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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeNov 27th 2018

Correct the characterization of nerves of groupoids.

• CommentRowNumber2.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 25th 2019

• CommentRowNumber3.
• CommentAuthorGuest
• CommentTimeDec 16th 2019
I don't believe the bar construction of a group is exactly the same, on-the-nose, as the nerve of the one-object group of a small category. In particular, the indexing is not the same - they are off by one. There are also a different number of maps in between G^n and G^n+1 at each n. If we are trying to understand these objects deeply we should be concerned with the details and not just identify objects when they look the same at a high level! Some nontrivial quotienting or downshifting or other fiddling has to be done to get them to agree.

To be precise, if you let G be a discrete group, and let G x - : Sets -> Sets be the monad that takes a set X and returns the "free G-set" G x X, then it's possible to carry out the bar resolution with respect to this monad described in the link in that section. This is clearly what is implied by the link. You would then apply the resulting series of functors to say, the singleton {*} equipped with the trivial group action, and get an augmented simplicial G-set. The empty set maps to *. The singleton [0] maps to the comonad iterated once, i.e. G x {*} \cong G. [n] maps to G^{n+1}.

But there is some confusion here. The nerve of a category is not augmented, unless you include the empty category; in this case there is an augmentation (but that's not the easy fix to this problem it might appear to be at first glance). In the diagram I wrote above, the ordinal [0] would map to G, and in general [n] to G^{n+1}. This is not what happens when you take the nerve of a one-object category. You get that [0] maps to the set of objects, which in the case of a one object of a category is a singleton, {*}. And [n] maps to G^n, the set of composable strings of n-morphisms.

A better insight into what's going on is provided by looking at for example Peter May's book "A Concise Course in Algebraic Topology." What the bar construction gives you, if I understand his writing correctly, is the simplicial G-set EG. The quotienting of this by the action of G is the desired BG.

Would love someone to write up a full exposition of the connection between these that takes this subtlety into account.
• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeDec 17th 2019

Guest, you’re right. What the section gives you is $E G$, not $B G$. I usually keep this straight in my head by using two-sided bar constructions, so $N B G = B(1, G, 1)$, whereas $E G = B(G, G, 1)$ gives you the contractible total space. I will fix in a moment.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeDec 17th 2019

Added a correction to the section on bar construction (h/t to Guest), linking instead to two-sided bar construction.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeDec 17th 2019

I’m confused; is the point that “the usual bar construction of $G$” would be interpreted by some people as referring to $E G$ rather than $B G$?

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeDec 17th 2019

I think the real point Guest was making is that the article had misidentified the nerve of $B G$ (or $B A$), by writing down instead the simplicial object for $E G$. The article also linked to bar construction, largely written by me, where the discussion is about various acyclic resolutions of structures, for example the machine which produces a standard acyclic free $\mathbb{Z}G$-resolution of $\mathbb{Z}$ for the purpose of defining group cohomology. Formally that would be more like $E G = B(G, G, 1)$ which “resolves a point”.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeDec 17th 2019

Got it, thanks.

1. Off by one error

Béranger Seguin

2. Added 2 pictures: one of a $2$-simplex of $N(\mathcal{C})$ and the other of a $3$-simplex.

3. Added a pointer to the page on the Street nerve of a tricategory.