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!



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


