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Basic definitions
Let ∆ be the category whose objects are finite, non-empty,
totally-ordered sets [n] = {0, 1, . . . , n} and morphisms are
order-preserving functions.
Definition
A simplicial object X in a category C is a functor ∆op → C.
Defining a simplicial object X amounts to taking a collection of
objects (Xn )n≥0 in C together with morphisms di : Xn → Xn−1
and si : Xn → Xn+1 for 0 ≤ i ≤ n subject to the relations
di dj = dj−1 di ,
i<j
si sj = sj+1 si , i ≤ j


i = j, j + 1
1,
di sj = sj−1 di , i < j


sj di−1 , i > j + 1
Basic definitions
Let ∆ be the category whose objects are finite, non-empty,
totally-ordered sets [n] = {0, 1, . . . , n} and morphisms are
order-preserving functions.
Definition
A simplicial object X in a category C is a functor ∆op → C.
A morphism of simplicial objects is a natural transformation of
the corresponding functors.
Defining a simplicial object X amounts to taking a collection of
objects (Xn )n≥0 in C together with morphisms di : Xn → Xn−1
and si : Xn → Xn+1 for 0 ≤ i ≤ n subject to the relations
di dj = dj−1 di ,
i<j
si sj = sj+1 si , i ≤ j


i = j, j + 1
1,
di sj = sj−1 di , i < j


Basic definitions
Let ∆ be the category whose objects are finite, non-empty,
totally-ordered sets [n] = {0, 1, . . . , n} and morphisms are
order-preserving functions.
Definition
A simplicial object X in a category C is a functor ∆op → C.
Defining a simplicial object X amounts to taking a collection of
objects (Xn )n≥0 in C together with morphisms di : Xn → Xn−1
and si : Xn → Xn+1 for 0 ≤ i ≤ n subject to the relations
di dj = dj−1 di ,
i<j
si sj = sj+1 si , i ≤ j


i = j, j + 1
1,
di sj = sj−1 di , i < j


sj di−1 , i > j + 1
Basic definitions
Definition
A simplicial object X in a category C is a functor ∆op → C.
Defining a simplicial object X amounts to taking a collection of
objects (Xn )n≥0 in C together with morphisms di : Xn → Xn−1
and si : Xn → Xn+1 for 0 ≤ i ≤ n subject to the relations
di dj = dj−1 di ,
i<j
si sj = sj+1 si , i ≤ j


