Simplicial set

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

A simplicial object in the category of sets $ \mathop{\rm Ens} $( cf. Simplicial object in a category), that is, a system of sets ( $ n $- fibres) $ K _ {n} $, $ n \geq 0 $, connected by mappings $ d _ {i} : K _ {n} \rightarrow K _ {n - 1 } $, $ 0 \leq i \leq n $( boundary operators), and $ s _ {i} : K _ {n} \rightarrow K _ {n + 1 } $, $ 0 \leq i \leq n $( degeneracy operators), satisfying the conditions

$$ \tag{* } \left . \begin{array}{ll} d _ {i} d _ {j} = d _ {j - 1 } d _ {i} \ \textrm{ if } i < j, &d _ {i} s _ {j} = \ \left \{ \begin{array}{} s _ {j - 1 } \\ \end{array} d _ {i} \ \textrm{ if } i < j, \\ s _ {i} s _ {j} = s _ {j + 1 } s _ {i} \ \textrm{ if } i \leq j; & \mathop{\rm id} \ \textrm{ if } i = j \textrm{ or } i = j + 1, \\ d _ {i} s _ {j} = \ \left \{ \begin{array}{} s _ {j - 1 } \\ \end{array} d _ {i} \ \textrm{ if } i < j, \\ \end{array} \right \} $$

The elements of the fibre $ K _ {n} $ are called the $ n $- dimensional simplices of the simplicial set $ K $. If only the operators $ d _ {i} $ are given, satisfying the relations $ d _ {i} d _ {j} = d _ {j - 1 } d _ {i} $, $ i < j $, then the system $ \{ K _ {n} , d _ {n} \} $ is called a semi-simplicial set.

A simplicial mapping $ f: K \rightarrow K ^ \prime $ between two simplicial sets $ K $ and $ K ^ \prime $ is a morphism of functors, i.e. a sequence of mappings $ f _ {n} : K _ {n} \rightarrow K _ {n} ^ \prime $, $ n \geq 0 $, satisfying the relations

$$ d _ {i} f _ {n + 1 } = f _ {n} d _ {i} ,\ \ 0 \leq i \leq n + 1; \ \ s _ {i} f _ {n} = f _ {n + 1 } s _ {i} ,\ \ 0 \leq i \leq n. $$

Simplicial sets and their simplicial mappings form a category, $ \Delta ^ {0} \mathop{\rm Ens} $. If all the $ f _ {n} $ are imbeddings, then $ K $ is called a simplicial subset of $ K ^ \prime $. In this case, the boundary and degeneracy operators in $ K $ are the restrictions to $ K $ of the corresponding operators in $ K ^ \prime $.

Given any topological space $ X $, one can define a simplicial set $ S ( X) $, called the singular simplicial set of the space $ X $. Its simplices are the singular simplices of $ X $( see Singular homology), i.e. continuous mappings $ \sigma : \Delta ^ {n} \rightarrow X $, where $ \Delta ^ {n} $ is the $ n $- dimensional geometric standard simplex:

$$ \Delta ^ {n} = \ \left \{ { ( t _ {0} \dots t _ {n} ) } : { 0 \leq t _ {i} \leq 1,\ \sum _ {i = 0 } ^ { n } t _ {i} = 1 } \right \} \subset \ \mathbf R ^ {n + 1 } . $$

The boundary operators $ d _ {i} $ and degeneracy operators $ s _ {i} $ of this simplicial set are defined by the formulas

$$ ( d _ {i} \sigma ) ( t _ {0} \dots t _ {n - 1 } ) = \ \sigma ( t _ {0} \dots t _ {i - 1 } ,\ 0, t _ {i} \dots t _ {n - 1 } ), $$

$$ ( s _ {i} \sigma ) ( t _ {0} \dots t _ {n + 1 } ) = $$

$$ = \ \sigma ( t _ {0} \dots t _ {i - 1 } , t _ {i} + t _ {i + 1 } , t _ {i+ 2 } \dots t _ {n + 1 } ). $$

The correspondence $ X \mapsto S ( X) $ is a functor (called the singular functor) from the category of topological spaces $ \mathop{\rm Top} $ into the category of simplicial sets $ \Delta ^ {0} \mathop{\rm Ens} $.

