What is the geometric realization of the the nerve of a fundamental groupoid of a space?

Question

It can be easily seen that there exists a functor $F:Top \rightarrow Grpd$ from the category of topological spaces to the category of groupoids defined as follows:

Obj: $X \mapsto \pi_{\leq 1}(X)$, where $\pi_{\leq 1}(X)$ is the fundamental groupoid of $X$.

Mor: ($f:X \rightarrow Y) \mapsto F(f):\pi_{\leq 1}(X) \rightarrow \pi_{\leq 1}(Y)$ where the functor $F(f)$ is defined as follows:

Obj: $x \mapsto f(x)$

Mor: $([\gamma]:x \rightarrow y) \mapsto [f(\gamma)]:f(x) \rightarrow f(y) $ where $[\gamma]$ is the homotopy class of path $\gamma$ in $X$ and $[f(\gamma)]$ is the homotopy class of path $f (\gamma)$ in $Y$.

Also it is not difficult to see that $F$ is well behaved with homotopy (for example in the chapter 6 of http://www.groupoids.org.uk/pdffiles/topgrpds-e.pdf)) that is in the sense that if $f,g: X \rightarrow Y$ are homotopic then the induced functors $F(f)$ and $F(g)$ are naturally isomorphic.

Also using this functor $F$ one can construct a 2-funntor $\tilde{F}: 1Type \mapsto Gpd$ where $1Type$ is the 2-category consisting of homotopy 1-types, maps and homotopy class of homotopies between maps and $Gpd$ is the 2-category consist of Groupoids, functors and natural transformations. Now according to Homotopy hypothesis of dimension 1 as mentioned in http://math.ucr.edu/home/baez/homotopy/homotopy.pdf this $\tilde{F}$ is an equivalence of 2-categories.

So from the above mentioned observations I felt that the functor $F$ is an interesting object of study.

Now if we consider the following sequence of functors: $$ X \stackrel{F}{\mapsto} \pi_{\le 1}(X) \stackrel{N}{\mapsto} N(\pi_{\le 1}(X)) \stackrel{r}{\mapsto} r(N(\pi_{\le 1}(X))) $$ where $N$ is the nerve functor and $r$ is the geometric realization functor.

My question is the following:

How the topological spaces $X$ and $r(N(\pi_{\leq 1}(X)))$ are related? It may be possible that my question does not make much sense when $X$ is a general topological space but then, does there exist any specific class of topological spaces $X$ which has a "good relation" with $r(N(\pi_{\leq 1}(X)))$?

I would be also very grateful if someone can refer some literature in this direction.

Thank you!

Answer 1

The inclusion of groupoids into simplicial sets is fully faithful. Its left adjoint, $\Pi_1$ is given by left Kan extension of the functor $\Delta\to \mathcal{Gpd}$ sending the n-simplex to the contractible groupoid with objects $\{0,...,n\}$.

The entirety of the data of the homotopy type of the space $X$ is contained in its singular simplicial set, which is canonically a Kan complex. In particular, the fundamental groupoid functor you've written above is canonically isomorphic to the composite $\Pi_1 \circ \operatorname{Sing}$. Then we have a universal natural transformation $$\operatorname{Sing} \to N\circ \Pi_1\circ \operatorname{Sing}$$ given by the unit of the adjunction $Π_1\dashv N$. Taking geometric realizations, we obtain a span

$$\lvert \Pi_1 \rvert \cong \lvert N\circ \Pi_1\circ \operatorname{Sing}\rvert \leftarrow \lvert \operatorname{Sing}\rvert \xrightarrow{\simeq} \operatorname{id_{\mathbf{Top}}},$$

where the righthand map is the counit of the adjunction between simplicial sets and topological spaces $\lvert \bullet \rvert \dashv \operatorname{Sing}$, and is a natural weak homotopy equivalence by a theorem of Quillen.

So the lefthand map exhibits the nerve of the fundamental groupoid as stage 1 of the Postnikov System as mentioned by Denis in the first comment.

Answer 2

A very rough argument that can be (easily) formalized is as follows:

We have a notion of $\infty$-groupoids. These are like groupoids, but they have homotopies between morphisms, homotopies between homotopies, and so on. Every topological space presents an infinity groupoid by taking the objects to be points, morphisms to be paths, morphisms between morphisms to be homotopies of paths, etc.

If one takes the connected components of this we get the path components of our space. If one takes the connected components of the automorphism group of a point, we get the fundamental group. If one takes the connected components of the morphisms from a constant path to itself we get the second homotopy group, and so on.

Hence this $\infty$-groupoid can be seen as presenting all the homotopical information of our space. Now the fundamental groupoid is given by taking this $\infty$-groupoid and taking connected components of the morphisms between points to get an actual groupoid. Now we just remarked that the higher homotopy information (homotopy groups after the first) are all contained in the connected components of the morphism sets. By discretizing these sets we are removing all higher homotopical information.

So what should we expect when we realize it? Well we should expect that $\pi_0 , \pi_1$ are that of are spaces, but $\pi_n$ for $n>1$ is trivial. This is exactly what happens, we get the first Postnikov space for $X$, i.e. $K(\pi_1(X),1)$ (or really a disjoint union of these for each path component).

Answer 3

I have now (May 13) partitioned the answer into the blocks 1,2, as I think 2 is the simpler answer!

1 I hope the book Nonabelian Algebraic Topology will answer the question for you.

A groupoid is level one of a structure called a crossed complex which is a kind of nonabelian chain complex but also with the groupoid structure in dimensions $\leqslant 1$, which operates on the higher dimensional stuff. There is a homotopically defined functor $\Pi$ from the category of filtered spaces to crossed complexes, using the fundamental groupoid and relative homotopy groups and also a functor $\mathbb B$ from crossed complexes to filtered spaces such that $\Pi \mathbb B$ is naturally equivalent to the identity. This setup is particularly useful for CW-complexes with their standard cellular filtration.

Part of the thesis of the book is to use structured spaces, in this case filtered spaces, to get to link various dimensions, and in this way to use strict algebraic structures. Also the proofs use higher cubical homotopy groupoids, and are non trivial, but can involve the intuitive idea of allowing "algebraic inverses to subdivision", that is generalising to dimension $n$ the usual composition of paths. This is more difficult to do simplicially.

Part I of the book deals with dimensions $0,1,2$ where it is easier to explain the intuitions, and history. Section 2.4 discusses the classifying space of a group and of a crossed module, but the groupoid case comes in Chapter 11.

2 But an answer can easily be put: a groupoid $G$ has a set of objects say $G_0$ and its classifying space $BG $ also contains the set $G_0$.The fundamental groupoid $\pi_1(BG, G_0)$ is naturally isomorphic to $G$! That is, you need the concept of the fundamental groupoid $\pi_1(X,S) $ on a set $S$ of base points, which is formed of homotopy classes rel end points of paths in $X$ with end points in $S$. You can find this developed in the book "Topology and Groupoids". The notion itself was published in my paper

``Groupoids and Van Kampen's theorem'', Proc. London Math. Soc. (3) 17 (1967) 385-40.

The use of this Van Kampen Theorem involving a set of base points was to allow a theorem which could compute fundamental groups of spaces, such as the circle, where the traditional theorem did not apply.

See also this mathoverflow link.