Zoran Skoda
homotopija lekcija9

Main topic today will be combinatorial simplicial complexes and simplicial sets.

A geometric nn-simplex is a convex hull of (n+1)(n+1) distinct points in a real vector space, called its vertices. It is said to be ordered if there is a total ordering on vertices of each simplex given, and the ordering on faces is the induced ordering. The standard nn-simplex Δ n n+1\Delta^n \subset\mathbb{R}^{n+1} is the subset

Δ n={x=(x 0,,x n) n+1|x 0+x 1++x n=1,0x i,i=0,,n}\Delta^n = \{x = (x_0,\ldots,x_n)\in\mathbb{R}^{n+1}\,|\,x_0+x_1+\ldots + x_n = 1, \,\,\,\,0\leq x_i,\,\,i=0,\ldots, n\}

All nn-dimensional polyhedra (bodies which are unions of pieces cut out by hyperplanes) can be made out of geometric simplices. Every geometric nn-simplex in a vector space VV is an image of the standard nn-simplex via a linear map from n\mathbb{R}^n to VV; this map is clearly unique up to the ordering of the simplices, hence for ordered geometric simplices there is a unique such map preserving the ordering of the simplices.

There are (n+1)(n+1) coordinate hyperplanes H n,iH_{n,i} in n+1\mathbb{R}^{n+1} namely H n,i={x n+1|x i=0}H_{n,i} = \{x\in\mathbb{R}^{n+1}\,|\,x_i = 0\}, each of which is a copy of the vector space n\mathbb{R}^n. The intersection Δ nH n,i\Delta^n\cap H_{n,i} is the ii-th face of the standard nn-simplex Δ n\Delta^n and is itself clearly a standard (n1)(n-1)-simplex ConvexHull{e 0,e 1,,e i1,e i+1,,e n}ConvexHull\{e_0,e_1,\ldots,e_{i-1},e_{i+1},\ldots,e_n\} in the corresponding copy of n\mathbb{R}^n namely in Span {e 0,e 1,,e i1,e i+1,,e n}Span_{\mathbb{R}}\{e_0,e_1,\ldots,e_{i-1},e_{i+1},\ldots,e_n\} where e ke_k are the standard basis (“unit”) vectors. Notice the omission of a unit vector e ie_i, hence the corresponding coordinate x ix_i, when taking a face.

Given a topological space, one is often interested in continuous maps f:Δ nXf:\Delta^n\to X; such continuous maps are called singular nn-simplices. The ii-th face of Δ n\Delta^n itself can be considered as a unique singular (n1)(n-1)-simplex d i:Δ n1Δ nd^i: \Delta^{n-1}\hookrightarrow\Delta^n preserving the order; postcomposing that map with ff we obtain a singular (n1)(n-1)-simplex i(f)=fd i:Δ n1X\partial_i (f) = f\circ d^i: \Delta^{n-1}\to X. One should in fact extend these considerations to faces of faces and so on. As each face corresponds to an omission of a coordinate, the faces of faces of faces and so on correspond to multiple omissions of coordinates. It is just important which coordinates are omited, not the order of omissions. However the label changes: in the singular simplex missing ii-th original coordinate, the ii-th coordinate is the one which was originally (i+1)(i+1)-st. Thus we have i1 j= j i\partial_{i-1} \partial_j = \partial_j\partial_i if i>ji\gt j.

There are (n+1)(n+1) linear surjections s i:Δ nΔ n1s^i:\Delta^n\to \Delta^{n-1}, i=0,,ni = 0,\ldots,n, namely s is^i is the unique linear map preserving the preorder on the vertices and sending ii-th and (i+1)(i+1)-st vertex to the vertex ii and being injective on the rest of the vertices. Thus the 1-dimensional space Span {e i+1e i}Span_{\mathbb{R}}\{e_{i+1}-e_i\} is sent to the origin, and correspondingly the image of the interval λe i+(1λ)e i+1\lambda e_i + (1-\lambda)e_{i+1}, 0λ10\leq \lambda\leq 1, degenerates to one point e ie_i.