An arbitrary simplicial complex $ K $ determines a simplicial set $ O ( K) $. Its $ n $- dimensional simplices are the $ ( n + 1) $- tuples $ ( x _ {0} \dots x _ {n} ) $ of vertices of $ K $ with the property that there is a simplex $ s $ in $ K $ such that $ x _ {i} \in s $ for $ i = 0 \dots n $. The operators $ d _ {i} $ and $ s _ {i} $ for this simplicial set are given by

$$ d _ {i} ( x _ {0} \dots x _ {n} ) = \ ( x _ {0} \dots \widehat{x} _ {i} \dots x _ {n} ), $$

$$ s _ {i} ( x _ {0} \dots x _ {n} ) = ( x _ {0} \dots x _ {i} , x _ {i} , x _ {i + 1 } \dots x _ {n} ), $$

where $ \widehat{ {}} $ means that the symbol below it is omitted. If $ K $ is ordered, then the simplices $ ( x _ {0} \dots x _ {n} ) $ for which $ x _ {0} \leq \dots \leq x _ {n} $ form a simplicial subset $ O ^ {+} ( K) $ of $ O ( K) $. The correspondence $ K \mapsto O ( K) $( $ K \mapsto O ^ {+} ( K) $) is a functor from the category of simplicial complexes (ordered simplicial complexes) into the category $ \Delta ^ {0} \mathop{\rm Ens} $.

For an arbitrary group $ \pi $ one can define a simplicial set $ K ( \pi ) $. Its $ n $- simplices are equivalence classes of $ ( n + 1) $- tuples $ ( x _ {0} : \dots : x _ {n} ) $, $ x _ {i} \in \pi $( where $ ( x _ {0} : \dots : x _ {n} ) \sim ( x _ {0} ^ \prime : \dots : x _ {n} ^ \prime ) $ if there is an element $ y \in \pi $ such that $ x _ {i} ^ \prime = yx _ {i} $ for all $ i = 0 \dots n $). The operators $ d _ {i} $ and $ s _ {i} $ of $ K ( \pi ) $ are given by

$$ d _ {i} ( x _ {0} : \dots : x _ {n} ) = \ ( x _ {0} : \dots : x _ {i - 1 } : \ x _ {i + 1 } : \dots : x _ {n} ), $$

$$ s _ {i} ( x _ {0} : \dots : x _ {n} ) = ( x _ {0} : \dots : x _ {i - 1 } : x _ {i} : x _ {i} : x _ {i + 1 } : \dots : x _ {n} ). $$

The simplicial set $ K ( \pi ) $ is actually a simplicial group.

Given an arbitrary Abelian group $ \pi $ and any integer $ n \geq 1 $, one can define a simplicial set (in fact, a simplicial Abelian group) $ E ( \pi , n) $. Its $ q $- dimensional simplices are the $ n $- dimensional cochains of the $ q $- dimensional geometric standard simplex $ \Delta ^ {q} $ with coefficients in $ \pi $( that is, $ E ( \pi , n) _ {q} = C ^ {n} ( \Delta ^ {q} ; \pi ) $). Denoting the vertices of $ \Delta ^ {q} $ by $ e _ {j} ^ {q} $, $ j = 0 \dots q $, one defines the simplicial mappings $ \delta _ {i} : \Delta ^ {q - 1 } \rightarrow \Delta ^ {q} $ and $ \sigma _ {i} : \Delta ^ {q} \rightarrow \Delta ^ {q - 1 } $ by the formulas

$$ \delta _ {i} ( e _ {j} ^ {q - 1 } ) = \ \left \{ \begin{array}{ll} e _ {j} ^ {q} &\textrm{ if } j < i, \\ e _ {j + 1 } ^ {q} &\textrm{ if } j \geq i; \\ \end{array} \right .$$

$$ \sigma _ {i} ( e _ {j} ^ {q} ) = \left \{ \begin{array}{ll} e _ {j} ^ {q - 1 } &\textrm{ if } j \leq i, \\ e _ {j - 1 } ^ {q - 1 } &\textrm{ if } j > i. \\ \end{array} \right .$$

