Geometric models for classifying spaces of a group

Question

For any given topological group $G$ we have Segal's construction/definition of $BG$. I'm recalling it in case the details turn out to be relevant.

Form the disjoint union of $G^n\times\Delta_n$ for $n\geq 0$ and identify points via $(d_i\cdot,\cdot)\sim (\cdot,\partial_i \cdot)$ where $\partial_i:\Delta_{n-1}\to\Delta_n$ are the face maps and $d_i :G^n\to G^{n-1}$ maps $(g_1,\ldots,g_n)\to (g_1,\ldots , g_i g_{i+1},\ldots,g_n)$ for $i\neq 0,n$. When $i=0$ (resp. $n$) the map $d_i$ is the projection on to the first (resp. last) $n-1$ factors.

On the other hand, one can consider $G$ as a topological category with one object and $G$ as its morphisms. One then considers the topological nerve $\mathcal{N}(G)$ which has one $0$-simplex, one $1$-simplex for each element of $G$, $2$-simplices are given by triangles labelled by $g_1,g_2,g_3$ such that the interior of the triangle corresponds to a path $h$ joining $g_1 g_2$ and $g_3$. Higher simplices are defined similarly. If one uses the natural gluing on this collection, one arrives at the geometric realization of $\mathcal{N}(G)$. Let's call it $\mathcal{B}G$. So here's my question, which may very well be known.

How are the two constructions $BG$ and $\mathcal{B}G$ related?

In fact, this prompts a more general question :

Given a topological category (in the sense of Segal) $\mathcal{C}$, i.e., $\mathit{Ob}$ and $\mathit{Mor}$ are topological spaces, one can form the topological nerve $\mathcal{N}(\mathcal{C})$, as explained above, and then take its realization. On the other hand, the usual nerve $N(\mathcal{C})$ of $\mathcal{C}$ is a simplicial object in $\mathbf{Top}$ and we can take its realization. How are these two realizations related?

I would love to know the answer to the above question for the usual notion of topological category too, viz., where only morphisms $\mathit{Mor}(x,y)$ is required to be a (compacty generated Hausdorff) space for any $x,y\in\mathit{Ob}$.

EDIT : May be I hadn't explained the topological nerve definition which Harry succintly does in his comment below. It is also the same definition given in Higher Topos Theory by J. Lurie.

Answer 1

The correct construction for a topological category is as follows:

If C is a topological category, we can replace it trivially with a simplicial category by taking the simplicial singular complex associated to each hom-space. By abuse of notation, we will call this functor $Sing$.

Now it suffices to give the answer for simplicial categories.

However, to find the classifying space of a simplicial category, we take its associated quasicategory by looking at the homotopy coherent nerve.

The homotopy coherent nerve is usually constructed formally as the adjoint of another functor called $\hat{FU}$, which is the extension of the bar construction $\bar{FU}$ for the associated comonad $FU:Cat\to Cat$ of the free-forgetful adjunction $U:Quiv\rightleftarrows Cat:F$.

Specifically, given any comonad based at $X$, we can form the bar construction, which gives us a functor from $X\to X^{\Delta^{op}}$. This is done by taking objects to be $F_k=F^{k+1}$ the degeneracies to be instances of the comultiplication map $s_i:F_k\to F_{k+1}=F^i\mu F^{k-i}:F^{k+1}\to F^{k+2}$ and faces given by the appropriate application of the counit (the idea is similar to the above, and I leave it as an exercise). In particular, we may take the whole simplicial object in $End(X)$, which gives us our functor $\bar{F}:X\to X^{\Delta^{op}}$

Back to our specific case, we resolve the comonad $FU:Cat\to Cat$ to a functor $\bar{FU}:Cat\to Cat_\Delta$ (since the resolution is trivial on objects , we can say this with a straight face). Restricting $\bar{FU}$ to $\Delta$, which can always be embedded as a full subcategory of $Cat$. By general abstract nonsense, any functor $X\to C$ where C is cocomplete lifts to a unique colimit preserving functor $Psh(X)\to C$ (since taking presheaves gives a "free" cocompletion). Standard notation suggests that we call this functor $\hat{FU}:sSet\to Cat_\Delta$, but following Lurie, we will call it $\mathfrak{C}$. In particular, this functor has a right adjoint called the homtopy coherent nerve, which we can compute as follows:

$$\mathcal{N}(C)_n:=Hom_{Cat_\Delta}(\mathfrak{C}(\Delta^n),C)$$.

for any simplicial category $C$.

Returning to your original case, $BC=\mathcal{N}(Sing(C))$ for a topological category $C$, and for a topological group, we need only notice that a topological group is identical to a one-object Top-enriched category, all of whose morphisms are invertible (something like this).

As for why this is the right definition, I fear I must refer you to Lurie's HTT. It relies on a proof of a certain Quillen equivalence, and alas, the margins are too small...

Edit: Alright, so the reason why they agree is covered in §4.2.4 of HTT, I'm pretty sure.

Answer 2

They're the same. You can construct the geometric realization of a simplicial space $X:\Delta^{op} \to Top$ by taking its co-end with the functor $\Delta \to Top$ which sends the $n$ to the "standard n-simplex" $\Delta^n$. Segal's construction is an explicit description of this co-end in the particular case that $X$ is the enriched nerve of a topological group.

There is something to be said however, since often this construction "isn't right". To be more specific, its really the "FAT geometric realization" which captures the correct homotopy type of a topological group/groupoid, in the sense of the existence of universal principal bundle over it, or, in a more abstract way, it is equal to the weak homotopy type of its corresponding topological stack. Under some mild conditions on your groupoid (e.g. the unit map being a neighborhood deformation retract), then you still get the right answer. That fat geometric realization is essentially what you get by keeping only the strictly increasing arrows in $\Delta$ when you form the co-end.

Answer 3

I came here looking for essentially an answer to the original question, but none of the replies so far actually provide a reference (HTT 4.2.4 does not contain the result, as far as I can tell.)

In [1] there is a proof that the ordinary and homotopy coherent nerve of a "fibrant groupoid" are homotopy equivalent (corollary 2.6.3). They also comment that this result can "probably" be deduced from work of Cordier-Porter, Lurie and Joyal.

Here "fibrant groupoid" means "fibrant simplicially enriched category with homotopy category a groupoid", i.e. a simplicially enriched category in which all morphism spaces are kan complexes, and all (0-)morphisms are equivalences. Given a topological group $G$, such a category is produced by taking the one-object category with mapping space the singular complex of $G$.

[1] V. Hinich, Homotopy coherent nerve in Deformation theory arXiv:0704.2503v4