Simplicial amount

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

motivation

A simplicial set is a categorical (i.e., purely algebraic) model that describes those topological spaces that arise from the gluing together of simplices or are homotopy-equivalent to such a space. Similarities exist for the description of certain topological spaces by means of CW complexes with the main difference that simplicial sets as purely algebraic constructs are not equipped with any topology (see also the formal definition below ).

In order to actually get topological spaces from simplicial sets, there is a functor geometric realization that maps into the category of compact Hausdorff spaces . Many classical homotopy-theoretical results for CW-complexes have correspondences in the category of simplicial sets.

Formal definition

In the language of category theory, a simplicial set is a contravariant functor ${\displaystyle X}$

${\displaystyle X\colon \Delta \to Set,}$

where is the simplicial category ; a small category whose objects are given by ${\displaystyle \Delta }$

${\displaystyle Ob(\Delta )=\{\mathbf {n} =\{0,1,...,n\}\mid n\in \mathbb {N} \}}$

and whose morphisms are the order- preserving maps between these sets. This means

${\displaystyle Hom_{\Delta }(\mathbf {m} ,\mathbf {n} )=\{\alpha :\mathbf {m} \rightarrow \mathbf {n} \mid i\leq j\Rightarrow \alpha (i)\leq \alpha (j)\}}$.

Here is the category of sets. ${\displaystyle Set}$

It is common to use simplicial sets as covariant functors of the oppositional category

${\displaystyle X\colon \Delta ^{op}\to Set}$

define. This definition is equivalent to the above.

Alternatively, simplicial sets can be thought of as simplicial objects (see below) in the category of sets , but this is just another language for the same definition above. If we use a covariant functor instead of a contravariant, we get the definition of a cosimplicial set. ${\displaystyle Set}$ ${\displaystyle X\colon \Delta \to Set}$

Simplicial sets form a category that is usually referred to as or simply . Their objects are simplicial sets and their morphisms are natural transformations . The corresponding category for cosimplicial sets is usually mentioned . ${\displaystyle sSet}$ ${\displaystyle S}$ ${\displaystyle cSet}$

These definitions derive from the relationship of the conditions of the edge maps and the degeneracy maps (also degeneration maps) to the category . ${\displaystyle \Delta }$

Edge and degeneracy images

In there are two important classes of pictures we edge illustrations and degeneration figures call. They describe the combinatorial structure of the underlying simplicial sets. ${\displaystyle \Delta }$

The degeneracy mapping for is given as the unique surjective morphism in , which meets the number twice. ${\displaystyle s_{i}\colon {\mathbf {n} }\to \mathbf {n-1} }$ ${\displaystyle i=0,\ldots ,n-1}$ ${\displaystyle Hom_{\Delta }(\mathbf {n} ,\mathbf {n-1} )}$ ${\displaystyle i\in \mathbf {n-1} }$

The edge mapping for is given as the unique injective morphism in that does not meet the number . ${\displaystyle d_{i}\colon {\mathbf {n} }\to \mathbf {n+1} }$ ${\displaystyle i=0,\ldots ,n+1}$ ${\displaystyle Hom_{\Delta }(\mathbf {n} ,\mathbf {n+1} )}$ ${\displaystyle i\in \mathbf {n+1} }$

By definition, these mappings meet the following simplicial identities :

1. ${\displaystyle d_{j}d_{i}=d_{i}d_{j-1}}$ if ${\displaystyle i<j}$
2. ${\displaystyle s_{j}d_{i}=d_{i}s_{j-1}}$ if ${\displaystyle i<j}$
3. ${\displaystyle s_{j}d_{j}=id=s_{j}d_{j+1}}$
4. ${\displaystyle s_{j}d_{i}=d_{i-1}s_{j}}$ if ${\displaystyle i>j+1}$
5. ${\displaystyle s_{j}s_{i}=s_{i}s_{j+1}}$ if ${\displaystyle i\leq j}$

The simplicial category has monotonous, non-falling functions as morphisms. Since the morphisms are generated by those who 'omit' or 'add' a single element, the above explicit relations form the basis of the topological applications. One can show that these relations are sufficient. ${\displaystyle \Delta }$

The standard n- simplex and the simplex category

Categorically, the standard simplex ${\displaystyle n}$ (denoted by ) is the functor , where the chain of the first nonnegative natural numbers is assumed . The geometrical realization is given by the topological standard simplex in a general position ${\displaystyle \delta ^{n}}$ ${\displaystyle hom(-,{\mathbf {n} })}$ ${\displaystyle {\mathbf {n} }}$ ${\displaystyle 0\to 1\to \ldots \to n}$ ${\displaystyle (n+1)}$ ${\displaystyle \mid \Delta ^{n}\mid }$ ${\displaystyle n}$

${\displaystyle |\Delta ^{n}|=\{(x_{0},\dots ,x_{n})\in \mathbb {R} ^{n+1}\mid 0\leq x_{i}\leq 1,\sum x_{i}=1\}.}$