The induced homomorphisms of cochain groups

$$ d _ {i} = \ \delta _ {i} ^ {*} : \ C ^ {n} ( \Delta ^ {q} ; \pi ) \rightarrow \ C ^ {n} ( \Delta _ {q - 1 } ; \pi ), $$

$$ s _ {i} = \sigma _ {i} ^ {*} : C ^ {n} ( \Delta ^ {q - 1 } , \pi ) \rightarrow C ^ {n} ( \Delta ^ {q} ; \pi ) $$

are, by definition, the boundary and degeneracy operators of the simplicial set $ E ( \pi , n) $. The simplices that are cocycles form a simplicial subset of $ E ( \pi , n) $, called the Eilenberg–MacLane simplicial set and denoted by $ K ( \pi , n) $. The coboundary operator on the groups $ C ^ {*} ( \Delta ^ {q} ; \pi ) $ defines a canonical simplicial mapping $ E ( \pi , n) \rightarrow K ( \pi , n + 1) $, denoted by $ \delta $. Since the concept of a one-dimensional cocycle also makes sense when $ \pi $ is non-Abelian (see Non-Abelian cohomology), the simplicial set $ K ( \pi , 1) $ can be defined without the assumption that $ \pi $ is Abelian. This simplicial set is isomorphic to the simplicial set $ K ( \pi ) $( by assigning to every simplex $ z \in K ( \pi , 1) _ {q} = Z ^ {1} ( \Delta ^ {q} , \pi ) $ the values at the vertices $ e _ {j} ^ {q} $ of a zero-dimensional cochain whose coboundary is $ z $).

By assigning to every fibre $ K _ {n} $ of a simplicial set $ K $ the free Abelian group generated by it, one obtains a simplicial Abelian group and thus a chain complex. This complex is denoted by $ C ( K) $ and is called the chain complex of $ K $. The (co)homology groups of $ C ( K) $( with coefficients in a group $ G $) are called the (co) homology groups $ H ( K; G) $ and $ H ^ {*} ( K; G) $ of $ K $. The (co)homology groups of a singular simplicial set $ S ( X) $ are the (co)homology groups of the space $ X $. The (co)homology groups of $ O ( K) $ and $ O ^ {+} ( K) $ are isomorphic and are called the (co) homology groups of the simplicial complex $ K $. The (co)homology groups of the simplicial set $ K ( \pi ) $ are the (co) homology groups of $ \pi $.

A simplex $ x \in K _ {n} $ of a simplicial set $ K $ is called degenerate if there is a simplex $ y \in K _ {n - 1 } $ and a degeneracy operator $ s _ {i} $ such that $ x = s _ {i} y $. The Eilenberg–Zil'ber lemma states that any simplex $ x \in K _ {n} $ can be uniquely written in the form $ x = K ( s) y $, where $ s $ is a certain epimorphism $ s _ {i} : [ n] \rightarrow [ m] $, $ m \leq n $, and $ y \in K _ {m} $ is a non-degenerate simplex. The smallest simplicial subset of a simplicial set $ K $ containing all its non-degenerate simplices of dimension at most $ n $ is denoted by $ K ^ {n} $ or $ \mathop{\rm Sk} ^ {n} K $, and is called the $ n $- dimensional skeleton or $ n $- skeleton of $ K $.

The standard geometric simplices (cf. Standard simplex)

$$ \Delta ^ {n} = \ \left \{ { ( t _ {0} \dots t _ {n} ) } : { 0 \leq t _ {i} \leq 1,\ \sum _ {i = 0 } ^ { n } t _ {i} = 1 } \right \} \subset \ \mathbf R ^ {n + 1 } $$

form a co-simplicial topological space with respect to the co-boundary operators $ \delta _ {i} $ and co-degeneracy operators $ \sigma _ {i} $, defined by the formulas

$$ \delta _ {i} ( t _ {0} \dots t _ {n - 1 } ) = \ ( t _ {0} \dots t _ {i - 1 } , 0,\ t _ {i} \dots t _ {n - 1 } ), $$

$$ \sigma _ {i} ( t _ {0} \dots t _ {n + 1 } ) = ( t _ {0} \dots t _ {i - 1 } , t _ {i} + t _ {i + 1 } , t _ {i + 2 } \dots t _ {n + 1 } ). $$

