Zoran Skoda homotopija lekcija9

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

A geometric $n$-simplex is a convex hull of $(n+1)$ 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 $n$-simplex $\Delta^n \subset\mathbb{R}^{n+1}$ is the subset

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

All $n$-dimensional polyhedra (bodies which are unions of pieces cut out by hyperplanes) can be made out of geometric simplices. Every geometric $n$-simplex in a vector space $V$ is an image of the standard $n$-simplex via a linear map from $\mathbb{R}^n$ to $V$; 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 $(n+1)$ coordinate hyperplanes $H_{n,i}$ in $\mathbb{R}^{n+1}$ namely $H_{n,i} = \{x\in\mathbb{R}^{n+1}\,|\,x_i = 0\}$, each of which is a copy of the vector space $\mathbb{R}^n$. The intersection $\Delta^n\cap H_{n,i}$ is the $i$-th face of the standard $n$-simplex $\Delta^n$ and is itself clearly a standard $(n-1)$-simplex $ConvexHull\{e_0,e_1,\ldots,e_{i-1},e_{i+1},\ldots,e_n\}$ in the corresponding copy of $\mathbb{R}^n$ namely in $Span_{\mathbb{R}}\{e_0,e_1,\ldots,e_{i-1},e_{i+1},\ldots,e_n\}$ where $e_k$ are the standard basis (“unit”) vectors. Notice the omission of a unit vector $e_i$, hence the corresponding coordinate $x_i$, when taking a face.

Given a topological space, one is often interested in continuous maps $f:\Delta^n\to X$; such continuous maps are called singular $n$-simplices. The $i$-th face of $\Delta^n$ itself can be considered as a unique singular $(n-1)$-simplex $d^i: \Delta^{n-1}\hookrightarrow\Delta^n$ preserving the order; postcomposing that map with $f$ we obtain a singular $(n-1)$-simplex $\partial_i (f) = f\circ d^i: \Delta^{n-1}\to X$. 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 $i$-th original coordinate, the $i$-th coordinate is the one which was originally $(i+1)$-st. Thus we have $\partial_{i-1} \partial_j = \partial_j\partial_i$ if $i\gt j$.

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

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

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

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

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

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

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

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

where $\Delta^m$ is the standard geometric simplex (see above), $X_m$ 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 $m$-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 $g:[m]\to [n]$ is a monotone map, then there is a unique linear map $\Delta^g : \Delta^m\to\Delta^n$ sending the unit vector $e_i\in\Delta^m$ to the unit vector $e_{f(i)}\in\Delta^n$. Now the equivalence relation above is the smallest equivalence relation such that $X_m\times\Delta^m\ni(x,t)\sim (y,s)\in X_n\times\Delta^n$ if there is a monotone map $f\in\mathbf{\Delta}([m],[n])$ such that $y = X(f)(x)$ and $s = \Delta^f(t)$.

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

More generally, if $C$ is a category, a functor $X:\mathbf{\Delta}^{op}\to C$ is called a simplicial object in $C$. 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 $C$ which are the covariant functors $\Delta\to C$.

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

$$\coprod_{m=0}^\infty X_m\times \Delta^m\to \coprod_{k=0}^\infty Y_k\times \Delta^k$$

by $(x,s)\mapsto (f(x),s)$ 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 $J : \mathbf{\Delta}\hookrightarrow Top$, $[n]\mapsto\Delta^n$. Composing it with $Hom_{Top}(-,X)$ gives the singular complex functor $S: Top\mapsto SSet$. Therefore the singular functor is given the composition

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

and the geometric realization is its left adjoint functor:

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

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 $\mathbf{\Delta}/X$ for a simplicial set $X$ whose elements are maps of simplicial sets of the form $\Delta[n]\to X$ and the morphisms are commutative triangles where $\Delta[n]$ is the simplicial set represented by $[n]$ (in other words, $\Delta[n]_m = \mathbf{\Delta}([m],[n])$). By Yoneda lemma, there is a natural bijection $Nat(\Delta[n],X)\cong X_n$ hence the objects of the $\mathbf{\Delta}/X$ are the simplices of $X$, hence it is called the category of simplices of $X$. There is a formula

