Simplicial set

(formerly called semi-simplicial complex, full semi-simplicial complex)

A simplicial object in the category of sets (cf. Simplicial object in a category), that is, a system of sets (-fibres) , , connected by mappings , (boundary operators), and , (degeneracy operators), satisfying the conditions

(*)

The elements of the fibre are called the -dimensional simplices of the simplicial set . If only the operators are given, satisfying the relations , , then the system is called a semi-simplicial set.

A simplicial mapping between two simplicial sets and is a morphism of functors, i.e. a sequence of mappings , , satisfying the relations

Simplicial sets and their simplicial mappings form a category, . If all the are imbeddings, then is called a simplicial subset of . In this case, the boundary and degeneracy operators in are the restrictions to of the corresponding operators in .

Given any topological space , one can define a simplicial set , called the singular simplicial set of the space . Its simplices are the singular simplices of (see Singular homology), i.e. continuous mappings , where is the -dimensional geometric standard simplex:

The boundary operators and degeneracy operators of this simplicial set are defined by the formulas

The correspondence is a functor (called the singular functor) from the category of topological spaces into the category of simplicial sets .

An arbitrary simplicial complex determines a simplicial set . Its -dimensional simplices are the -tuples of vertices of with the property that there is a simplex in such that for . The operators and for this simplicial set are given by

where means that the symbol below it is omitted. If is ordered, then the simplices for which form a simplicial subset of . The correspondence () is a functor from the category of simplicial complexes (ordered simplicial complexes) into the category .

For an arbitrary group one can define a simplicial set . Its -simplices are equivalence classes of -tuples , (where if there is an element such that for all ). The operators and of are given by

The simplicial set is actually a simplicial group.

Given an arbitrary Abelian group and any integer , one can define a simplicial set (in fact, a simplicial Abelian group) . Its -dimensional simplices are the -dimensional cochains of the -dimensional geometric standard simplex with coefficients in (that is, ). Denoting the vertices of by , , one defines the simplicial mappings and by the formulas

The induced homomorphisms of cochain groups

are, by definition, the boundary and degeneracy operators of the simplicial set . The simplices that are cocycles form a simplicial subset of , called the Eilenberg–MacLane simplicial set and denoted by . The coboundary operator on the groups defines a canonical simplicial mapping , denoted by . Since the concept of a one-dimensional cocycle also makes sense when is non-Abelian (see Non-Abelian cohomology), the simplicial set can be defined without the assumption that is Abelian. This simplicial set is isomorphic to the simplicial set (by assigning to every simplex the values at the vertices of a zero-dimensional cochain whose coboundary is ).

By assigning to every fibre of a simplicial set the free Abelian group generated by it, one obtains a simplicial Abelian group and thus a chain complex. This complex is denoted by and is called the chain complex of . The (co)homology groups of (with coefficients in a group ) are called the (co) homology groups and of . The (co)homology groups of a singular simplicial set are the (co)homology groups of the space . The (co)homology groups of and are isomorphic and are called the (co) homology groups of the simplicial complex . The (co)homology groups of the simplicial set are the (co) homology groups of .

A simplex of a simplicial set is called degenerate if there is a simplex and a degeneracy operator such that . The Eilenberg–Zil'ber lemma states that any simplex can be uniquely written in the form , where is a certain epimorphism , , and is a non-degenerate simplex. The smallest simplicial subset of a simplicial set containing all its non-degenerate simplices of dimension at most is denoted by or , and is called the -dimensional skeleton or -skeleton of .

The standard geometric simplices (cf. Standard simplex)

form a co-simplicial topological space with respect to the co-boundary operators and co-degeneracy operators , defined by the formulas

In the disjoint union , where all the are regarded as discrete sets, the formulas

generate an equivalence relation, the quotient space by which is a complex (a cellular space) whose cells are in one-to-one correspondence with the non-degenerate simplices of . This complex is denoted by or and is called the geometric realization in the sense of Milnor of . Any simplicial mapping induces a continuous mapping , given by

and the correspondence , defines a functor . This functor is left adjoint to the singular functor . The corresponding natural isomorphisms

are defined by the formulas

where

For any topological space the adjunction morphism is a weak homotopy equivalence (which proves that any topological space is weakly homotopy equivalent to a complex).

The construction of the geometric realization extends to the case of a simplicial topological space . One can also define the geometric realization in the sense of Giever–Hu by taking only the boundary operators into account (in this model there are cells for all the simplices of , not just for the non-degenerate ones). If every degeneracy operator is a closed cofibration (a condition which holds automatically in the case of a simplicial set), then the natural mapping is a homotopy equivalence.

The category admits products: given simplicial sets and , their product is the simplicial set for which

In particular, given any simplicial set , one can define its product with the simplicial segment . The projections and define a bijective mapping

which is a homeomorphism if the product is a complex (for example, if both simplicial sets and are countable or if one of the complexes , is locally finite). In particular, it follows that the geometric realization of any countable simplicial monoid (group, Abelian group) is a topological monoid (group, Abelian group).

