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I am getting re-introduced to Algebraic Topology. To motivate things up, the lifting problem was stated:

Given a map $p\colon E\to B$ and a map $f\colon X\to E$, does there exist a map making the following diagram commute?
enter image description here

Then, homotopies were introduced, and it was mentioned that algebraic topology interests itself with the homotopic flavored question of the above problem, that is, we relax "$p\circ g = f$" to "$p\circ g$ is homotopic to $f$". Let's call this property of $p$ as lifting property up to homotopy.

Then, the definition of HLP was mentioned:

$p\colon E\to B$ is said to have homotopy lifting property with respect to a space $X$ iff for any commutative diagram (solid arrows) of the following kind, there exists a diagonal arrow (dashed) making the whole diagram commute:
enter image description here

The homotopy lifting is simultaneously "stronger" and "weaker" than the original lifting in the following sense:

  1. Weaker because the domain is no longer a general topological space, but of the form $X\times I$.

  2. Stronger because we are lifting $G$ subject to an additional constraint, namely $G(\cdot, 0) = g$ for a given $g$.


This raises the natural question: Is there a relation between HLP and the "lifting property up to homotopy"? Does one imply the other? I have not been able to say anything conclusive.

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The map $p\colon E\rightarrow B$ has the "lifting property up to homotopy" if and only if it has a section up to homotopy, i.e. a map $s\colon B\rightarrow E$ s.t. $p\circ s\simeq\mathrm{id}_B$ (apply the property to the identity of $B$). In particular, this implies $p$ induces a surjection on all homotopy groups. There are plenty of fibrations (maps satisfying the HLP for all spaces) that do not satisfy this property, e.g. you can take the path-space fibration $PX\rightarrow X$ for any based space $X$ that is not weakly contractible (since $PX$ is always contractible). In the converse direction, any homotopy equivalence has the "lifting property up to homotopy", but need not be a fibration, e.g. the inclusion $\{0\}\rightarrow I$ (this doesn't even satisfy path lifting).

As a general remark, I think your contextualization is a bit off. The problems we are classically interested in are bona fide lifting problems. The point of the HLP is not to "replace" these by other lifting problems. Instead, it allows us to study the the lifting problem for a map $f$ by instead studying it for a map $f^{\prime}$ homotopic to it (and if the lifting problem is solvable, the resulting lifts will be homotopic in a compatible manner). This is systematically exploited in areas like obstruction theory to solve genuine lifting problems via homotopy-theoretic methods.

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  • $\begingroup$ Thanks! I haven't come across path-space fibration yet. Will come back to this when I do. But I nevertheless got the $\{0\}\hookrightarrow I$ example. $\endgroup$
    – Atom
    Commented May 23 at 5:30

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