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

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

An important class of examples of simplicial sets comes from the so-called nerve construction due to Grothendieck. To every category $C$ one associated a simplicial set $N(C)$, the nerve of category $C$, whose $n$-simplices are the composable $n$-tuples of morphisms in $C$. In other words, the $n$-simplex is a sequence $(f_n,\ldots,f_1)$ of $n$ morphisms which can be sequentially composed: the chain $A_0\stackrel{f_1}\to A_1\stackrel{f_2}\to\ldots\stackrel{f_n}\to A_{n}$ composes. Given a monotone map $g: [m]\to [n]$, one defines a map $N(g):N(C)_n\to N(C)_m$ by its value on each $(n+1)$-tuple $(f_ n,\ldots,f_1)$: $N(g)(f)(k)$ is the composition of $g(k)-g(k-1)$ morphisms forming the chain from $A_{g(k-1)}$ to $A_{g(k)}$ if they are different and the identity morphism otherwise. This is obvious once we fix the objects $A_{g(k)}$ to be the $k$-th vertex of $N(g)(f)$. For example, let the monotone function $g:[2]\to [3]$ be given by $g(0) = 1$, $g(1)=g(2)=3$, then

$$N(g)(A_0\stackrel{f_1}\to A_1\stackrel{f_2}\to A_2\stackrel{f_3}\to A_3) = A_1\stackrel{f_3\circ f_2}\to A_3\stackrel{id_{A_3}}\to A_3$$

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 $A$ to an object $B$ iff $A\leq B$. 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 $\Delta$ of finite nonempty ordinals and monotone maps is a small subcategory of the large category $Cat$ of small categories and functors. Denote the inclusion functor by $j:\Delta\hookrightarrow Cat$. The nerve functor is simply taking the hom set $Cat(-,C)$, as a contravariant functor of the first argument (but restricted to the subcategory $\Delta$),

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

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

Theorem. The categorical nerve functor $N:Cat\to sSet$ 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 $n$-simplex $[n]$ can be identified simply with the standard geometric $n$-simplex in $\mathbb{R}^{n+1}$. This correspondence $[n]\mapsto\Delta^n$ easily extends to a functor $\mathbf{\Delta}\to Top$.

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

$$\partial^j_n\partial^i_{n-1} = \partial^i_n\partial^{j-1}_{n-1}, \,\,\,i\lt j$$

$$\sigma^{j}_{n-1} \sigma^{i}_n = \sigma^{i}_{n-1} \sigma^{j+1}_n,\,\,\,\,\,i\leq j$$

$$\sigma^j_{n+1} \partial^i_n = \partial^i_{n+1}\sigma^{j-1}_{n+2},\,\,\,\,i\lt j$$

$$\sigma^j_{n+1} \partial^i_n = id,\,\,\,\,i = j,\,\,\,\mathrm{or}\,\,\,\,i = j+1$$

$$\sigma^j_{n+1} \partial^i_n = \partial^{i-1}_{n+1}\sigma^{j}_{n+2},\,\,\,\,i\gt j+1$$

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

$$\partial_i\partial_j = \partial_{j-1}\partial_i,\,\,\,\,i\lt j,$$

$$\sigma_i\sigma_j = \sigma_{j+1}\sigma_i,\,\,\,\,i\leq j,$$

$$\partial_i\sigma_j = \left\lbrace \array{\sigma_{j-1}\partial_i,&i\lt j\\ id,& i = j,j+1\\ \sigma_j \partial_{i-1},&i\gt j+1}\right.$$

Theorem. Every morphism $f:[m]\to[n]$ in the category of combinatorial simplices $\Delta$ can be written as $f = je$ where $e$ is an epimorphism and $j$ a monomorphism. Moreover the monic $j$ can be written as a finite composition $j = \partial^{i_1}_{n}\partial^{i_2}_{n-1}\cdots\partial^{i_r}_{n-r+1}$ of face maps which is also unique if we require the ordering $0\leq i_r\lt i_{r-1}\lt\ldots \lt i_1\leq n$ and the epic $e$ is a finite composition $e =\sigma^{k_s}_{m-s}\sigma^{k_{s-1}}_{m-s+1}\cdots\sigma^{k_2}_{m-2}\sigma^{k_1}_{m-1}$ of degeneracies which is also unique if we require the ordering $0\leq k_s\lt k_{s-1}\lt \ldots \lt k_1 \lt m$. Labels $i_1\gt \ldots\gt i_r$ are precisely the labels $i_\alpha$ omited in $Im f = f([m])\subset [n]$ and $k_1\gt \ldots\gt k_s$ are precisely the labels $k_\beta$ in $[m]$ such that $f(k_\beta)=f(k_\beta+1)$.

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