Timeline for Loop space in Topological sense v.s. Categorical sense

5 events

when toggle format	what		by	license	comment
Nov 8, 2017 at 11:50	comment	added	Denis Nardin		Another way of seeing it. is that a map into the homotopy pullback of $A\leftarrow B\rightarrow C$ is the same thing as a map to $A$, a map to $C$ and a homotopy of their postcompositions to $B$. Plugging in the diagram $\to X\leftarrow $ you see that a map to the homotopy pullback is a homotopy of the constant map to itself, that is a map to $\Omega X$.
Nov 8, 2017 at 11:17	comment	added	Sam Gunningham		Not sure if you have already understood this, but if you want to compute the homotopy pullback correctly, you can replace one of the $\{\ast\}$ with the path space $PX$ (paths starting at $\ast$). The path space is contractible, and $PX\to X$ (which sends a path to its endpoint) is a fibration. Thus the homotopy pullback can be computed as the naive fiber product of $PX \times_X \{\ast\}$ which is exactly $\Omega X$. The homotopy pullback models the pullback in the sense of $(\infty,1)$-categories.
Nov 8, 2017 at 5:09	comment	added	Y. S		That's the misunderstanding I made!
Nov 8, 2017 at 4:53	comment	added	მამუკა ჯიბლაძე		$(\infty,1)$-categories have $(\infty,1)$-pullbacks, in general you should not expect them to have "ordinary" pullbacks
Nov 8, 2017 at 4:40	history	asked	Y. S	CC BY-SA 3.0