homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
A model category is a homotopical category equipped with extra good control over the weak equivalences. In particular every object of the category is weakly equivalent to an object that is particularly well behaved for forming derived hom-spaces into it – these are the fibrant objects, as well as weakly equivalent to a object that is particularly well-behaved for forming derived hom-spaces out of it – these are the cofibrant objects.
Hence fibrant and cofibrant objects are particularly good representatives of objects, which are the “same” as the given objects up to weak equivalence.
These concepts exists also in homotopical categories with less extra structure than that of a full model category. For instance a category of fibrant objects implements roughly half of the model category axioms, namely those for fibrations and, as the name indicates, it has a concept of weakly equivalent replacement by fibrant objects, but in general not by cofibrant object. And dually, in a cofibration category there is a notion of cofibrant objects but not necessarily of fibrant objects.
In a model category, an object $X$ is said to be fibrant if the unique morphism $X\to 1$ to the terminal object is a fibration.
Dually, $X$ is said to be cofibrant if the unique morphism $0\to X$ from the initial object is a cofibration.
Hence the axiom that every morphism in a model category factors
as an acyclic cofibration followed by a fibration
as a cofibration followed by an acyclic fibration
implies fibrant resolution and cofibrant resolution of objects:
For $X$ any object then
the factorization of the terminal morphism as an acyclic cofibration followed by a fibration yields a fibrant object $X_{fib}$ weakly equivalent to $X$
the factorization of the initial morphism as a cofibration followed by an acyclic fibration yields a cofibrant object $X_{cof}$ weakly equivalent to $X$
The standard examples appear
in the classical model structure on topological spaces, here every object $X$ is fibrant (namely the continuous function $X \to \ast$ to the point space is a Serre fibration), and the cofibrant objects are the retracts of cell complexes, in particular the CW-complexes;
in the classical model structure on simplicial sets, here every object is cofibrant, and the fibrant objects are the Kan complexes.
Other examples include:
In the projective model structure on dgc-algebras in non-negative degree, the cofibrant objects are the Sullivan algebras (see there). This plays a key role in rational homotopy theory
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