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I have several questions, but they all are mostly definition-centered and, I assume, are easy for a person with good understanding of fibrations (unfortunately, I am not one).

Since there are several questions (although they seem very interconnected) to answer, I will put a bounty on this question as soon as it becomes possible, so that the person who will spend his time on sharing knowledge with me gets at least some additional reward.

I remember the following definition of a homotopy fibration. If we have a continuous map $f:X\to Y$, then there exists a homotopy equivalence $h:X\to P_f$ and a (Hurewicz) fibration $p:P_f\to Y$ such that $f=p\circ h$. Let $F$ be a fiber of $p$. In this case, we call the space $F$ a homotopy fiber of the map $f$ (and denote it by $\operatorname{hofib}(f)$) and we call $$F\xrightarrow{g} X\xrightarrow{f} Y$$ a homotopy fibration. Furthermore, the space $P_f$ (the mapping path space) is constructed explicitly as a subspace of $X\times PY$, which allows us to write explicit formulae for $h$ and $p$.

Question 1: What is the map $g: F\to X$ in the homotopy fibration explicitly? Is it $(x,\gamma)\mapsto x$? Does it mean that map $g$ is always surjective?

As far as I understand, if we have any other factorization $f=p'\circ h'$, the fiber of $p'$ is homotopy equivalent to $\operatorname{hofib}(f)$, i.e., the homotopy fiber is defined correctly. So, assume we have $h':X\to \widetilde{X}$ and fibration $p':\widetilde{X}\to Y$ with fiber $F'$.

Question 2: Is $$F'\xrightarrow{g'} X\xrightarrow{f} Y$$ also called a homotopy fibration? What is the map $g'$ in this case?

I have recently ecountered a book "Introduction to homotopy theory" by Paul Selick. There, on pages 53-54, he defines homotopy fibration differently. He says that $$F\xrightarrow{g} X\xrightarrow{f} Y$$ is called a homotopy fibration if there exists a homotopy commutative diagram

\begin{equation} \require{AMScd} \begin{CD} F @>{g}>> X @>{f}>> Y\\ @V{h_1}VV @V{h_2}VV @V{h_3}VV\\ W @>{i}>> E @>{p}>> B \end{CD} \end{equation} such that the bottom row is a fibration, and the vertical arrows are homotopy equivalences.

Question 3:

How are these 2 definitions equivalent?

There is this question. So, we get that the homotopy fiber of the inclusion $$i: X \vee X \rightarrow X \times X$$ is $\Sigma\left(\Omega X\wedge \Omega X\right)$.

Question 4: What is the first arrow (that is, $g$) in the homotopy fibration $$\Sigma\left(\Omega X\wedge \Omega X\right) \xrightarrow{g} X \vee X \xrightarrow{i} X \times X$$ explicitly?

Thank you!

Haldot
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    These are many different questions at once. It would be better to focus on one question at a time... – Jonas Linssen May 08 '23 at 21:26
  • I thought about it, but you can notice that all these questions are very much connected. For example, questions 1 and 2 are very close to each other, the answer to the question 2 seems to be using the answers to both Q1 and Q2, and answer to Q4 should be just implementation of the answers to Q1 and Q2 in the case of a concrete homotopy fibration. If I posted this as several questions, I think each of them would have to have the same setting written in the question's description, and the answers could not quote one another, which would not be great. – Haldot May 08 '23 at 21:36
  • Having said that, I fully understand that responsibly answering all of this is a bit lengthier than usual. That is why I plan not to choose the best answer until I am able to set the bounty on the question (that is, I will choose the best answer in 2 days). I hope that this way the person who answers my question will get at least some compensation for his effort. – Haldot May 08 '23 at 21:39
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    For 1 and 2, the map is $F \to \widetilde X \to X$, where the second map is the homotopy inverse. For 1, $\widetilde X = P_f \to X$ is the projection. If $X$ is path connected then path fibration construction will give you a surjective $g$, but this is not so meaningful up to homotopy. For 3, you need that if you change $X$ and $Y$ up to homotopy, the $F$ you get out of 1, or indeed 2, changes only up to homotopy. 1-3 should be direct from definitions/constructions (and explained in textbooks), but 4 probably requires unwrapping the cube theorem in this case. – ronno May 09 '23 at 06:03

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