In the disjoint union $ \cup _ {n = 0 } ^ \infty K _ {n} \times \Delta ^ {n} $, where all the $ K _ {n} $ are regarded as discrete sets, the formulas

$$ ( d _ {i} x, u) \sim \ ( x, \delta _ {i} u),\ \ x \in K _ {n} ,\ \ u \in \Delta ^ {n - 1 } ; $$

$$ ( s _ {i} x, u) \sim ( x, \sigma _ {i} u),\ x \in K _ {n} ,\ u \in \Delta ^ {n + 1 } , $$

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 $ K $. This complex is denoted by $ | K | $ or $ RK $ and is called the geometric realization in the sense of Milnor of $ K $. Any simplicial mapping $ f: K \rightarrow L $ induces a continuous mapping $ Rf: RK \rightarrow RL $, given by

$$ Rf [ x, u] = \ [ f ( x), u] , $$

and the correspondence $ K \mapsto RK $, $ f \mapsto Rf $ defines a functor $ R: \Delta ^ {0} \mathop{\rm Ens} \rightarrow \mathop{\rm Top} $. This functor is left adjoint to the singular functor $ S: \mathop{\rm Top} \rightarrow \Delta ^ {0} \mathop{\rm Ens} $. The corresponding natural isomorphisms

$$ \phi : \Delta ^ {0} \mathop{\rm Ens} ( K, S ( X)) \rightarrow \ \mathop{\rm Top} ( RK, X), $$

$$ \psi : \mathop{\rm Top} ( RK, X) \rightarrow \Delta ^ {0} \mathop{\rm Ens} ( K, S ( X)) $$

are defined by the formulas

$$ \phi ( f ) [ x, u] = f ( x) ( u), $$

$$ ( \psi ( g) ( x)) ( u) = g [ x, u], $$

where

$$ x \in K _ {n} ,\ \ u \in \Delta ^ {n} ,\ \ f \in \Delta ^ {0} \mathop{\rm Ens} ( K, S ( X)),\ \ g \in \mathop{\rm Top} ( RK, X). $$

For any topological space $ X $ the adjunction morphism $ \Phi ( X): RS ( X) \rightarrow X $ is a weak homotopy equivalence (which proves that any topological space is weakly homotopy equivalent to a complex).

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

The category $ \Delta ^ {0} \mathop{\rm Ens} $ admits products: given simplicial sets $ K = \{ K _ {n} , d _ {i} ^ {K} , s _ {i} ^ {K} \} $ and $ L = \{ L _ {n} , d _ {i} ^ {L} , s _ {i} ^ {L} \} $, their product is the simplicial set $ K \times L $ for which

$$ ( K \times L) _ {n} = K _ {n} \times L _ {n} , $$

$$ d _ {i} ^ {K \times L } = d _ {i} ^ {K} \times d _ {i} ^ {L} , $$

$$ s _ {i} ^ {K \times L } = s _ {i} ^ {K} \times s _ {i} ^ {L} . $$

In particular, given any simplicial set $ K $, one can define its product with the simplicial segment $ \Delta ^ {1} $. The projections $ \pi _ {1} : K \times L \rightarrow K $ and $ \pi _ {2} : K \times L \rightarrow L $ define a bijective mapping

$$ R \pi _ {1} \times R \pi _ {2} : R ( K \times L) \rightarrow RK \times RL, $$

which is a homeomorphism if the product $ RK \times RL $ is a complex (for example, if both simplicial sets $ K $ and $ L $ are countable or if one of the complexes $ RK $, $ RL $ 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 $ f, g: K \rightarrow L $ are called homotopic if there is a simplicial mapping (a homotopy) $ F: K \times \Delta ^ {1} \rightarrow L $ such that

