Let $A\rightarrow D\leftarrow C$ a diagram of connected pointed toplogical space where $A\rightarrow D$ is a fibration. Denote $P=A\times_{D}C$. We obtain a homotopy fiber sequence $$ \Omega D\rightarrow P\rightarrow A\times C $$

If we suppose that $D=\Omega X$ for some pointed topological space $X$. Do we obtain a homotopy fiber sequence $$ P\rightarrow A\times C\rightarrow D ?$$ where the map $A\times C\rightarrow D$ is obtained as a composition $A\times C\rightarrow D\times D\rightarrow D$ (the second map is a concatenation of loops)

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    $\begingroup$ The hofiber of $A\times C\to D$ has a model consisting of points $(a,c)$ together with a homotopy $f(a)g(c)^{-1} \sim *$, so points $(a,c)$ together with a homotopy $f(a)\sim g(c)$, so it amounts to a point in a model of the homotopy pullback $A\times^h_D C$, but $P$ is precisely that homotopy pullback, because $A\to D$ is a fibration. $\endgroup$ – Maxime Ramzi Jan 18 at 13:42
  • $\begingroup$ @Max if i understand correctly your comment then the answer is YES ? $\endgroup$ – lab Jan 18 at 13:49
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    $\begingroup$ This is true if one of the loops is inverted before the concatenation in the definition of the map $D \times D \to D$ and the chosen point of $D$ is the trivial loop. $\endgroup$ – Valery Isaev Jan 18 at 14:24
  • $\begingroup$ @ValeryIsaev Now I see thanks! $\endgroup$ – lab Jan 18 at 14:32

Yes. Here are some details.

  1. The space $P$ sits in homotopy pullback diagram $\require{AMScd}$ $$ \begin{CD} P @>>> D \\ @VVV@VVV \\ A\times C @>>> D\times D \end{CD} $$ where the the right vertical map is the diagonal. In fact, one can see this by replacing the latter map with the free path fibration $D^I \to D\times D$. After this replacement the diagram becomes a pullback and a pullback with one of the terminal maps a fibration is always a homotopy pullback.

  2. When $D = \Omega X$, there the diagonal is induced by the map $m:D\times D \to D$ given by $(\gamma_1,\gamma_2) \mapsto \gamma_1 \cdot \bar\gamma_2$, where the bar means loop inversion. This means that there is a commutative homotopy pullback $$ \begin{CD} D @>\text{diag} >> D\times D \\ @VVV @VV m V \\ * @>>> D \end{CD} $$ where $*$ is some contractible space. (Perhaps the easiest way to see this is to note that the diagonal $X\to X \times X$ has homotopy fiber $D$.)

  3. The base change of a map which is induced in also induced: this means that there is a commutative homotopy pullback $$ \begin{CD} P @>>> A \times C \\ @VVV @VVV \\ * @>>> D \end{CD} $$ where the bottom map is the same as in the previous diagram.


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