I have seen on the nLab that we can [view the loop space as a particular homotopy pullback](https://ncatlab.org/nlab/show/loop+space+object), and that it is even the way a "loop space object" is defined in general (when it exists). 


Can someone give me some intuition of why it is so? 
To which construction in geometry does this correspond (up to homotopy)? 


Any reference would also be welcome.



The homotopy pullback of a point along itself is the loop space
This can be seen from the following general formula$^1$ for homotopy pullbacks of topological spaces.


**Proposition.** Let
$$X\xrightarrow{f}Z\xleftarrow{g}Y$$
be a diagram of topological spaces. Then
$$\mathrm{hocolim}\left(X\xrightarrow{f}Z\xleftarrow{g}Y\right)=\frac{X\times Z^{[0,1]}\times Y}{\left((x,\alpha\colon [0,1]\rightarrow Z,y)\ \middle|\ f(x)\sim\alpha(0)\text{ and }g(y)\sim\alpha(1)\right)}.$$


**Corollary.** Let $(X,x\_0)$ be a pointed space. Then the loop space of $X$ is the homotopy pullback of the diagram
$$\*\hookrightarrow X\hookleftarrow\*,$$
where the inclusion maps send the one-point space $\*$ to $x\_0$.


*Proof.* We have
$$\mathrm{hocolim}(\*\hookrightarrow X\hookleftarrow\*)=\frac{\*\times\mathrm{Hom}\left([0,1],X\right)\times \*}{\left((\*,f\colon[0,1]\rightarrow X,\*)\ \middle|\ f(0)\sim f(1)\sim x\_0\right)}\cong\frac{\mathcal{L}X}{\left(f\in\mathcal{L}X\ \middle|\ f(0)\sim f(1)\sim x\_0\right)}=\Omega X.$$
(Here $\mathcal{L}X$ is the unbased loop space of $X$.)




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$^1$See Section 2.1 of [these notes](http://www.staff.science.uu.nl/~meier007/TMF-Lecture.pdf) by Lennart Meier.