A combinatorial simplicial complex XX is a set X 0X_0 of vertices or 00-simplices, together with a choice of sets X nX_n, n=1,2,n = 1,2,\ldots of subsets of X 0X_0 of cardinality (n+1)(n+1), called the (combinatorial) nn-simplices, so that if σ\sigma is an nn-simplex, every rr-element subset (where r<nr\lt n) of σ\sigma is an rr-simplex. Here we do not allow that vertices coincide. We assume a total ordering on the vertices.

Every topological space XX gives rise to a canonical (combinatorial) simplicial complex whose nn-simplices are precisely the singular nn-simplices in XX.

Every simplicial complex gives rise to a more general structure of a simplicial set.

The labels of vertices of a combinatorial nn-simplex form an (n+1)(n+1)-element ordered set, which can be viewed as an image of a unique order-preserving bijective map also denoted X n:[n]SetX_n:[n]\to Set from a standard ordinal n+1={0<1<2<<n}=:[n]n+1 = \{0\lt 1\lt 2\lt \ldots\lt n\} =: [n] to the category of sets. Looking at its ii-th face now corresponds to the composition with a monotone map i:[n1][n]\partial^i:[n-1]\to [n] omitting ii. Similarily, degenerating nn-simplex to (n1)(n-1)-simplex corresponds to the composition with a map σ i:[n][n1]\sigma^i:[n]\to[n-1] of ordinals repeating ii twice. More generally, any monotone map g:[m][n]g: [m]\to [n] induces a map X(g):X nX mX(g):X_n\to X_m obtained by precomposition; what is a contravariantly functorial prescription.

The category whose objects are the nonempty finite ordinals [0],[1],[2],[0],[1],[2],\ldots and whose morphisms are (not necessarily strictly) monotone maps (= maps of sets preserving the order) is called the category of combinatorial simplices Δ\Delta (topological simplex category). A simplicial set is a contravariant functor X:Δ opSetX:\Delta^{op}\to Set.

For every simplicial set XX one defines a topological space |X||X|, its geometric realization, by the following formula

|X|=( m=0 X m×Δ m)/ |X| = \left(\coprod_{m=0}^\infty X_m\times \Delta^m\right)/\sim

where Δ m\Delta^m is the standard geometric simplex (see above), X mX_m is taken with the discrete topology, \sim is a relation of equivalence to be described below and the product and the quotient are understood as categorical in the category of all topological spaces. The formula is obvious: for every mm-simplex we form a real simplex. The relation of equivalence hence has to identify the points of the faces with the appropriate points of the simplex whose face is in question. As we reasoned before, the linear maps of geometric simplices are determined by the maps among the sets of vertices. More precisely, if g:[m][n]g:[m]\to [n] is a monotone map, then there is a unique linear map Δ g:Δ mΔ n\Delta^g : \Delta^m\to\Delta^n sending the unit vector e iΔ me_i\in\Delta^m to the unit vector e f(i)Δ ne_{f(i)}\in\Delta^n. Now the equivalence relation above is the smallest equivalence relation such that X m×Δ m(x,t)(y,s)X n×Δ nX_m\times\Delta^m\ni(x,t)\sim (y,s)\in X_n\times\Delta^n if there is a monotone map fΔ([m],[n])f\in\mathbf{\Delta}([m],[n]) such that y=X(f)(x)y = X(f)(x) and s=Δ f(t)s = \Delta^f(t).

A map of simplicial sets F:XYF:X\to Y is a natural transformation of functors, i.e. a family F n:X nY nF_n:X_n\to Y_n of maps, indexed by [n]Δ[n]\in\mathbf{\Delta}, such that the naturality requirement F nX(f)=Y(f)F mF_n\circ X(f)= Y(f)\circ F_m for all f:[m][n]f:[m]\to[n] holds. Simplicial sets and their maps form a category sSetsSet of simplicial sets.

