Talk:Whitehead theorem

Model categories

What is the meaning of the (single-sentence unsourced) section on the generalization to model categories? There is no definition of "homotopy equivalent" in the general model-categorical context, to my understanding. I think the section should be deleted. Briarsong (talk) 22:28, 28 December 2024 (UTC)

You are right in saying that there is - in general - no concept of homotopy equivalences in model categories between arbitrary objects because one has to distinguish left and right homotopies. But in the case given, we are considering fibrant-cofibrant objects, for which these concepts align and are equivalences, thus we can define homotopy equivalences in the usual sense. Regarding the missing source, one could quote the book "Model Categories and Their Localizations" by Hirschhorn (or similar one). Marco Eibrink (talk) 23:49, 4 August 2025 (UTC)

Clarification requested by TakuyaMurata

I am a confused by the comment from TakuyaMurata. Although I can see how the section might seem unconnected, it still states a valid generalisation of Whitehead's Theorem for CW-complexes. I also do not understand what is meant by a "matter of definition", since - at least from the definition of model category given - this is not a completely trivial result and - as far as I can see - not an equivalent definition. Marco Eibrink (talk) 00:29, 5 August 2025 (UTC)