i = j, j + 1
1,
di sj = sj−1 di , i < j


sj di−1 , i > j + 1
X0 d j
s0
d0
d1
*
X1X d j
s0
s1
d0
d1
d2
*%
X2Z _ kf
.
'
X3
...
Examples
1) Trivial simplicial object.
Let d ∈ C. The corresponding constant functor ∆op → C is a
simplicial object.
2) Combinatorial simplicial complex.
Let K be a collection of nonempty finite subsets of some ordered
set V (“vertex set“) such that if σ ∈ K and τ ⊂ σ, then τ ∈ K.
We define a functor SS(K) : ∆op → Sets by letting
n
o
SS(K)[n] = (v0 , v1 , . . . , vn )|{v0 , v1 , . . . , vn } ∈ K
α
and for a morphism [m] → [n] in ∆,
(SS(K)α)(v0 , v1 , . . . vn ) = (vα (0), vα (1), . . . vα (n))
Examples
1) Trivial simplicial object.
Let d ∈ C. The corresponding constant functor ∆op → C is a
simplicial object.
2) Combinatorial simplicial complex.
Let K be a collection of nonempty finite subsets of some ordered
set V (“vertex set“) such that if σ ∈ K and τ ⊂ σ, then τ ∈ K.
We define a functor SS(K) : ∆op → Sets by letting
n
o
SS(K)[n] = (v0 , v1 , . . . , vn )|{v0 , v1 , . . . , vn } ∈ K
α
and for a morphism [m] → [n] in ∆,
(SS(K)α)(v0 , v1 , . . . vn ) = (vα (0), vα (1), . . . vα (n))
Examples
1) Trivial simplicial object.
Let d ∈ C. The corresponding constant functor ∆op → C is a
simplicial object.
2) Combinatorial simplicial complex.
Let K be a collection of nonempty finite subsets of some ordered
set V (“vertex set“) such that if σ ∈ K and τ ⊂ σ, then τ ∈ K.
We define a functor SS(K) : ∆op → Sets by letting
n
o
SS(K)[n] = (v0 , v1 , . . . , vn )|{v0 , v1 , . . . , vn } ∈ K
α
and for a morphism [m] → [n] in ∆,
(SS(K)α)(v0 , v1 , . . . vn ) = (vα (0), vα (1), . . . vα (n))
Examples
3) Total singular complex of a topological space.
Let Y be a topological space. For n ≥ 0 we denote by ∆n the
standard affine simplex
{(x0 , x1 , . . . , xn ) ∈ Rn+1 |x0 + x1 + · · · + xn = 1, xi ≥ 0}.
and by din : ∆n → ∆n+1 , sin : ∆n → ∆n−1 the maps
din (x0 , x1 , . . . , xn ) = (x0 , x1 , . . . , xi−1 , 0, xi , . . . xn )
sin (x0 , x1 , . . . , xn ) = (x0 , x1 , . . . , xi−1 + xi , . . . xn ), 0 ≤ i ≤ n
To define the simplicial set SY , we take
di : Top(∆n , Y ) → Top(∆n−1 , Y ),
φ 7→ din ◦ φ
si : Top(∆n , Y ) → Top(∆n+1 , Y ),
φ 7→ φ ◦ sin
Examples
3) Total singular complex of a topological space.
Let Y be a topological space. For n ≥ 0 we denote by ∆n the
standard affine simplex
{(x0 , x1 , . . . , xn ) ∈ Rn+1 |x0 + x1 + · · · + xn = 1, xi ≥ 0}.
and by din : ∆n → ∆n+1 , sin : ∆n → ∆n−1 the maps
din (x0 , x1 , . . . , xn ) = (x0 , x1 , . . . , xi−1 , 0, xi , . . . xn )
sin (x0 , x1 , . . . , xn ) = (x0 , x1 , . . . , xi−1 + xi , . . . xn ), 0 ≤ i ≤ n
To define the simplicial set SY , we take
di : Top(∆n , Y ) → Top(∆n−1 , Y ),
φ 7→ din ◦ φ
si : Top(∆n , Y ) → Top(∆n+1 , Y ),
φ 7→ φ ◦ sin
Examples
3) Total singular complex of a topological space.
Let Y be a topological space. For n ≥ 0 we denote by ∆n the
standard affine simplex
{(x0 , x1 , . . . , xn ) ∈ Rn+1 |x0 + x1 + · · · + xn = 1, xi ≥ 0}.
and by din : ∆n → ∆n+1 , sin : ∆n → ∆n−1 the maps
din (x0 , x1 , . . . , xn ) = (x0 , x1 , . . . , xi−1 , 0, xi , . . . xn )
sin (x0 , x1 , . . . , xn ) = (x0 , x1 , . . . , xi−1 + xi , . . . xn ), 0 ≤ i ≤ n
To define the simplicial set SY , we take
SYn = {continuous maps ∆n → Y }
di : Top(∆n , Y ) → Top(∆n−1 , Y ),
φ 7→ din ◦ φ
si : Top(∆n , Y ) → Top(∆n+1 , Y ),
φ 7→ φ ◦ sin
Examples
3) Total singular complex of a topological space.
Let Y be a topological space. For n ≥ 0 we denote by ∆n the
standard affine simplex
{(x0 , x1 , . . . , xn ) ∈ Rn+1 |x0 + x1 + · · · + xn = 1, xi ≥ 0}.
and by din : ∆n → ∆n+1 , sin : ∆n → ∆n−1 the maps
din (x0 , x1 , . . . , xn ) = (x0 , x1 , . . . , xi−1 , 0, xi , . . . xn )
sin (x0 , x1 , . . . , xn ) = (x0 , x1 , . . . , xi−1 + xi , . . . xn ), 0 ≤ i ≤ n
To define the simplicial set SY , we take
SYn = Top(∆n , Y )
di : Top(∆n , Y ) → Top(∆n−1 , Y ),
φ 7→ din ◦ φ
si : Top(∆n , Y ) → Top(∆n+1 , Y ),
φ 7→ φ ◦ sin
Examples
3) Total singular complex of a topological space.
Let Y be a topological space. For n ≥ 0 we denote by ∆n the
standard affine simplex
