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!

14more comments