# What's the intuition for weak homotopy equivalence?

Defintition: Two topological spaces $$X$$ and $$Y$$ are said to be homotopy equivalent if there exist continuous maps $$f: X \to Y$$ and $$g: Y \to X$$, such that the composition $$f \circ g$$ is homotopic to $$\text{id}_Y$$ and $$g\circ f$$ is homotopic to $$\text{id}_X$$.

Definition: A continuous map $$f: X \to Y$$ is called a weak homotopy equivalence if it induces isomorphisms $$\pi_n (X, x_0) \to \pi_n(Y,f(x_0))$$ for all $$n \geq 0$$ and all choices of basepoint $$x_0$$.

One can think of homotopy equivalent spaces as spaces, which can be deformed continuously into one another. However, I have absolutely no idea how to think about weak homotopy equivalences. How can one intuitively think about them?

Example: Warsaw circle; any map between the Warsaw circle and a point induces a weak homotopy equivalence. However these two spaces are not homotopy equivalent (link)

How can one intuitively see why we have weak homotopy equivalence, but not homotopy equivalence? Which spaces have (intuitively) the same weak homotopy type?

What are the properties that weak homotopy equivalences preserve? What are the properties that weak homotopy equivalence does not preserve, but homotopy ewuivalence preserves? Are there concrete examples of applications of weak homotopy equivalences?

Also, why bother defining weak homotopy equivalence? How do we benefit from it and where is it useful? Especially, what is the motivation for it outside of CW complexes (since for them, it is equivalent to homotopy equivalence by Whitehead's theorem).

• Every space X is weakly equivalent to a CW complex, but not homotopy equivalent to one, iirc this is constructed inductively using the homotopy groups of X. I believe weak equivalence removes a lot of pathologies like warsaw circle. I don't think topologists work with non CW complexes (might be wrong). I think it's kind of like among projection resolutions, every quasi isomorphism is a homotopy equivalence and every complex has a projective resolution, both this case and for space are examples of cofibrant replacement (or fibrant replacement, forgot which). Jun 5, 2022 at 11:52
• @TychusFindlay: Topologists work with non CW complexes all the time. Spaces that are homotopy equivalent but not homeomorphic to CW complexes are of particular importance. But there are plenty of spaces not even homotopy equivalent to CW complexes that topologists encounter, ponder, and study. Jun 5, 2022 at 13:22
• While I do not know how to answer your question about "benefit", here is a post which explains somewhat why mathematicians are satisfied with weak homotopy equivalence as being a useful concept, although the explanation goes quite far beyond topology per se. Jun 5, 2022 at 14:37
• @LeeMosher Thanks for the response, I said that because when I was at uni I was told topologists work in the "homotopy category" of spaces, and simplicial sets are another model for this (unstable) homotopy category in addition to CW complexes. Simplicial sets are more restrictive than CW complexes on the point set level, so I got the impression topologists don't care about non CW complexes. Jun 6, 2022 at 8:03

Let’s take it for granted that we are interested in spaces up to homotopy equivalence. But how can we prove that two given spaces $$X,Y$$ are actually homotopy equivalent? One option is of cause to provide an actual homotopy equivalence, but this might be actually quite hard to do (we have to specify a lot of maps).
There are a lot of functors associating to a space $$X$$ some algebraic object (usually a group, for example singular homology, or homotopy groups) in a homotopy invariant way in the sense that homotopy equivalent spaces have isomorphic associated groups. Since algebraic objects are easier to handle than topological ones (for example a map is injective iff the kernel is zero) we might ask ourselves, if we can reverse this process: Given a map $$X\rightarrow Y$$ of spaces, which induces an isomorphism of the associated algebraic objects, is it maybe already an homotopy equivalence?