

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 25th 2019

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

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

    Jens Hemelaer

    diff, v34, current

    • CommentRowNumber18.
    • CommentAuthorzskoda
    • CommentTimeSep 25th 2022
    • 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

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