{(x0 , x1 , . . . , xn ) ∈ Rn+1 |x0 + x1 + · · · + xn = 1, xi ≥ 0}.
and by din : ∆n → ∆n+1 , sin : ∆n → ∆n−1 the maps
din (x0 , x1 , . . . , xn ) = (x0 , x1 , . . . , xi−1 , 0, xi , . . . xn )
sin (x0 , x1 , . . . , xn ) = (x0 , x1 , . . . , xi−1 + xi , . . . xn ), 0 ≤ i ≤ n
To define the simplicial set SY , we take
SYn = Top(∆n , Y )
di : Top(∆n , Y ) → Top(∆n−1 , Y ),
φ 7→ din ◦ φ
si : Top(∆n , Y ) → Top(∆n+1 , Y ),
φ 7→ φ ◦ sin
Examples
3) Total singular complex of a topological space.
Equivalently, SY can be defined as the functor Top(ρ(−), Y ),
where ρ : ∆ → Top is the (covariant) functor sending an
ordinal [n] to the affine simplex ∆n .
So the correspondence Y 7→ SY is functorial:
S : Top → sSets = Sets∆
op
Examples
3) Total singular complex of a topological space.
Equivalently, SY can be defined as the functor Top(ρ(−), Y ),
where ρ : ∆ → Top is the (covariant) functor sending an
ordinal [n] to the affine simplex ∆n .
So the correspondence Y 7→ SY is functorial:
S : Top → sSets = Sets∆
op
Examples
4) Simplicial abelian groups.
A simplicial abelian group is a simplicial object in the category
Ab of abelian groups.
ii) Let A be a simplicial abelian group. We can construct a
chain complex associated with A.
A0 d j
A0 j
where ∂n =
n
P
∂1
s0
d0
d1
*
A1X d j
A1 j
∂2
s0
s1
d0
d1
d2
*$
A2Z _ kf
A2 j
∂3
.
' A3
A3 j
∂3
...
...
(−1)i di . One can check that ∂∂ = 0.
i=1
yields a functor D : sAb → ChZ .
This
Examples
4) Simplicial abelian groups.
A simplicial abelian group is a simplicial object in the category
Ab of abelian groups.
i) Let F : Sets → Ab be a functor assigning a to a set X the
free group generated on X. It induces a functor
F∗ : sSets → sAb
Σ 7→ F ◦ Σ
Intuitively, one can think of F∗ (Σ)n as of the free abelian group
on the n-simplicies of the ”simplcial complex“ Σ.
ii) Let A be a simplicial abelian group. We can construct a
chain complex associated with A.
s0
A0 d j
A0 j
∂1
d0
d1
*
s0
s1
A1X d j
A1 j
∂2
d0
d1
d2
*$
A2Z _ kf
A2 j
∂3
.
' A3
A3 j
∂3
...
...
Examples
4) Simplicial abelian groups.
A simplicial abelian group is a simplicial object in the category
Ab of abelian groups.
i) Let F : Sets → Ab be a functor assigning a to a set X the
free group generated on X. It induces a functor
F∗ : sSets → sAb
Σ 7→ F ◦ Σ
Intuitively, one can think of F∗ (Σ)n as of the free abelian group
on the n-simplicies of the ”simplcial complex“ Σ.
ii) Let A be a simplicial abelian group. We can construct a
chain complex associated with A.
s0
A0 d j
A0 j
∂1
d0
d1
*
s0
s1
A1X d j
A1 j
∂2
d0
d1
d2
*$
A2Z _ kf
A2 j
∂3
.
' A3
A3 j
∂3
...
...
Examples
4) Simplicial abelian groups.
A simplicial abelian group is a simplicial object in the category
Ab of abelian groups.
ii) Let A be a simplicial abelian group. We can construct a
chain complex associated with A.
A0 d j
A0 j
where ∂n =
n
P
∂1
s0
d0
d1
*
A1X d j
A1 j
∂2
s0
s1
d0
d1
d2
*$
A2Z _ kf
A2 j
∂3
.
' A3
A3 j
∂3
...
...
(−1)i di . One can check that ∂∂ = 0.
i=1
yields a functor D : sAb → ChZ .
This
Examples
4) Simplicial abelian groups.
A simplicial abelian group is a simplicial object in the category
Ab of abelian groups.
ii) Let A be a simplicial abelian group. We can construct a
chain complex associated with A.
A0 d j
A0 j
where ∂n =
n
P
∂1
s0
d0
d1
*
A1X d j
A1 j
∂2
s0
s1
d0
d1
d2
*$
A2Z _ kf
A2 j
∂3
.
' A3
A3 j
∂3
...
...
(−1)i di . One can check that ∂∂ = 0.
i=1
yields a functor D : sAb → ChZ .
This
Examples
4) Simplicial abelian groups.
A simplicial abelian group is a simplicial object in the category
Ab of abelian groups.
ii) Let A be a simplicial abelian group. We can construct a
chain complex associated with A.
A0 d j
A0 j
where ∂n =
n
P
∂1
s0
d0
d1
*
A1X d j
A1 j
∂2
s0
s1
d0
d1
d2
*$
A2Z _ kf
A2 j
∂3
.
' A3
A3 j
∂3
...
...
(−1)i di . One can check that ∂∂ = 0.
i=1
yields a functor D : sAb → ChZ .
This
Examples
Application.
Combining the constructions of examples 3 and 4, we can define
singular homology in one line:
S
F
D
H
Hn (−, Z) : Top → sSets →∗ sAb → ChZ →n Ab.
Remark.
In fact, for any abelian category A there is an equivalence of
categories


