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
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model category, model -category
Definitions
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Universal constructions
Refinements
Producing new model structures
Presentation of -categories
Model structures
for -groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant -groupoids
for rational -groupoids
for rational equivariant -groupoids
for -groupoids
for -groups
for -algebras
general -algebras
specific -algebras
for stable/spectrum objects
for -categories
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for -operads
for -categories
for -sheaves / -stacks
A model category is a homotopical category equipped with especially nice 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 is said to be fibrant if the unique morphism to the terminal object is a fibration.
Dually, is said to be cofibrant if the unique morphism 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 any object then
the factorization of the terminal morphism as an acyclic cofibration followed by a fibration yields a fibrant object weakly equivalent to
the factorization of the initial morphism as a cofibration followed by an acyclic fibration yields a cofibrant object weakly equivalent to
The standard examples appear
in the classical model structure on topological spaces, here every object is fibrant (namely the continuous function 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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Last revised on May 14, 2023 at 12:29:39. See the history of this page for a list of all contributions to it.