I know that the loop space of given pointed topological space $(X,\ast)$ is the set of pointed maps $\mathrm{Map}_\ast(S^1,X)$. I would denote it by $\Omega X$.
On the other hand, I saw an article in $n$Lab which contains a categorical definition of loop space: In a $(\infty,1)$-category with a zero-object $0$, the loop space of object $X$ is the pullback of the diagram $0\rightarrow X\leftarrow 0$ (,as far as I have understood).
I have tried to apply this definition to the category $\mathrm{Top}_\ast$, which is a $(\infty,1)$-category (as $n$-morphisms are $n$-homotopies) with the zero-object $\{\ast\}$. However, the pullback of $\{\ast\}\rightarrow X\leftarrow\{\ast\}$ seems to be $\{\ast\}$, not $\Omega X$. And, even if we accept that $\Omega X$ is the pullback, then there have to be an up-to-homotopy unique pointed map $Y\rightarrow \Omega X$ for any pointed space $Y$, because two zero-maps $Y\rightrightarrows X$ are homotopic by definition. I cannot imagine such a map $Y\rightarrow \Omega X$ except the trivial zero-map. This is not desirable for me because originally I am interested in $\Omega$-spectrum and thus the map $Y\rightarrow \Omega X$ should be a homotopy equivalence. So I got confused completely.
Is not the categorical object defined as above the categorical counterpart of $\Omega X$? Is the explanation of the article a bit incorrect? Or have I misunderstood something?
(I glanced at a referenced paper by Lurie, but I couldn't read the construction because of my poor knowledge about category theory...)
