Simplicial set

In mathematics, a simplicial set is a construction in categorical homotopy theory that 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 that can be built up (or faithfully represented up to homotopy) from simplices 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 do not naturally exist.

Formal definition

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

X: Δ → Set

where Δ denotes the simplex category whose objects are finite strings of ordinal numbers of the form

[n] = 0 → 1 → ... → n

(or in other words non-empty totally ordered finite sets) and whose morphisms are order-preserving functions between them, and Set is the category of small sets.

It is common to define simplicial sets as a covariant functor from the opposite category, as

X: Δ^op → Set

That is, as a presheaf. This definition is clearly equivalent to the one immediately above.

Alternatively, one can think of a simplicial set as a simplicial object (see below) in the category Set, but this is only 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 maps and degeneracy maps to the category Δ.

Face and degeneracy maps

The simplex category Δ is generated by two particularly important families of morphisms (maps), whose images under a given simplicial set functor are called face maps and degeneracy maps of that simplicial set.

The face maps of a simplicial set are the images in that simplicial set of the morphisms $\delta ^{1},\dotsc ,\delta ^{n}\colon [n-1]\to [n]$ , where $\delta ^{i}$ is the only injection $[n-1]\to [n]$ that "misses" $i$ . Let us denote these face maps by $d_{1},\dotsc ,d_{n}$ respectively.

The degeneracy maps of a simplicial set are the images in that simplicial set of the morphisms $\sigma ^{1},\dotsc ,\sigma ^{n}\colon [n+1]\to [n]$ , where $\sigma ^{i}$ is the only surjection $[n+1]\to [n]$ that "hits" $i$ twice. Let us denote these degeneracy maps by $s_{1},\dotsc ,s_{n}$ respectively.

The defined maps satisfy the following simplicial identities:

d_i d_j = d_j−1 d_i if i < j
d_i s_j = s_j−1 d_i if i < j
d_i s_j = id if i = j or i = j + 1
d_i s_j = s_j d_i−1 if i > j + 1
s_i s_j = s_j+1 s_i if i ≤ j.

The standard n-simplex and the category of simplices

Categorically, the standard n-simplex, denoted Δⁿ, is the functor hom(-, [n]) where [n] denotes the string 0 → 1 → ... → n of the first (n + 1) nonnegative integers and the homset is taken in the category Δ. In many texts, it is written instead as hom([n],-) where the homset is understood to be in the opposite category Δ^op.^[1]

The geometric realization |Δⁿ| is just defined to be the standard topological n-simplex in general position given by

|\Delta ^{n}|=\{(x_{0},\dots ,x_{n})\in \mathbb {R} ^{n+1}:0\leq 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(Δⁿ, X).^[2] The n-simplices of X are then collectively denoted by X_n. Furthermore, there is a category of simplices, denoted by $\Delta \downarrow {X}$ whose objects are maps (i.e. natural transformations) Δⁿ → X and whose morphisms are natural transformations Δⁿ → Δ^m over X arising from maps [n] → [m] in Δ. That is, $\Delta \downarrow {X}$ is a slice category of Δ over X. The following isomorphism shows that a simplicial set X is a colimit of its simplices:^[3]

X\cong \varinjlim _{\Delta ^{n}\to X}\Delta ^{n}

where the colimit is taken over the category of simplices 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 Hausdorff topological spaces.

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

X\times Y

is realized as a product

|X|\times _{Ke}|Y|

of the corresponding topological spaces, where $\times _{Ke}$ denotes the Kelley space product. This product is the right adjoint functor that takes X to X_C as described here, applied to the ordinary topological product |X| × |Y|. 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 right adjoint to the geometric realization functor described above, i.e.:

hom_Top(|X|, Y) ≅ hom_S(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 closed simplicial model category.

A key turning point of the theory 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 abstract nonsense. Furthermore, the geometric realization and singular functors give a Quillen equivalence of closed model categories inducing 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. It is part of the general definition of a Quillen adjunction that the right adjoint functor (in this case, the singular set functor) carries fibrations (resp. trivial fibrations) to fibrations (resp. trivial fibrations).

Simplicial objects

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

X: Δ → C

or equivalently a covariant functor

X: Δ^op → C

When C is the category of sets, we are just talking about simplicial sets. Letting C be the category of groups or category of abelian groups, we obtain the categories sGrp of simplicial groups 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 simplicial abelian groups can be computed 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.

Notes

^ S. Gelfand, Yu. Manin, "Methods of Homological Algebra"
^ Specifically, consider $\Delta ^{n}=\Delta ^{\mathrm {op} }(\mathbf {n} ,-)$ , then the Yoneda lemma gives $\mathrm {Nat} (\Delta ^{\mathrm {op} }(\mathbf {n} ,-),X)\cong X(\mathbf {n} )$
^ Goerss & Jardine, p.7

References

Goerss, P. G.; Jardine, J. F. (1999). Simplicial Homotopy Theory. Progress in Mathematics. Vol. 174. Basel, Boston, Berlin: Birkhäuser. ISBN 978-3-7643-6064-1.

Gelfand, S.; Manin, Yu. Methods of homological algebra.

Dylan G.L. Allegretti, Simplicial Sets and van Kampen's Theorem (An elementary introduction to simplicial sets).

simplicial set at the nLab

[1] S. Gelfand, Yu. Manin, "Methods of Homological Algebra"

[2] Specifically, consider $\Delta ^{n}=\Delta ^{\mathrm {op} }(\mathbf {n} ,-)$ , then the Yoneda lemma gives $\mathrm {Nat} (\Delta ^{\mathrm {op} }(\mathbf {n} ,-),X)\cong X(\mathbf {n} )$

[3] Goerss & Jardine, p.7

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