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I have done a fair bit of editing to décalage.
expanded the Definition-section to make it contain explcitly the three different perspectives and their relation:
a) in components, b) as a restriction of ordinary décalage, c) in terms of joins and cones;
changed notation from “$Dec$” to “$Dec_0$”, not to collide with the notation for total décalage;
finally added the explicit discussion and proof that for $X$ a Kan complex, $Dec_0 X \to X$ is a Kan fibration replacement of the canonical effective epimorphism $X_0 \to X$. (you may remember – but hopefully not – that we had a bit of confused discussion about this a few years back, which I don’t want to keep from falling into oblivion by linking to it again… ;-)
added more references with some of the lemmas that I am citing, mostly by Danny Stevenson, and also a technical result in Joyal’s lectures.
In definition 3 I think $\square$ is defined on $sSet\times sSet$ instead of on $sSet$?
Yes, thanks! I have fixed it now.
Stephan,
I have further expanded the Idea-section at décalage. When you have a minute, could you look over it and tell me if this is the kind of introduction that you would find useful? (Or would have found useful, had I mentioned it earlier ;-)
:-) Yes, absolutely. This would have spared me a couple of hours laboring with an inconvenient fibration replacement.
I have rearranged things a bit in that introductory section, saying informally what the construction is before going into what it does.
Thanks, Tim, that does look better. I have just broken the long sentence in half, where it leads over to my previous sentence.
I wonder if the type of use I made of decalage in that Topology paper might not interpret nicely in other contexts, e.g. in simplicial sets. I think Simona Paoli looked at that but cannot remember what her conclusions were.
The statement of Proposition 2 implicitly assumes that $X$ is connected, right? (Since the proof uses corollary 2 to lemma 1 which assumes this.)
No, $X$ can be arbitrary. We are just using that the standard $n$-simplex is connected.
So we want to know what the $n$-cells
$\Delta[n] \to Dec_0 X$are. Since $C$ is defined to denote the left adjoint of $Dec_0 : sSet \to sSet$, this are equivalently the morphisms
$C(\Delta[n]) \to X \,.$To evaluate this we look at corollary 1 and find that $C(\Delta[1]) = \Delta[1] \star \Delta[0]$.
Ah, I see what may be confusing, the variable in corollary 1 is called “$X$”, but here it takes the value $\Delta[n]$. I’ll change the name of that variable in corollary 1 .
[edit: okay, I have renamed it to “$S$”]
Ok, I see.
Slightly misleading!? I’m not sure what you mean; more precisely, I don’t know where on the page you are looking.
Conceptually, I find it simplest to work in the algebraists’ category of simplices, which is the category of finite ordinals and order-preserving maps. This is a monoidal category under ordinal sum $+$. There the terminal object $[0]$ is automatically a monoid. Hence $[0]$ seen in the opposite category is a comonoid, which induces a comonad $\Delta^{op} \to \Delta^{op}$ by $[0] +(-)$. This restricts to a comonad on the topologists’ category of simplices, i.e., the category of finite nonempty ordinals. That’s the decalage comonad.
Anyway, I wish you had asked before performing such a substantial edit, which not only erased the flow of reasoning which had been there (which was not IMO misleading, even though the notation doesn’t quite match the notation introduced in edits that came after revision #5, the point in history where that section was first written), but introduces other infelicities (such as “natural functor” and “(co)monoidal product”, among others). I think the section will simply have to be rewritten again.
Since this appears to be your first edit, let me say that except for minor typos, the general informal rule is not to excise other people’s work to such an extent, before discussing the issue here first.
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