Two simplicial mappings are called homotopic if there is a simplicial mapping (a homotopy) such that

for any simplex and for any composition (of length ) of degeneracy operators. This definition (modelled on the usual definition of homotopy of continuous mappings) is equivalent to the interpretation in simplicial sets of the general definition of homotopy of simplicial mappings between arbitrary simplicial objects (see Simplicial object in a category).

Given the notion of homotopy, it is possible to develop a homotopy theory for simplicial sets similar to that for polyhedra. It turns out that these two theories are completely parallel; this finds expression in the fact that the corresponding homotopy categories are equivalent (the equivalence being induced by the geometric realization functor). In particular, geometric realizations of homotopic simplicial mappings are homotopic and, for example, the geometric realization of is the Eilenberg–MacLane space . However, the actual construction of the homotopy theory for simplicial sets differs slightly in its details from the construction of the homotopy theory for topological spaces. The main difference is that the relation of homotopy for simplicial mappings is not, in general, an equivalence relation. This difficulty is overcome in the following way.

A simplicial mapping of the standard horn (see Standard simplex) into a simplicial set is called a horn in . Every horn is uniquely defined by an -tuple of -simplices , for which for all , . One says that a horn fills out if one can find an -dimensional simplex such that for every . The simplicial set is said to be full (or to satisfy the Kan condition) if all its horns fill out.

The singular simplicial set of an arbitrary topological space is always full, and so is every simplicial group; in particular, the Eilenberg–MacLane simplicial sets and are full. The importance of full simplicial sets lies in the fact that the relation of homotopy between simplicial mappings from an arbitrary simplicial set to a full simplicial set is an equivalence relation. Therefore, in the subcategory of full simplicial sets, the construction of a homotopy theory involves no major difficulties. Moreover, there is a functor (see [4]) assigning to every simplicial set a full simplicial set, , whose geometric realization is homotopy equivalent to the geometric realization of and which can therefore be used in place of in all questions of homotopy.

Two -simplices and of a simplicial set are called comparable if , . Two such simplices are said to be homotopic if there is an -dimensional simplex such that , and , . For full simplicial sets this is an equivalence relation, and two simplices are homotopic if and only if their characteristic simplicial mappings are homotopic .

A simplicial set is said to be pointed if it contains a distinguished zero-dimensional simplex (where the symbol is also used to denote all degenerations of this simplex as well as the simplicial set generated by it, which is usually referred to as the distinguished point of ). For a full pointed simplicial set , the set of homotopy classes of -dimensional simplices comparable with is a group when . This group is called the -dimensional homotopy group of ; this terminology is justified by the fact that and, in particular, and for . A simplicial set for which for all is called an -connected set; in particular, a -connected simplicial set is called connected, and a -connected simplicial set simply connected. For , the addition in is induced by the operation which assigns to two simplices and (comparable with ) the simplex , where is a simplex of dimension , filling the horn , , , . If is a simplicial monoid with unit , then the addition is also induced by the multiplication in this monoid (the product of two simplices comparable with is comparable with ).

Since any simplex comparable with is a cycle (of the chain complex defined by ), there is a natural Hurewicz homomorphism , which induces an isomorphism

when (Poincaré's theorem), and for it is an isomorphism if is -connected (Hurewicz' theorem). For full simplicial sets both variants of Whitehead's theorem hold, that is, a simplicial mapping of full simplicial sets is a homotopy equivalence if and only if it induces an isomorphism of homotopy groups; in the simply-connected case this condition is equivalent to the induced homomorphisms of the homology groups being isomorphisms.

In the case when is a simplicial group, the homotopy group is isomorphic to the homology group of the (not necessarily Abelian) chain complex for which

and the boundary operator is the restriction to of . If is Abelian, then is a subcomplex of , regarded as a chain complex, and also a chain deformation retract of it, and hence a direct summand of it. It turns out that the subcomplex generated by the degenerate simplices can be taken as the other direct summand. Therefore, the corresponding quotient complex of is chainwise equivalent to it. For example, it follows that the cohomology groups of an arbitrary simplicial set are isomorphic to the normalized cohomology groups (the normalization theorem), that is, the groups obtained from the cochains that vanish on all degenerate simplices. Furthermore, .

The functor induces an equivalence between the homotopy theory of simplicial Abelian groups and the homology theory of chain complexes. In particular, it follows that any connected simplicial Abelian group is homotopy equivalent to a product of Eilenberg–MacLane simplicial sets .

A full simplicial set is called minimal when comparable simplices are homotopic if and only if they coincide. The simplicial set is minimal. Every homotopy equivalence of minimal simplicial sets is an isomorphism. Every full simplicial set has a minimal subset. It is a deformation retract, and is thus uniquely defined up to isomorphism.