More generally, if CC is a category, a functor X:Δ opCX:\mathbf{\Delta}^{op}\to C is called a simplicial object in CC. Thus simplicial set is a simplicial object in the category of sets. Another important special cases are the simplicial groups and simplicial abelian groups. One also uses the theory of cosimplicial objects in CC which are the covariant functors ΔC\Delta\to C.

The correspondence X|X|X\mapsto |X| actually extends to a functor, called the geometric realization functor ||:sSetS| |:sSet\to S. Namely, given a map f:XYf:X\to Y of simplicial sets one first defines a map

m=0 X m×Δ m k=0 Y k×Δ k \coprod_{m=0}^\infty X_m\times \Delta^m\to \coprod_{k=0}^\infty Y_k\times \Delta^k

by (x,s)(f(x),s)(x,s)\mapsto (f(x),s) and one shows that this induces a well defined continuous map after we quotient out by the equivalence relation on both sides. It is then a straightfoward check that this prescription on morphisms is functorial. This gives in fact an embedding J:ΔTopJ : \mathbf{\Delta}\hookrightarrow Top, [n]Δ n[n]\mapsto\Delta^n. Composing it with Hom Top(,X)Hom_{Top}(-,X) gives the singular complex functor S:TopSSetS: Top\mapsto SSet. Therefore the singular functor is given the composition

Δ op[n]Δ nTop opHom(,X)Set \mathbf{\Delta^{op}}\stackrel{[n]\mapsto\Delta^n}\to Top^{op}\stackrel{Hom(-,X)}\to Set

and the geometric realization is its left adjoint functor:

Top(|X|,Y)=sSet(X,SY). Top(|X|,Y) = sSet(X,SY).

There is also a colimit formula (following either by direct check or by the theory of Kan extensions) for the geometric realization. To this aim, one considers the overcategory Δ/X\mathbf{\Delta}/X for a simplicial set XX whose elements are maps of simplicial sets of the form Δ[n]X\Delta[n]\to X and the morphisms are commutative triangles where Δ[n]\Delta[n] is the simplicial set represented by [n][n] (in other words, Δ[n] m=Δ([m],[n])\Delta[n]_m = \mathbf{\Delta}([m],[n])). By Yoneda lemma, there is a natural bijection Nat(Δ[n],X)X nNat(\Delta[n],X)\cong X_n hence the objects of the Δ/X\mathbf{\Delta}/X are the simplices of XX, hence it is called the category of simplices of XX. There is a formula

|X|=colim (Δ[n]X)Δ/X|Δ[n]|. |X|= colim_{(\Delta[n]\to X)\in\mathbf{\Delta}/X} |\Delta[n]|.

Of course, |Δ[n]|=Δ n|\Delta[n]| = \Delta^n in our notation.

An important class of examples of simplicial sets comes from the so-called nerve construction due to Grothendieck. To every category CC one associated a simplicial set N(C)N(C), the nerve of category CC, whose nn-simplices are the composable nn-tuples of morphisms in CC. In other words, the nn-simplex is a sequence (f n,,f 1)(f_n,\ldots,f_1) of nn morphisms which can be sequentially composed: the chain A 0f 1A 1f 2f nA nA_0\stackrel{f_1}\to A_1\stackrel{f_2}\to\ldots\stackrel{f_n}\to A_{n} composes. Given a monotone map g:[m][n]g: [m]\to [n], one defines a map N(g):N(C) nN(C) mN(g):N(C)_n\to N(C)_m by its value on each (n+1)(n+1)-tuple (f n,,f 1)(f_ n,\ldots,f_1): N(g)(f)(k)N(g)(f)(k) is the composition of g(k)g(k1)g(k)-g(k-1) morphisms forming the chain from A g(k1)A_{g(k-1)} to A g(k)A_{g(k)} if they are different and the identity morphism otherwise. This is obvious once we fix the objects A g(k)A_{g(k)} to be the kk-th vertex of N(g)(f)N(g)(f). For example, let the monotone function g:[2][3]g:[2]\to [3] be given by g(0)=1g(0) = 1, g(1)=g(2)=3g(1)=g(2)=3, then

