TITLE: What is the homotopy fiber of a fold map?
QUESTION [8 upvotes]: If $X$ and $Y$ are based spaces, let $p_X: X\vee Y\to X$ be the fold map, or projection, onto $X$.

What is the homotopy fiber $F$ of $p_X$?

I think I have an argument that $F$ is the half-smash product $$Y\rtimes\Omega X := (Y\times\Omega X)/(\ast\times \Omega X).$$
But I'm not entirely sure of this and would like a reference, if one exists.

REPLY [6 votes]: The synthetic summary of pretty much every approach described (this is just to give an easier forumalation of the actual calculation) is sometimes called Mather's Cube theorem, sometimes Ganea's (fiber) theorem, sometimes "flattening"; I like to call it distributivity, because it decategorifies to the equation $ x ( y + z) = (xy) + (xz)$.
Call the fiber you want $F$. The wedge sum $X\vee Y$ is a pushout
\begin{array}{c}
\bullet & \to & X \\
\downarrow & \ulcorner & \downarrow\\
Y & \to & X\vee Y
\end{array}
and the map $X\vee Y\to X$ (usually called "pinch") restricts as $1_X$ on $X$ and as $0: Y\to X$ on $Y$.  By the Pasting Lemmas, the fibers of the composites $Y \to X$ and $ X \to X $ are the pullbacks of $F$ along $Y \to X\vee Y$ and along $X \to X\vee Y$, respectively, and of course the fiber of $ \bullet\to X$ is $\Omega X$, either way you factor it; but these fibers we know because they are easy to describe; by the universality of this last fiber, there is a coherent cube:
\begin{array}{c}
\Omega X & & & \to & & & \bullet \\
& \searrow & & & & \swarrow \\
& & \bullet & \to & X \\
\downarrow & &\downarrow & \ulcorner & \downarrow & & \downarrow \\
& & Y & \to & X\vee Y\\
& \nearrow & & & & \nwarrow \\
Y \Omega X & & & \to & & & F
\end{array}
What distributivity (flattening/Mather/Ganea) says is that, because the bottom square (small, inner square) is a pushout, and the vertical (distorted) squares are all pull-backs, and the whole cube is coherent, so the upper square (large) is, as the bottom square, a pushout.  That is, $F\sim Y \Omega X / \Omega X $ is the half-smash you were suspecting.
It is important to make sure you understand the maps among the known fibers, when doing these calculations; not all maps $A B \to B$ are the obvious projection!  In the present case, there just aren't too many maps $\Omega X \to \bullet$, and you can choose the structure of $\Omega X$ via the fiber-of $Y\Omega X \to Y$, as in the leftmost square.
As "some guy on the street", I once asked here how broadly does the distributive law hold, and Jacob Lurie's answer was that, according to Rezk, among homotopical categories those with the distributive law are probably homotopy toposes.  I don't know how much that helps you, but searching for "Mather cube", or "Ganea fiber" or "flattening" will provide lots of relevant references.