Model category

In the mathematical model of a homotopy category is a category with the selected sub- classes of arrows, the "weak equivalences ", " fibrations " and are called " Kofaserungen ". The requirements for these classes represent an abstraction of the corresponding topological concepts and allow the construction of an associated homotopy category not only for the category of topological spaces, but also about the category of chain complexes. In the latter case it is called the associated Homotopiekategorien -derived categories.

The term was introduced in 1967 by Daniel G. Quillen.

• 2.1 linting and cofasernde objects

• 3.1 Topological spaces
• 3.2 Chain complexes

Definition

In one category, three subcategories same object were excellent:

• Weak equivalences
• Fibrations
• Kofaserungen.

We call ( co- ) acyclic fibrations and trivial if they are both weak equivalences.

Ie model category if the following axioms are satisfied:

MC1 ( (Co- ) limites )

Is finite bivollständig.

MC2 ( "2 out of 3")

If f, g, gf arrows in and two of them weak equivalences, so is the third.

MC3 ( Retrakte )

If f is a retract of an arrow g, belongs to one of the excellent sub, so f belongs to the same sub-category.

MC4 ( added)

If in the commutative diagram

I Cofaserung, p fibration and i or p acyclic, so there is an arrow which commutes with the diagram.

MC5 ( decomposition )

1 Each arrow can be represented as i for a fibration p and an acyclic Kofaserung.

2 Each arrow can be represented as an acyclic fibration for i p and a Kofaserung.

Properties

• The definition is self-dual: the dual category also carries the structure of a model category, in which only the classes of fibrations and Cofaserungen are reversed.
• The axiom MC4 characterizes the classes of fibrations and Cofaserungen: An arrow 'p' if and only fibration, if for every diagram in the acyclic i Cofaserung, an elevation h is ( according to Cofaserungen ). A model category structure is therefore already clearly defined by specifying the weak equivalences and one of the classes of fibrations and Cofaserungen.
• The class of fibrations is stable under base change, which is stable under the Cofaserungen Cobasiswechsel.

Linting and cofasernde objects

After MC1 contains in particular an initial object and a final object. An object 'X' is, linting, if fibrillation is cofasernd if Cofaserung is.

Examples

Topological spaces

In the category of topological spaces usually the following model category structure is considered: As the weak equivalences are the weak homotopy equivalences, fibrations chosen as the Serre- fibrations.

The topological spaces can be also provided with a model structure in which the weak equivalences are the homotopy equivalences.

Chain complexes

The category of chain complexes of R-modules with non-negative indices, the following model category structure:

• As the weak equivalences arrows (ie, degree preserving homomorphisms which respect the derivative operator ) can be selected that induce isomorphisms in homology.
• Fibrations are the arrows, whose components are for each degree monomorphisms with projective cokernel.
• Cofaserungen the arrows, for are injective in positive degrees.

Homotopy

In order to transfer the notion of homotopy in any model categories, cylinder objects and Wegobjekte be defined with the help of left and Rechtshomotopien be defined.

These two Homotopiebegriffe are generally neither equivalence relations nor do they agree with each other. In the event that the sources and destinations of the observed arrows are fraying and kofasernd, both definitions describe the same equivalence relation. One can therefore proceed to a homotopy as follows: First, arrows are functorial replaced by those that differ only by weak equivalences, but have lint and kofasernde sources and destinations. Then you can equivalence classes left or rechtshomotoper arrows summarize to homotopy classes and obtains the homotopy category.

Since one can describe the transition to the homotopy category as a localization with respect to the weak equivalences, one needs no knowledge of the fibrations and Kofaserungen for the construction of the homotopy category.