$$ F ( x, sd _ {0} t _ {1} ) = f ( x), $$

$$ F ( x, sd _ {1} t _ {1} ) = g ( x) $$

for any simplex $ x \in K _ {n} $ and for any composition $ s $( of length $ n $) 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 $ K ( \pi , n) $ is the Eilenberg–MacLane space $ K ( \pi , n) $. 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 $ \Lambda _ {k} ^ {n} \rightarrow K $ of the standard horn (see Standard simplex) into a simplicial set $ K $ is called a horn in $ K $. Every horn is uniquely defined by an $ ( n + 1) $- tuple of $ n $- simplices $ x _ {0} \dots x _ {k - 1 } , x _ {k + 1 } \dots x _ {n + 1 } $, for which $ d _ {i} x _ {j} = d _ {j - 1 } x _ {i} $ for all $ i < j $, $ i \neq k $. One says that a horn fills out if one can find an $ ( n + 1) $- dimensional simplex $ x $ such that $ d _ {i} x = x _ {i} $ for every $ i \neq k $. The simplicial set $ K $ is said to be full (or to satisfy the Kan condition) if all its horns fill out.

The singular simplicial set $ S ( X) $ of an arbitrary topological space $ X $ is always full, and so is every simplicial group; in particular, the Eilenberg–MacLane simplicial sets $ K ( \pi ) $ and $ K ( \pi , n) $ 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]) $ \mathop{\rm Ex} ^ \infty : \Delta ^ {0} \mathop{\rm Ens} \rightarrow \Delta ^ {0} \mathop{\rm Ens} $ assigning to every simplicial set $ K $ a full simplicial set, $ \mathop{\rm Ex} ^ \infty K $, whose geometric realization is homotopy equivalent to the geometric realization of $ K $ and which can therefore be used in place of $ K $ in all questions of homotopy.

Two $ n $- simplices $ x $ and $ x ^ \prime $ of a simplicial set $ K $ are called comparable if $ d _ {i} x = d _ {i} x ^ \prime $, $ 0 \leq i \leq n $. Two such simplices are said to be homotopic if there is an $ ( n + 1) $- dimensional simplex $ y $ such that $ d _ {n} y = x $, $ d _ {n + 1 } y = x ^ \prime $ and $ d _ {i} y = s _ {n - 1 } d _ {i} x = s _ {n - 1 } d _ {i} x ^ \prime $, $ 0 \leq i \leq n $. For full simplicial sets this is an equivalence relation, and two simplices are homotopic if and only if their characteristic simplicial mappings are homotopic $ \mathop{\rm rel} \mathop{\rm Sk} ^ {n - 1 } \Delta ^ {n} $.

A simplicial set $ K $ is said to be pointed if it contains a distinguished zero-dimensional simplex $ \theta $( where the symbol $ \theta $ 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 $ K $). For a full pointed simplicial set $ K $, the set $ \pi _ {n} ( K) $ of homotopy classes of $ n $- dimensional simplices comparable with $ \theta $ is a group when $ n \geq 1 $. This group is called the $ n $- dimensional homotopy group of $ K $; this terminology is justified by the fact that $ \pi _ {n} ( K) = \pi _ {n} (| K |) $ and, in particular, $ \pi _ {n} ( K ( \pi , n)) = \pi $ and $ \pi _ {i} ( K ( \pi , n)) = 0 $ for $ i \neq n $. A simplicial set $ K $ for which $ \pi _ {i} ( K) = 0 $ for all $ i \leq n $ is called an $ n $- connected set; in particular, a $ 0 $- connected simplicial set is called connected, and a $ 1 $- connected simplicial set simply connected. For $ n \geq 1 $, the addition in $ \pi _ {n} ( K) $ is induced by the operation which assigns to two simplices $ x $ and $ y $( comparable with $ \theta $) the simplex $ d _ {n} z $, where $ z $ is a simplex of dimension $ n + 1 $, filling the horn $ x _ {i} = \theta $, $ i \leq n - 2 $, $ x _ {n - 1 } = x $, $ x _ {n + 1 } = y $. If $ K $ is a simplicial monoid with unit $ \theta $, then the addition is also induced by the multiplication in this monoid (the product of two simplices comparable with $ \theta $ is comparable with $ \theta $).

