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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).