Simplicial set

A simplicial set is a construction in categorical homotopy theory. It is a purely algebraic model for " nice " topological spaces. This model derives from the combinatorial topology, in particular the idea of ​​simplicial complexes.

Motivation

A simplicial set is a categorical (ie, purely algebraic ) model, which describes those topological spaces that arise from gluing of simplices or are homotopy equivalent to such a space. Similarities exist to describe certain topological spaces by means of CW - complexes with the main difference that simplicial sets are equipped as a purely algebraic construct with no topology ( see also below formal definition ).

In order to actually get out of simplicial sets are topological spaces, there is a geometric realization functor, which maps to the category of compactly generated Hausdorff spaces. Many classical homotopietheoretische results for CW - complexes have equivalents in the category of simplicial sets.

Formal definition

In the language of category theory is a simplicial set X is a contravariant functor

Where Δ is ​​the simplicial category; a small category whose objects are given by

And whose morphisms are the order preserving mappings between these levels. that is

This Set is the category of small amounts.

It is common for simplicial sets as covariant functors from the oppositional category

To define. This definition is equivalent to the above.

Alternatively, you can simplicial sets as simplicial objects (see below) present in the category of sets Set, but this is just a different language for the same above definition. If we have a covariant functor X: Δ → use Set instead of a contravariant, we get the definition of a kosimplizialen amount.

Simplicial sets form a category, with the SSET or simply S is commonly referred to. Your objects are simplicial sets and their morphisms are natural transformations. The appropriate category for kosimpliziale amounts usually called CSET.

These definitions stem from the relationship of the terms of edge images and the degeneracy illustrations ( pictures degeneration ) unto the category Δ.

Boundary and degeneracy pictures

In Δop there are two important classes of mappings, which we call edge illustrations and degeneracy maps. They describe the combinatorial structure of the underlying simplicial sets.

The boundary map di: n → n - 1 for i = 0, ..., n-1 is given as the unique surjective morphism in, hitting the number twice.

The degeneracy Figure si: n → n 1 for i = 0, ..., n 1 is given as the unique injective morphism in that does not meet the number.

By definition, these images meet the following simplicial identities:

The simplicial category Δ has as morphisms monotone nondecreasing functions. Since the morphisms are generated by a single element where the ' omit ' or ' add ', are the above explicit relations based on the topological applications. It can be shown that these ratios are sufficient.

The standard n- simplex, and the simplex category

Categorical the standard n- simplex is (designated An) of functor hom (-, n), where n is the chain 0 → 1 → ... → the first n (n 1) is non-negative integers. The geometric realization |? N | is just given the standard topological n- simplex in general position by

Via Yoneda lemma, the n - simplices of a simplicial set X is classified by natural transformations in hom (An, X). The set of n- simplices of X is then denoted by Xn. There is also a simplex category referred whose objects are pictures? N → X and whose morphisms natural transformations Dm →? N on X induced by pictures n → m in Δ. The following isomorphisms that a simplicial set X is a colimit of its simplices is:

Where the colimit over the simplex category of X is taken.

Geometric realization

There is a functor | • |: S → CGHaus, called the geometric realization, which performs a simplicial set X into its corresponding realization in the category of compactly generated Hausdorff spaces.

This larger category is used as Funktorziel, especially as a product of simplicial sets

As a product

Is implemented the corresponding topological spaces, the Kelley - space product is. To define the Realisierungsfunktor, we define it first to n - simplices? N than the corresponding topological n- simplex |? N |. This definition consists in a natural way to any simplicial set X continued by

Sets, where the colimit is taken over the n - simplex category of X. The geometric realization is functorial in S.

Capable of realizing concretely to the geometrical implementation | X | as follows: It is a copy of the standard n- simplex, for each element of ( for each n ) and identified ( " glued " ) to each of then each connected to the i-th page area of ( by means of the canonical homeomorphism between the standard ( n-1) - simplex and the side surface of the standard n- simplex), as well as in each case with ( by means of the canonical projection of the standard (n 1) - simplex of the standard N simplex, the i-th and (i 1 )-th area of the (n 1 ) simplex maps to both the i-th area of the n- simplex) for all i

Singular limits of a space

The singular of a topological space Y is the simplicial set defined by S ( Y): n → hom ( |? N |, Y) for each object n ∈ Δ, with the obvious functoriality on the morphisms. This definition is analogous to the standard idea in singular homology, a topological space (with standard n- simplices ) " debug target " as. In addition, the singular functor S is rechtsadjungiert to the above geometric realization, ie:

For every simplicial set X and any topological space Y.

Homotopy theory of simplicial sets

In the category of simplicial sets fibrations are Kan fibrations. A map between simplicial sets is defined as a weak equivalence if the geometric realization is a weak equivalence of spaces. An illustration is a Kofaserung if it is a monomorphism of simplicial sets. It's a tricky set of Quillen that the category of simplicial sets closed with these Morphismenklassen the axioms of a real (proper ) Model Category met.

The crux of this theory is that the realization of a Kan fibration is a Serre fibration of spaces. With the above model structure is a homotopy theory of simplicial sets can be developed. Furthermore induce functors " geometric realization " and " singular quantities " is an equivalence of Homotopiekategorien

Between the homotopy category of simplicial sets and the usual homotopy category of CW - complexes (with associated homotopy classes of figures).

Simplicial objects

A simplicial object X in a category C is a contravariant functor

Or a covariant functor

If C is the category of sets, we speak of simplicial sets. If C is the category of groups, or abelian groups, we obtain the categories SgRP ( simplicial groups ) or sAb ( simplicial abelian groups).

Simplicial groups and simplicial abelian groups continue to have the structure of a closed model category induced by the underlying simplicial sets.

The homotopy fiber Direction simplicial abelian groups is obtained by applying the Dold - Kan correspondence, via an equivalence of categories between simplicial abelian groups and chain complexes limited the functors

And

Supplies.