Since any simplex $ x $ comparable with $ \theta $ is a cycle (of the chain complex $ C ( K) $ defined by $ K $), there is a natural Hurewicz homomorphism $ h: \pi _ {n} ( K) \rightarrow H _ {n} ( K) $, which induces an isomorphism

$$ \pi _ {1} ( K)/[ \pi _ {1} ( K), \pi _ {1} ( K)] \rightarrow H _ {1} ( K) $$

when $ n = 1 $( Poincaré's theorem), and for $ n > 1 $ it is an isomorphism if $ K $ is $ ( n - 1) $- connected (Hurewicz' theorem). For full simplicial sets both variants of Whitehead's theorem hold, that is, a simplicial mapping $ f: K \rightarrow L $ 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 $ K $ is a simplicial group, the homotopy group $ \pi _ {n} ( K) $ is isomorphic to the homology group $ H _ {n} ( \overline{K}\; ) $ of the (not necessarily Abelian) chain complex $ \overline{K}\; $ for which

$$ \overline{K}\; _ {n} = \ K _ {n} \cap \mathop{\rm Ker} d _ {0} \cap \dots \cap \mathop{\rm Ker} d _ {n - 1 } , $$

and the boundary operator is the restriction to $ \overline{K}\; _ {n} $ of $ (- 1) ^ {n} d _ {n} $. If $ K $ is Abelian, then $ \overline{K}\; $ is a subcomplex of $ K $, 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 $ K $ is chainwise equivalent to it. For example, it follows that the cohomology groups of an arbitrary simplicial set $ K $ 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, $ \pi _ {n} ( C ( K)) = H _ {n} ( K) $.

The functor $ K \mapsto \overline{K}\; $ 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 $ K $ is homotopy equivalent to a product of Eilenberg–MacLane simplicial sets $ K ( \pi _ {n} ( K), n) $.

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

A simplicial mapping $ p: E \rightarrow B $ is called a Kan fibration if any horn $ f: \Lambda _ {k} ^ {n} \rightarrow E $ in $ E $ can be filled whenever $ p \circ f: \Lambda _ {k} ^ {n} \rightarrow B $ can be, and for any filling $ g: \Delta ^ {n + 1 } \rightarrow B $ of $ p \circ f $ there is a filling $ \widetilde{f} : \Delta ^ {n + 1 } \rightarrow E $ of $ f $ such that $ p \circ \widetilde{f} = g $. Kan fibrations are the simplicial analogue of Serre fibrations (cf. Serre fibration), and they satisfy the following homotopy lifting theorem: If the simplicial mappings $ \widetilde{f} : K \rightarrow E $ and $ \Phi : K \times \Delta ^ {1} \rightarrow B $ satisfy the equation $ \Phi \circ ( \mathop{\rm id} \times \delta _ {1} ) = p \circ \widetilde{f} $, then there is a simplicial mapping $ \widetilde \Phi : K \times \Delta ^ {1} \rightarrow E $ such that $ \widetilde \Phi \circ ( \mathop{\rm id} \times \delta _ {1} ) = \widetilde{f} $ and $ p \circ \widetilde \Phi = \Phi $. If the fibration $ p $ is surjective, then $ E $ is full if and only if $ B $ is full. The fibre of $ p: E \rightarrow B $ is the (automatically full) simplicial set $ F = p ^ {-} 1 ( \theta ) $, where $ \theta $ is the distinguished point of $ B $. For any Serre fibration $ p: E \rightarrow B $ the simplicial mapping $ S ( p): S ( E) \rightarrow S ( B) $ is a Kan fibration, and for any Kan fibration $ p: E \rightarrow B $ the mapping $ R p : RE \rightarrow RB $ is a Serre fibration (see [5]).