 chain complexes in A 
simplicial
concentrated in
sA =
←→ Ch≥0 (A) =
.
objects in A


non-negative degrees
This is known as Dold-Kan correspondence.
Examples
Application.
Combining the constructions of examples 3 and 4, we can define
singular homology in one line:
S
F
D
H
Hn (−, Z) : Top → sSets →∗ sAb → ChZ →n Ab.
Remark.
In fact, for any abelian category A there is an equivalence of
categories


 chain complexes in A 
simplicial
concentrated in
sA =
←→ Ch≥0 (A) =
.
objects in A


non-negative degrees
This is known as Dold-Kan correspondence.
Geometric realization
For a simplicial set X we construct a topological space |X|
called the geometric realization of X as follows:
1) for each n ≥ 0 topologize the product Xn × ∆n as the
disjoint union of copies of the affine simples ∆n indexed by
elements of Xn ;
`
2) on the union
(Xn × ∆n ) define the equivalence relation
n≥0
by declaring
(x, s) ∼ (y, t)
iff there exists α : [m] → [n] such that α∗ (y) = x, α∗ (s) = t.
`
3) take |X| to be
(Xn × ∆n )/ ∼.
n≥0
Examples
5) Classifying space. Let G be a group. We define a
simplicial set BG as follows:
BG0 = {1}, BG1 = G, BG2 = G, . . . , BGn = Gn , . . .
si (g1 , . . . , gn ) = (g1 , . . . , gi , 1, gi+1 , . . . , gn )


i=0
(g2 , . . . , gn ),
di (g1 , . . . , gn ) = (g1 , . . . , gi gi+1 , . . . , gn ), 0 < i < n


(g1 , . . . , gn−1 ),
i=n
Geometric realization |BG| is called the classifying space of G.
It has the following property:


 isomorphism classes of 
homotopy classes
principal G − bundles
←→
.
of maps X → |BG|


over X
Examples
6) Nerve of a category. Let C be a small category. We define
the nerve of C to be the simplicial set N C constructed as
follows:
N C0 = objects of C
N C1 = {• −→ •} - morphisms in C
N C2 = {• −→ • −→ •}
N C3 = {• −→ • −→ • −→ •}
...
N Cn = {• −→ • −→ . . . −→ •}
...
The degeneracy map si : N Cn → N Cn+1 inserts the identity
arrow at the i-th spot and the face map di : N Cn → N Cn−1
composes i-th and (i + 1)-th morphisms.
Examples
7) Nerve of a cover. Let U = (Ui )i∈I be an open cover of a
space X. We define the nerve of U to be the simplicial set N U
constructed as follows:
N Un = {(α0 , . . . , αn )|Uα0 ∩ Uαn 6= ∅}
and for an order preserving function f : [m] → [n],
(N Uf )(α0 , . . . , αn ) = (αf (0) , . . . , αf (m) ).
Application. If U is a good cover, then the Cech cohomology
Ȟ n (X, R) can be defined as H n (|N U|, R).
Examples
7) Nerve of a cover. Let U = (Ui )i∈I be an open cover of a
space X. We define the nerve of U to be the simplicial set N U
constructed as follows:
N Un = {(α0 , . . . , αn )|Uα0 ∩ Uαn 6= ∅}
and for an order preserving function f : [m] → [n],
(N Uf )(α0 , . . . , αn ) = (αf (0) , . . . , αf (m) ).
Application. If U is a good cover, then the Cech cohomology
Ȟ n (X, R) can be defined as H n (|N U|, R).