Simplicial set

In mathematics, a simplicial set is a construction in category (mathematics) homotopy theory which is a purely algebraic model of the notion of a well-behaved topological space. Historically, this model arose from earlier work in combinatorial topology and in particular from the notion of simplicial complexes.

=Motivation=

A simplicial set is a categorical (that is, purely algebraic) model capturing those topological spaces which can be built up (or faithfully represented up to homotopy) from simplex (mathematics) and their incidence relations. This is similar to the approach of CW complexes to modeling topological spaces, with the crucial difference that simplicial sets are purely algebraic and do not carry any actual topology (this will become clear in the formal definition).

To get back to actual topological spaces, there is a geometric realization functor available which takes values in the category of compactly generated Hausdorff spaces. Most classical results on CW complexes in homotopy theory have analogous versions for simplicial sets which generalize these results. While algebraic topologists largely continue to prefer CW complexes, there is a growing contingent of researchers interested in using simplicial sets for applications in algebraic geometry where CW complexes would not typically apply.

=Formal definition=

Using the language of s.

Alternatively, one can think of a simplicial set as a simplicial object (see below) in the category Set, but this is just different language for the definition just given. If we use a covariant functor X instead of a contravariant one, we arrive at the definition of a cosimplicial set.

Simplicial sets form a category usually denoted sSet or just S whose objects are simplicial sets and whose morphisms are natural transformations between them. There is a corresponding category for cosimplicial sets as well, denoted by cSet.

These definitions arise from the relationship of the conditions imposed on the face and degeneracy maps to the category Δ.

=Face and degeneracy maps=

In Δop, there are two particularly important classes of maps called face maps and degeneracy maps which capture the underlying combinatorial structure of simplicial sets.

The face maps di : n → n − 1 are given by

: di (0 → … → n ) = (0 → … → i  − 1 → i  + 1 → … → n ).

The degeneracy maps si : n → n + 1 are given by

: si (0 → … → n ) = (0 → … → i  − 1 → i → i → i  + 1 → … → n ).

By definition, these maps satisfy the following simplicial identities:

# di dj = d j −1 d i if i < j # di sj = s j −1 di if i < j # dj sj = id = d j +1 sj # di sj = s j d i −1 if i > j  + 1 # si sj = s j +1 s i if i ≤ j .

=The standard n -simplex and the simplex category=

Categorically, the standard n -simplex, denoted Δ n , is the functor hom(-, n ) where n denotes the string 0 → 1 → ... → n of the first ( n + 1) nonnegative integers. The geometric realization |Δ n | is just the standard topological n -simplex in general position given by

:|Delta^n| = {(x_0, dots, x_n) in mathbb{R}^{n+1}: 0leq x_i leq 1, sum x_i = 1 }.

By the Yoneda lemma, the n -simplices of a simplicial set X are classified by natural transformations in hom(Δ n , X ). The n -simplices of X are then collectively denoted by Xn . Furthermore, there is a simplex category, denoted by Deltadownarrow{X} whose objects are maps Δ n → X and whose morphisms are natural transformations Δ m → Δ n over X arising from maps n → m in Δ. The following isomorphism shows that a simplicial set X is a colimit of its simplices:

: X ≅ limΔ n → X Δ n

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

=Geometric realization=

There is a functor |•|: S → CGHaus called the geometric realization taking a simplicial set X to its corresponding realization in the category of compactly-generated space Hausdorff topological spaces.

This larger category is used as the target of the functor because, in particular, a product of simplicial sets

:X imes Y

is realized as a product

:|X| imes_{Ke} |Y|

of the corresponding topological spaces, where imes_{Ke} denotes the Kelley space product. To define the realization functor, we first define it on n-simplices Δn as the corresponding topological n-simplex | Δn |. The definition then naturally extends to any simplicial set X by setting

:|X| = lim Δn → X | Δn |

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

=Singular set for a space=

The singular set of a topological space Y is the simplicial set defined by S(Y): n → hom( | Δn |, Y) for each object n ∈ Δ , with the obvious functoriality condition on the morphisms. This definition is analogous to a standard idea in singular homology of probing a target topological space with standard topological n-simplices. Furthermore, the singular functor S is adjoint functor to the geometric realization functor described above, i.e.:

:homTop(| X |, Y ) ≅ homS( X , SY )

for any simplicial set X and any topological space Y .

=Homotopy theory of simplicial sets=

In the category of simplicial sets one can define Fibrations to be Kan fibrations. A map of simplicial sets is defined to be a weak equivalence if the geometric realization is a weak equivalence of spaces. A map of simplicial sets is defined to be a cofibration if it is a monomorphism of simplicial sets. It is a difficult theorem of Daniel Quillen that the category of simplicial sets with these classes of morphisms satisfies the axioms for a proper model category closed model category simplicial model category model category. A key turning point of the theorem is that the realization of a Kan fibration is a Serre fibration of spaces. With the model structure in place, a homotopy theory of simplicial sets can be developed using standard homotopical algebra abstract nonsense. Furthermore, the geometric realization and singular functors induce an equivalence of homotopy categories

:|•|: Ho(S) ↔ Ho(Top) : S

between the homotopy category for simplicial sets and the usual homotopy category of CW complexes with homotopy classes of maps between them.

=Simplicial objects=

A simplicial object X in a category C is a contravariant functor X: Δop → C . When C = Set we are just talking about simplicial sets. Letting C = Grp or Ab we obtain the categories sGrp of simplicial group (mathematics)s and sAb of simplicial abelian groups, respectively. Simplicial groups and simplicial abelian groups also carry closed model structures induced by that of the underlying simplicial sets. The homotopy groups of fibrant object simplicial abelian groups can be recovered by making use of the Dold-Kan correspondence which yields an equivalence of categories between simplicial abelian groups and bounded chain complexes and is given by functors N: sAb → Ch+ and Γ: Ch+ → sAb.

=References=

P.G. Goerss and J.F. Jardine, Simplicial Homotopy Theory , Progress in Mathematics Vol. 174, Birkhäuser Basel-Boston-Berlin (1999) ISBN 376436064X