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the standard bar complex of a bimodule in homological algebra is a special case of the bar construction of an algebra over a monad. I have added that as an example to bar construction.
I also added the crucial remark (taken from Ginzburg’s lecture notes) that this is where the term “bar” originates from in the first place: the original authors used to write the elements in the bar complex using a notaiton with lots of vertical bars (!).
(That’s a bad undescriptive choice of terminoiogy. But still not as bad as calling something a “triple”. So we have no reason to complain. ;-)
If I see correctly, all occurences of “$\Delta$” at bar construction refer to the augmented simplex category, which elsewhere is denoted “$\Delta_+$” or “$\Delta_a$”. I vote for changing the notation to one of these notations. While it is true that this will add a little bit of notational overhead to the entry, it serves to avoid confusion for all readers who happen to jump into the entry at some point and miss some paragraph at the beginning.
I changed all the $\Delta$’s in bar construction to $\Delta_a$’s (and introduced the notation at the exact point where augmented simplex category of mentioned).
Thanks, Todd!!
I agree; there’s a mistake there. I’ll fix it a bit later.
Hi Patrick,
welcome to the $n$Lab!
Just to say that to make your comments come out with intended formatting, choose “Mardown+Itex” below the edit box. Then things here work mostly as on the nLab itself.
In particular, for quotation marks just type
"
as usual, and for hyperlinks do
[link text](url)
Hi Patrick,
Those notes were written a long time ago, and I didn’t know then that John would be posting them publicly (not that I really mind). You are right that the end result in $Top$ (or in your favorite convenient category of spaces) should be called the mapping cylinder of the canonical quotient $X \to \pi_0(X)$, not the mapping cone. It amounts to a coproduct of cones, one for each connected component $x \in \pi_0(X)$. Does this help, or do you need something more?
Added:
The original reference for bar constructions in the generality of monads is
added various pointers to simplicial classifying space, where this is discussed.
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