Let $ K $ be a full pointed simplicial set and let $ n \geq 0 $. Write $ x \sim ^ {n} y $ for $ x, y \in K _ {q} $ when $ d _ {i} x = d _ {i} y $ for all $ i \leq n $, that is, when

$$ \left . \chi _ {x} \right | _ { \mathop{\rm Sk} ^ {n} \Delta ^ {q} } = \left . \chi _ {y} \right | _ { \mathop{\rm Sk} ^ {n} \Delta ^ {q} } $$

(see Standard simplex). This is an equivalence relation, and the quotient sets $ ( \mathop{\rm Cosk} ^ {n} K) _ {q} = K _ {q} / \sim ^ {n} $ form a simplicial set $ \mathop{\rm Cosk} ^ {n} K $( with respect to the induced boundary and degeneracy operators), called the $ n $- co-skeleton of $ K $. By definition, $ \mathop{\rm Cosk} ^ \infty K = K $. For any $ n \geq 0 $, the simplicial set $ \mathop{\rm Cosk} ^ {n} K $ is full and $ \pi _ {q} ( \mathop{\rm Cosk} ^ {n} K) = 0 $ when $ q > n $. Moreover, for any $ m \leq n $ the natural surjective simplicial mapping

$$ p _ {m} ^ {n} : \ \mathop{\rm Cosk} ^ {n} K \rightarrow \ \mathop{\rm Cosk} ^ {m} K $$

is a fibration inducing an isomorphism of homotopy groups in dimensions less than or equal to $ m $. In particular, the fibre of $ p _ {n - 1 } ^ {n} $ is homotopy equivalent to the Eilenberg–MacLane simplicial set $ K ( \pi _ {n} ( K), n) $. The sequence of fibrations

$$ K \rightarrow \dots \rightarrow \ \mathop{\rm Cosk} ^ {n + 1 } K \rightarrow \ \mathop{\rm Cosk} ^ {n} K \rightarrow \ \mathop{\rm Cosk} ^ {n - 1 } K \rightarrow \dots $$

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

The construction of the Postnikov system admits a direct generalization to an arbitrary fibration $ p: E \rightarrow B $ of a full simplicial set $ E $ over a full simplicial set $ B $. Let $ \mathop{\rm Cosk} ^ {\ } p $ be the simplicial set whose fibres $ ( \mathop{\rm Cosk} ^ {n} p) _ {q} $ are the quotient sets of the fibres $ E _ {q} $ by the relation $ x \sim ^ {n} y $, which holds if and only if $ p ( x) = p ( y) $ and $ d _ {i} x = d _ {i} y $ for all $ i \leq n $. By definition, $ \mathop{\rm Cosk} ^ \infty p = E $. Note that $ \mathop{\rm Cosk} ^ {0} p = B $. For $ m \leq n \leq \infty $ the natural simplicial mapping

$$ p _ {m} ^ {n} : \mathop{\rm Cosk} ^ {n} p \rightarrow \mathop{\rm Cosk} ^ {m} p $$

is a fibration inducing an isomorphism of homotopy groups in dimensions less than or equal to $ m $ or greater than $ n + 1 $. In particular, the fibre of $ p _ {n - 1 } ^ {n} $ is homotopy equivalent to the Eilenberg–MacLane simplicial set $ K ( \pi _ {n} ( F ), n) $. The fibre of $ p _ {0} ^ {n} : \mathop{\rm Cosk} ^ {n} p \rightarrow B $ is the simplicial set $ \mathop{\rm Cosk} ^ {n} F $, where $ F $ is the fibre of $ p: E \rightarrow B $. The sequence of fibrations

$$ E \rightarrow \dots \rightarrow \ \mathop{\rm Cosk} ^ {n + 1 } p \rightarrow \ \mathop{\rm Cosk} ^ {n} p \rightarrow \ \mathop{\rm Cosk} ^ {n - 1 } p \rightarrow \dots \rightarrow B $$

is called the Moore–Postnikov system of $ p: E \rightarrow B $.

