Homotopy Category

The homotopy category is a Category construction in algebraic topology that formalizes the idea of treating topological spaces as equivalent when they can be Homotopy Theory into each other, i.e., when they are homotopy equivalent. The naive homotopy category turns homotopy equivalence into categorical isomorphism, giving rise to homotopy-invariant concepts such as homology and Homotopy Group. However, in this approach, not every object behaves ‘nicely’. In particular,

• pullbacks,
• limits/colimits,
• etc. May not behave homotopically the way we want. For example, in the naive homotopy, the pullback of $p:S^1\to S^1$, and $i:\ast \to S^1$, where $p(e^{ix})=e^{2ix}$ only captures pointwise data giving the space with 2 points ($S^0$).

More generally, the homotopy category can be defined for any model category (such as spaces, chain complexes, or simplicial sets) by formally inverting weak equivalences, following Quillen’s construction. In this sense, the homotopy category is obtained by localizing the original category with respect to weak equivalences, whereby any morphism that is a weak (homotopy) equivalence becomes an isomorphism in the homotopy category. For topological spaces with the Quillen model structure, this yields the homotopy category frequently studied in topology, where morphisms between spaces are regarded up to weak homotopy equivalence.​

See also

homotopy 2-category

Sources

https://chatgpt.com/c/69010aae-d3a4-8322-ba91-3122e6c99b55 https://www.perplexity.ai/search/define-homotopy-category-in-2-MDqDterzTZaen15sPgIUrQ