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Under definition 1 of salamander lemma, I fixed a mistake in the definition of $A_\Box$ where there was a direct sum of two submodules, where there needed to be a sum (i.e., join) instead.
Thanks for catching this.
And thanks for looking into writing out a proof of the braid lemma, if I am guessing correctly that this is what you are doing?! :-)
I hope to be doing that, yes; it may take a little while because I’ve never looked carefully at the salamander lemma – I’ve only had an idea what it was good for.
Also in salamander lemma, looking at the zig-zags after remark 3, they look off. The \swarrows should pertain to vertical differentials according to 2. under remark 3, but in the zig-zags below the \swarrows go from $X_{k+1,l}_\Box$ to $^\Box X_{k, l}$, i.e., an extramural map corresponding to a horizontal differential $X_{k+1, l} \to X_{k, l}$. So I assume the arrows in both zig-zags should be switched to the opposite directions.
Thanks, Todd. Right now I don’t have the leisure to check. I could check later today. But if you looked at it and think it’s a typo, I suppose you should feel free to fix it.
In the proof of #IntramuralIsos, replace 0 objects in the zigzag diagrams with bullets to represent an “unknown” object, since from the double complexes in the statement, we don’t know that the corresponding objects are actually 0 (and what they are is immaterial to the proof).
Mark S Davis
It would be good to have a preview mode, but it needs somebody to go and program it.
For the time being, one can use the Sandbox for testing/preview purposes.
OK so I made an nForum account just to make some edits here! There were a few times in the proof of the Sharp 3x3 Lemma (Prop 3.1) and the next that we refer to the zigzag of morphisms as being made up of intramural rather than extramural morphisms, which is fixed, along with some minor misspellings here and there.
Looks good. Thanks!
I have a few other remarks which I want to make about the page but are more seriously mathematical so wanted to ask on here if they make sense before sending the edits through:
In the proof of the Salamander Lemma, I think there are two really minor issues. In point 3, it is said that $b$ has to satisfy $\partial^{\text{hor}} b = 0$ which is true but I think what’s relevant is that it satisfies $\partial^{\text{vert}} b = 0$ so that $\partial^{\text{vert}} \partial^{\text{hor}} a = 0$ rather than what’s written. In point 4, isn’t the representative we are looking for $b - \partial^{\text{hor}} a$ (rather than +)?
Another point is that in the proof of both Sharp 3x3 Lemmas we show stuff like the donor of $B'^{\text{hor}}$ appearing in the zigzag of extramural maps as well as the donor of $A^{\text{cor}}$… Technically we just want the donor of $B'$ and $A$ right? Or was this just to sort of informally point out that in our argument we will use the face that, for example, there’s an isomorphism between $A^{\text{vert}}$ and its donor?
Thanks again.
Have now looked through the proof of the Salamander lemma and agree that there were these two glitches you point out (in item 3 and 4 of the proof).
I went ahead and fixed them.
Haven’t looked at the other issue that you highlight. But since you clearly know what you are talking about and if you have the energy, please feel invited to edit!
Much appreciated.
Added the edits discussed about the zigzags of extramural isomorphisms. Thanks for looking over what I wrote and thus helping with the first time I get my feet wet in the categorical community :) Definitely hope to be doing more categorical stuff in the future so who knows, might see more edits later down the line.
Also might at some point add more homological algebra statements and their proofs using S-Lemma but this does seem pretty illustrative for now.
Thanks for joining in, Isky! Much appreciated.
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