It is convenient to define spectra in the language of simplicial sets. A simplicial spectrum is a sequence $ \{ X _ {(} q) \} $ of pointed sets (whose elements are called simplices, and the distinguished simplex is denoted by $ \theta $) defined for any integer $ q $, and equipped with mappings $ d _ {i} : X _ {(} q) \rightarrow X _ {( q - 1) } $, $ i \geq 0 $( boundary operators), and $ s _ {i} : X _ {(} q) \rightarrow X _ {( q - 1) } $, $ i \geq 0 $( degeneracy operators), which satisfy the relations (*) together with the following condition: For every simplex $ x \in X $ there is an integer $ n $ such that $ d _ {i} x = \theta $ when $ i > n $. To any spectrum $ X $ and integer $ n $ one can assign the simplicial set $ X _ {n} $ defined by

$$ ( X _ {n} ) _ {q} = \ \{ {x \in X _ {( q - n) } } : { d _ {i} x = \theta \textrm{ for } i > q,\ d _ {0} \dots d _ {q} x = 0 } \} . $$

These simplicial sets $ X _ {n} $ are equipped with imbeddings $ SX _ {n} \subset X _ {n + 1 } $, where $ S $ is the suspension functor. From the sequence of simplicial sets $ X _ {n} $ and imbeddings $ SX _ {n} \subset X _ {n + 1 } $, the simplicial spectrum $ X $ can in turn be uniquely recovered. If every member of $ X $ is full, then $ X _ {n} = \Omega X _ {n + 1 } $, where $ \Omega $ 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 $ K $ is called a Kan complex if it satisfies the Kan condition, [2], p. 2.

Let $ B $ be the set of all monomorphisms $ \Lambda ^ {k} [ n] \rightarrow \Delta [ n] $ of horns.

A class of monomorphisms $ {\mathcal M} $ in a category is called saturated if it satisfies the following conditions:

i) all isomorphisms belong to $ {\mathcal M} $;

ii) let

$$ \begin{array}{rcl} X & \rightarrow & Y \\ {size - 3 {m } } \downarrow &{} &\downarrow {size - 3 {m ^ \prime } } \\ {X ^ \prime } & \rightarrow &{Y ^ \prime } \\ \end{array} $$

be a co-Cartesian square. Then if $ m \in {\mathcal M} $, also $ m ^ \prime \in {\mathcal M} $( stability of $ {\mathcal M} $ under pushouts; a co-Cartesian square is a Cartesian square in the dual category);

iii) given a commutative diagram

$$ \begin{array}{rcccl} X & \rightarrow ^ { u } & Y & \rightarrow ^ { v } & X \\ {size - 3 {m ^ \prime } } \downarrow &{} &\downarrow {size - 3 {m } } &{} &\downarrow {size - 3 {m ^ \prime } } \\ {X ^ \prime } & \mathop \rightarrow \limits _ { {u ^ \prime }} &{Y ^ \prime } & \mathop \rightarrow \limits _ { {v ^ \prime }} &{X ^ \prime } \\ \end{array} $$

with $ v \circ u = \mathop{\rm id} $, $ v ^ \prime \circ u ^ \prime = \mathop{\rm id} $ and $ m \in {\mathcal M} $, then $ m ^ \prime \in {\mathcal M} $( stability of $ {\mathcal M} $ under retractions);

iv) $ {\mathcal M} $ is stable under countable compositions and arbitrary direct sums.

Let $ \widehat{B} $ be the saturated closure of $ B $, i.e. the intersection of all saturated classes containing $ B $. These are called the anodyne extensions in [1].

A morphism $ p : E \rightarrow X $ of $ \Delta ^ {0} \mathop{\rm Ens} $ is called a Kan fibration if for each anodyne extension $ i: K \rightarrow L $ and commutative square

$$ \begin{array}{rcl} K &\rightarrow ^ { u } & E \\ {size - 3 {i } } \downarrow &{} &\downarrow {size - 3 {p } } \\ L & \rightarrow _ { v } & X \\ \end{array} $$

there exists a morphism $ w : L \rightarrow E $ such that $ w \circ i = u $ and $ p \circ w = v $. A simplicial set $ X $ is a Kan complex if and only if the unique morphism $ X \rightarrow \Delta [ 0] $, where $ \Delta [ 0] $ is the standard zero simplex, is a Kan fibration.