A simplicial mapping is called a Kan fibration if any horn in can be filled whenever can be, and for any filling of there is a filling of such that . Kan fibrations are the simplicial analogue of Serre fibrations (cf. Serre fibration), and they satisfy the following homotopy lifting theorem: If the simplicial mappings and satisfy the equation , then there is a simplicial mapping such that and . If the fibration is surjective, then is full if and only if is full. The fibre of is the (automatically full) simplicial set , where is the distinguished point of . For any Serre fibration the simplicial mapping is a Kan fibration, and for any Kan fibration the mapping is a Serre fibration (see [5]).

Let be a full pointed simplicial set and let . Write for when for all , that is, when

(see Standard simplex). This is an equivalence relation, and the quotient sets form a simplicial set (with respect to the induced boundary and degeneracy operators), called the -co-skeleton of . By definition, . For any , the simplicial set is full and when . Moreover, for any the natural surjective simplicial mapping

is a fibration inducing an isomorphism of homotopy groups in dimensions less than or equal to . In particular, the fibre of is homotopy equivalent to the Eilenberg–MacLane simplicial set . The sequence of fibrations

is called the Postnikov system of a full simplicial set . If is minimal, then this sequence is a resolution of (see Homotopy type).

The construction of the Postnikov system admits a direct generalization to an arbitrary fibration of a full simplicial set over a full simplicial set . Let be the simplicial set whose fibres are the quotient sets of the fibres by the relation , which holds if and only if and for all . By definition, . Note that . For the natural simplicial mapping

is a fibration inducing an isomorphism of homotopy groups in dimensions less than or equal to or greater than . In particular, the fibre of is homotopy equivalent to the Eilenberg–MacLane simplicial set . The fibre of is the simplicial set , where is the fibre of . The sequence of fibrations

is called the Moore–Postnikov system of .

It is convenient to define spectra in the language of simplicial sets. A simplicial spectrum is a sequence of pointed sets (whose elements are called simplices, and the distinguished simplex is denoted by ) defined for any integer , and equipped with mappings , (boundary operators), and , (degeneracy operators), which satisfy the relations (*) together with the following condition: For every simplex there is an integer such that when . To any spectrum and integer one can assign the simplicial set defined by

These simplicial sets are equipped with imbeddings , where is the suspension functor. From the sequence of simplicial sets and imbeddings , the simplicial spectrum can in turn be uniquely recovered. If every member of is full, then , where is the loop functor. The geometric realization functor gives an equivalence of the category of simplicial spectra and the category of topological spectra. Simplicial spectra can be defined for an arbitrary category. The category of Abelian group spectra is isomorphic to the category of (Abelian) chain complexes.

References

[1]	P. Gabriel, M. Zisman, "Calculus of fractions and homotopy theory" , Springer (1967)
[2]	J.P. May, "Simplicial objects in algebraic topology" , v. Nostrand (1967)
[3]	K. Lamotke, "Semisimpliziale algebraische Topologie" , Springer (1968)
[4]	D.M. Kan, "On c.s.s. complexes" Amer. J. Math. , 79 (1957) pp. 449–476
[5]	D.G. Quillen, "The geometric realization of a Kan fibration is a Serre fibration" Proc. Amer. Math. Soc. , 19 (1968) pp. 1499–1500
[6]	E.H. Brown, "Finite computability of Postnikov complexes" Ann. of Math. (2) , 65 (1957) pp. 1–20
[7]	D.M. Kan, "A combinatorial definition of homotopy groups" Ann. of Math. (2) , 67 (1958) pp. 282–312
[8]	D.M. Kan, "On homotopy theory and c.s.s. groups" Ann. of Math. (2) , 68 (1958) pp. 38–53
[9]	D.M. Kan, "An axiomatization of the homotopy groups" Illinois J. Math. , 2 (1958) pp. 548–566
[10]	D.M. Kan, "A relation between CW-complexes and free c.s.s. groups" Amer. J. Math. , 81 (1959) pp. 512–528

Comments

The "Kan condition" that every horn fills out is also called the extension condition.

A simplicial set or simplicial complex is called a Kan complex if it satisfies the Kan condition, [2], p. 2.

Let be the set of all monomorphisms of horns.

A class of monomorphisms in a category is called saturated if it satisfies the following conditions:

i) all isomorphisms belong to ;

ii) let

be a co-Cartesian square. Then if , also (stability of under pushouts; a co-Cartesian square is a Cartesian square in the dual category);

iii) given a commutative diagram

with , and , then (stability of under retractions);

iv) is stable under countable compositions and arbitrary direct sums.

Let be the saturated closure of , i.e. the intersection of all saturated classes containing . These are called the anodyne extensions in [1].

A morphism of is called a Kan fibration if for each anodyne extension and commutative square

there exists a morphism such that and . A simplicial set is a Kan complex if and only if the unique morphism , where is the standard zero simplex, is a Kan fibration.