N(g)(A 0f 1A 1f 2A 2f 3A 3)=A 1f 3f 2A 3id A 3A 3 N(g)(A_0\stackrel{f_1}\to A_1\stackrel{f_2}\to A_2\stackrel{f_3}\to A_3) = A_1\stackrel{f_3\circ f_2}\to A_3\stackrel{id_{A_3}}\to A_3

This explicit description of a nerve can be expressed more economically in categorical language, with an advantage that some properties are easy to establish. Each partially ordered set can be considered a small category: there is a morphism (unique when it exists) from an object AA to an object BB iff ABA\leq B. This object is an identity if the objects are equal and the composition is obvious, by uniqueness of the morphisms between given objects: it is enough to know the source and target to compose. Moreover, the order-preserving maps clearly correspond to functors. Thus the category Δ\Delta of finite nonempty ordinals and monotone maps is a small subcategory of the large category CatCat of small categories and functors. Denote the inclusion functor by j:ΔCatj:\Delta\hookrightarrow Cat. The nerve functor is simply taking the hom set Cat(,C)Cat(-,C), as a contravariant functor of the first argument (but restricted to the subcategory Δ\Delta),

N(C):=Cat(,C)j op:Δ opj opCat opCat(,C)Set,C([n]Hom Cat([n],C)). N(C) := Cat(-,C)\circ j^{op} :\mathbf{\Delta}^{op}\stackrel{j^{op}}\hookrightarrow Cat^{op}\stackrel{Cat(-,C)}\to Set,\,\,\,C\mapsto ([n]\mapsto Hom_{Cat}([n],C)).

It is not difficult to see that the two descriptions coincide.

Theorem. The categorical nerve functor N:CatsSetN:Cat\to sSet is a fully faithful functor.

In other words, we preserve all essential categorical information on a category when replacing it by its nerve.

In fact the geometric realization can also be understood in a formal way very alike the nerve. Namely, the geometric realization of the combinatorial nn-simplex [n][n] can be identified simply with the standard geometric nn-simplex in n+1\mathbb{R}^{n+1}. This correspondence [n]Δ n[n]\mapsto\Delta^n easily extends to a functor ΔTop\mathbf{\Delta}\to Top.

Regarding that Δ\Delta may be viewed as a subcategory of category of sets too where just the monotone functions among sets [n][n] are considered, we can decompose each map [m][n][m]\to [n] into a composition of an epimorphism followed by a monomorphism. Moreover each epimorphism is a composition of the degeneracy maps σ n i:[n+1][n]\sigma^i_n:[n+1]\to[n] for 0in0\leq i\leq n which are the surjections which repeat (only) ii, and each monomorphism is a composition of the face maps n i:[n1][n]\partial^i_n:[n-1]\to[n] which are the injections which omit only ii. So, Δ\Delta can be described by generators and relations which will be studied next; moreover there is a standard basis. If XX is a simplicial set denote by i= i n\partial_i = \partial_i^n the map X( n i)X(\partial^i_n). Notice that the key index ii is now lower.

