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
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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:
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The boundary operators and degeneracy operators
of this simplicial set are defined by the formulas
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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
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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
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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
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The induced homomorphisms of cochain groups
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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)
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form a co-simplicial topological space with respect to the co-boundary operators and co-degeneracy operators
, defined by the formulas
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In the disjoint union , where all the
are regarded as discrete sets, the formulas
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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
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and the correspondence ,
defines a functor
. This functor is left adjoint to the singular functor
. The corresponding natural isomorphisms
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are defined by the formulas
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where
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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
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In particular, given any simplicial set , one can define its product with the simplicial segment
. The projections
and
define a bijective mapping
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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
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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
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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
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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
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(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
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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
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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
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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
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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
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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
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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
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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.
Simplicial set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simplicial_set&oldid=48711