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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.
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.
I noticed that any mention of Diaconescu’s theorem was missing from the entry flat functor, so I added a section.
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.
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 $v$.
Maybe we could write $h^* T$ or the like.
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 $h$.
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.
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?
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.
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.
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.
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”.
re #10,
thanks, of course. What was I thinking?
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.
updated to
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