Via the Yoneda lemma , the -implices of a simplicial set are classified by natural transformations into . The set of -Simplices of is then denoted by. There is also a category with Simplex designated whose objects pictures and their morphisms natural transformations over induced by pictures in are. The following isomorphisms show that a simplicial set is a colimes of its simplices: ${\displaystyle n}$ ${\displaystyle X}$ ${\displaystyle hom(\Delta ^{n},X)}$ ${\displaystyle n}$ ${\displaystyle X}$ ${\displaystyle X_{n}}$ ${\displaystyle \Delta \downarrow {X}}$ ${\displaystyle \Delta ^{n}\to X}$ ${\displaystyle \Delta ^{m}\to \Delta ^{n}}$ ${\displaystyle X}$ ${\displaystyle {\mathbf {n} }\to {\mathbf {m} }}$ ${\displaystyle \Delta }$ ${\displaystyle X}$

${\displaystyle X\simeq \lim _{\Delta ^{N}\to X}\Delta ^{n}}$

Whereby the Kolimes is taken over the simplex category of . ${\displaystyle X}$

Geometric realization

There is a functor | • | , called the geometric realization , which transfers a simplicial set into its corresponding realization in the category of compactly generated Hausdorff spaces. ${\displaystyle \colon S\to CGHaus}$ ${\displaystyle X}$

This larger category is used as a functor goal, especially because it is a product of simplicial sets

${\displaystyle X\times Y}$

as a product

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

of the corresponding topological spaces is realized, where the Kelley space product is. To define the realization functor, we first define it on n-simplices as the corresponding topological n-simplex . This definition naturally continues to any simplicial set by one ${\displaystyle \times _{Ke}}$ ${\displaystyle \Delta ^{n}}$ ${\displaystyle |\Delta ^{n}|}$ ${\displaystyle X}$

${\displaystyle |X|=\lim _{\Delta ^{n}\to X}|\Delta ^{n}|}$

sets, whereby the Kolimes is taken from via the -Simplex category . The geometrical realization is functionally based . ${\displaystyle n}$ ${\displaystyle X}$ ${\displaystyle S}$

The geometrical realization can be realized concretely as follows: One takes a copy of the standard simplex for each element (for each n) and identifies ("glued") to each one with the -th side surface of (by means of the canonical homeomorphism between the standard simplex and the side surface of the standard simplex) and in each case with (by means of the canonical projection of the standard simplex on the standard simplex which the -th and -th corner of the simplex both the - the corner of the Simplex) for everyone . ${\displaystyle |X|}$ ${\displaystyle n}$ ${\displaystyle X_{n}}$ ${\displaystyle x\in X_{n}}$ ${\displaystyle d_{i}x}$ ${\displaystyle i}$ ${\displaystyle x}$ ${\displaystyle (n-1)}$ ${\displaystyle n}$ ${\displaystyle s_{i}x}$ ${\displaystyle x}$ ${\displaystyle (n+1)}$ ${\displaystyle n}$ ${\displaystyle i}$ ${\displaystyle (i+1)}$ ${\displaystyle (n+1)}$ ${\displaystyle i}$ ${\displaystyle n}$ ${\displaystyle i}$

Singular sets for a space

The singular set of a topological space is the simplicial set defined by for each object , with the obvious functoriality on the morphisms. This definition is analogous to the standard idea in singular homology of "testing out" a topological space (with standard -simplices) as the "target". In addition, the singular functor is right adjoint to the above geometrical realization, i.e. H.: ${\displaystyle Y}$ ${\displaystyle S(Y)\colon {\mathbf {n} }\to hom(|\Delta ^{n}|,Y)}$ ${\displaystyle {\mathbf {n} }\in \Delta }$ ${\displaystyle n}$ ${\displaystyle S}$

${\displaystyle hom_{Top}(|X|,Y)\simeq hom_{S}(X,SY)}$

for every simplicial set and every topological space . ${\displaystyle X}$ ${\displaystyle Y}$

Homotopy theory of simplicial sets

In the category of simplicial sets, grains are Kan grains. A mapping between simplicial sets is defined as a weak equivalence if the geometric realization is a weak equivalence of spaces. A mapping is a cofibre if it is a monomorphism of simplicial sets. It is a tricky theorem from Quillen that the category of simplicial sets together with these morphism classes fulfill the axioms of a proper closed model category.

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

| • | ${\displaystyle \colon Ho(S)\leftrightarrow Ho(Top)\colon S}$

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

Simplicial objects

A simplicial object in a category is a contravariant functor ${\displaystyle X}$ ${\displaystyle C}$

${\displaystyle X\colon \Delta \to C}$

or a covariant functor

${\displaystyle X\colon \Delta ^{op}\to C}$.

If the category is sets, we speak of simplicial sets. If the category is the groups or the Abelian groups, then we get the categories (simplicial groups) or (simplicial Abelian groups). ${\displaystyle C}$ ${\displaystyle C}$ ${\displaystyle sGrp}$ ${\displaystyle sAb}$

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

The homotopy groups of fibrous simplicial Abelian groups are obtained by applying the Dold - Kan correspondence, which provides an equivalence of categories between simplicial Abelian groups and bounded chain complexes via the functors

${\displaystyle N\colon sAb\to Ch_{+}}$

and

${\displaystyle \Gamma \to Ch_{+}\to sAb}$

supplies.

literature

• Paul G. Goerss, John F. Jardine: Simplicial Homotopy Theory. Birkhäuser, Basel et al. 1999, ISBN 3-7643-6064-X ( Progress in Mathematics 174).
• Peter May: Simplicial Objects in Algebraic Topology (a concrete and elementary introduction)
• Greg Friedman: An elementary illustrated introduction to simplicial sets (PDF file; 3.1 MB), Rocky Mountain Journal of Mathematics 42 (2012), 353–424.