Exercise. Make the direct check that the following bilinear relations (called the cosimplicial identities) between the face maps and the degeneracies hold in Δ\Delta:

n j n1 i= n i n1 j1,i<j \partial^j_n\partial^i_{n-1} = \partial^i_n\partial^{j-1}_{n-1}, \,\,\,i\lt j
σ n1 jσ n i=σ n1 iσ n j+1,ij \sigma^{j}_{n-1} \sigma^{i}_n = \sigma^{i}_{n-1} \sigma^{j+1}_n,\,\,\,\,\,i\leq j
σ n+1 j n i= n+1 iσ n+2 j1,i<j \sigma^j_{n+1} \partial^i_n = \partial^i_{n+1}\sigma^{j-1}_{n+2},\,\,\,\,i\lt j
σ n+1 j n i=id,i=j,ori=j+1 \sigma^j_{n+1} \partial^i_n = id,\,\,\,\,i = j,\,\,\,\mathrm{or}\,\,\,\,i = j+1
σ n+1 j n i= n+1 i1σ n+2 j,i>j+1 \sigma^j_{n+1} \partial^i_n = \partial^{i-1}_{n+1}\sigma^{j}_{n+2},\,\,\,\,i\gt j+1

It is not difficult now to write the simplicial identities for the operators X,i n\partial^n_{X,i}, σ X,i n\sigma^n_{X,i} for any simplicial object XX by applying XX to the above identities (functor will send identgities to identities; but be careful as ocntravariance reverses the order in the compositions). Thus, skipping XX and nn we get the simplicial identities

i j= j1 i,i<j,\partial_i\partial_j = \partial_{j-1}\partial_i,\,\,\,\,i\lt j,
σ iσ j=σ j+1σ i,ij,\sigma_i\sigma_j = \sigma_{j+1}\sigma_i,\,\,\,\,i\leq j,
iσ j={σ j1 i, i<j id, i=j,j+1 σ j i1, i>j+1 \partial_i\sigma_j = \left\lbrace \array{\sigma_{j-1}\partial_i,&i\lt j\\ id,& i = j,j+1\\ \sigma_j \partial_{i-1},&i\gt j+1}\right.

Theorem. Every morphism f:[m][n]f:[m]\to[n] in the category of combinatorial simplices Δ\Delta can be written as f=jef = je where ee is an epimorphism and jj a monomorphism. Moreover the monic jj can be written as a finite composition j= n i 1 n1 i 2 nr+1 i rj = \partial^{i_1}_{n}\partial^{i_2}_{n-1}\cdots\partial^{i_r}_{n-r+1} of face maps which is also unique if we require the ordering 0i r<i r1<<i 1n0\leq i_r\lt i_{r-1}\lt\ldots \lt i_1\leq n and the epic ee is a finite composition e=σ ms k sσ ms+1 k s1σ m2 k 2σ m1 k 1e =\sigma^{k_s}_{m-s}\sigma^{k_{s-1}}_{m-s+1}\cdots\sigma^{k_2}_{m-2}\sigma^{k_1}_{m-1} of degeneracies which is also unique if we require the ordering 0k s<k s1<<k 1<m0\leq k_s\lt k_{s-1}\lt \ldots \lt k_1 \lt m. Labels i 1>>i ri_1\gt \ldots\gt i_r are precisely the labels i αi_\alpha omited in Imf=f([m])[n]Im f = f([m])\subset [n] and k 1>>k sk_1\gt \ldots\gt k_s are precisely the labels k βk_\beta in [m][m] such that f(k β)=f(k β+1)f(k_\beta)=f(k_\beta+1).

Theorem. Let X nX_n for all integers n0n\geq 0 be some objects of a fixed category AA and let some morphisms σ i n:X nX n+1\sigma_i^n : X_n\to X_{n+1}, i n:X nX n1\partial_i^n:X_n\to X_{n-1} be given for all 0in0\leq i\leq n and satisfy the simplicial identities. Then there exists a unique simplicial object XX in AA satisfying X([n])=X nX([n]) = X_n, X( n i)= i nX(\partial^i_n) = \partial_i^n and X(σ n j)=σ j nX(\sigma^j_n) = \sigma^n_j.

Last revised on February 2, 2010 at 18:34:17. See the history of this page for a list of all contributions to it.