homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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see also algebraic topology
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on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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Given a category with weak equivalences $(\mathcal{C},W)$, then its homotopy category $Ho(\mathcal{C})$ is, if it exists, the result of universally forcing the weak equivalences to become actual isomorphisms, also called the localization at the weak equivalences
The classical example is the category of topological spaces with weak equivalences those continuous functions which are homotopy equivalences or weak homotopy equivalences. The corresponding homotopy category is often referred to as “the homotopy category”, by default, or the “classical homotopy category” for emphasis. This turns out to be equivalent to the category of topological spaces or (for weak homotopy equivalences) of just those homeomorphic to CW-complexes with left homotopy-classes of continuous functions between them, whence the name “homotopy category”.
The existence of a homotopy category, as well as tractable presentations of it typically require extra properties of the class of weak equivalences (such as that they admit a calculus of fractions) or even extra structure (such as fibration category/cofibration category structure, or full model category structure, or further enhancements of that to simplicial model category structures, etc). See at homotopy category of a model category for more on this.
More generally, to every (∞,1)-category is associated a homotopy category, whose morphisms are literally the homotopy classes of the original morphisms. See at homotopy category of an (∞,1)-category for more on this.
These two concepts of “homotopy category” are compatible: to a category with weak equivalences is associated, if it exists, an (∞,1)-category obtained by universally forcing the weak equivalences to become actual homotopy equivalences, also called the simplicial localization $L_W \mathcal{C}$ at the weak equivalences. The homotopy categories of $(\mathcal{C},W)$ and of $L_W \mathcal{C}$ coincide, which justifies the terminology “homotopy category” generally.
Given a simplicially enriched category $C$, we can form for each pair of objects, $x,y$, of objects of $C$, the set, $\pi_0C(x,y)$, of connected components of the ‘function space’ $C(x,y)$. As $\pi_0$ preserves finite limits, this gives a category, denoted $\pi_0(C)$. As 1-simplices in $C(x,y)$ can be often interpreted as being homotopies, this category $\pi_0(C)$ is often called the homotopy category of $C$, and then the notation $Ho(C)$ may be used.
This notions is closely related to the next, by using, say the hammock localisation of Dwyer and Kan, as then $\pi_0$ of that simplicially enriched category, coincides with the following.
Given a category with weak equivalences (such as a model category), its homotopy category $Ho(C)$ is – if it exists – the category which is universal with the property that there is a functor
that sends every weak equivalence in $C$ to an isomorphism in $Ho(C)$.
One also writes $Ho(C) := W^{-1}C$ or $C[W^{-1}]$ and calls it the localization of $C$ at the collection $W$ of weak equivalences.
More in detail, the universality of $Ho(C)$ means the following:
The second condition implies that the functor $F_Q$ in the first condition is unique up to unique isomorphism.
If it exists, the homotopy category $Ho(C)$ is unique up to equivalence of categories.
As described at localization, in general, the morphisms of $Ho(C)$ must be constructed using zigzags of morphisms in $C$ in which the backwards-pointing arrows are weak equivalences. This means that in general, $Ho(C)$ need not be locally small even if $C$ is. However, in many cases (such as any model category) there is a more direct description of the morphisms in $Ho(C)$ as homotopy classes of maps in $C$ between suitably “good” (fibrant and cofibrant) objects.
In 2-categorical terms, the homotopy category $Ho(C)$ is the coinverter of the canonical 2-cell
where $W$ is the category whose objects are morphisms in $W$ and whose morphisms are commutative squares in $C$.
In classical homotopy theory, the homotopy category refers to the homotopy category Ho(Top) of Top with weak equivalences taken to be weak homotopy equivalences.
Ho(Top) is often restricted to the full subcategory of spaces of the homotopy type of a CW-complex (the full subcategory of CW-complexes in $Ho(Top)$). This is equivalent to $Ho(sSet_{Quillen})$, the homotopy category of the standard Quillen-model structure on simplicial sets. This equivalence is one aspect of the homotopy hypothesis.
In homological algebra the localization of the category of chain complexes at the quasi-isomorphisms is called the derived category. But see also at homotopy category of chain complexes.
In stable homotopy theory one considers the stable homotopy category of spectra.
In equivariant stable homotopy theory one considers the equivariant stable homotopy category of spectra.
See the references at model category.