Visualizing Simplicial Objects

Definition. Let [latex][n][/latex] denote the set [latex]\{0,1,\dots,n\}[/latex]. We define the category [latex]\Delta\subseteq \mathcal{Set}[/latex] (called the category of combinatorial simplices) as follows: The objects of [latex]\Delta[/latex] are the sets [latex][n][/latex]. The morphisms in [latex]\Delta[/latex] are the order-preserving set functions.

Remark. For each [latex]n[/latex], the set [latex][n][/latex] is a category using the linear ordering on [latex]n[/latex]. To be precise, the objects are the elements of [latex][n][/latex] and for [latex]i,j\in [n][/latex],

\leq:i\to j, &\mbox{if }i\leq j,\\
\emptyset, &\mbox{otherwise}.

Definition. A simplicial object (resp. cosimplicial object) in a category [latex]\mathcal{C}[/latex] is a contravariant (resp. covariant) functor [latex]K:\Delta^{\operatorname{op}}\to \mathcal{C}[/latex] (resp. [latex]K:\Delta\to \mathcal{C}[/latex]). In particular, if [latex]\mathcal{C}=\mathcal{Set}[/latex], simplicial objects are called simplicial sets and similarly for cosimplicial objects.

Put simply, a simplicial object [latex]K[/latex] is completely determined by the following data:

  • An object [latex]K_n:= K([n])[/latex] called the [latex]n[/latex]-skeleton of [latex]K[/latex],
  • A [latex]\mathcal{C}[/latex]-map [latex]f^*:=K(f):K_n\to K_m[/latex] for each [latex]f\in \hom([m],[n])[/latex] called the attachment maps of [latex]K[/latex].

For each [latex]n\geq 0[/latex], we have a natural simplicial set [latex]\Delta^n[/latex] called the [latex]n[/latex]-simplex. This simplicial set is defined as follows:

  • [latex]\Delta^n_i :=\hom_\Delta ([i],[n])[/latex].
  • If [latex]f:[i]\to [j][/latex], then [latex]f^*:\hom_\Delta ([j],[n])\to \hom_\Delta ([i],[n])[/latex] is given by [latex]g\mapsto g\circ f[/latex].

Some basic properties are:

  • If [latex]i>n[/latex], then [latex]\Delta^n_i=\emptyset[/latex].
  • If [latex]m\leq n[/latex], then [latex]\Delta^m\subseteq \Delta^n[/latex]. (More precisely, there is an isomorphic copy of [latex]\Delta^m[/latex] in [latex]\Delta^n[/latex].)
  • If [latex]K[/latex] is a simplicial set, [latex]K_n\simeq \hom(\Delta^n, K)[/latex] where morphisms are taken in the category of simplicial sets.

For all [latex]n[/latex], there is a canonical visualization of [latex]\Delta^n[/latex]. For simplicity, we consider the [latex]n=2[/latex] case:

  • For [latex]\Delta^2_0[/latex], label [latex]0\mapsto 0[/latex] by [latex](1,0,0)[/latex], [latex]0\mapsto 1[/latex] by [latex](0,1,0)[/latex], and [latex]0\mapsto 2[/latex] by [latex](0,0,1)[/latex].
  • For [latex]\Delta^2_1[/latex], label
    [latex]0\mapsto 0,1\mapsto 1[/latex] by a path [latex]\gamma_0:[0,1]\to \mathbb{R}^3[/latex] taking [latex](1,0,0)[/latex] to [latex](0,1,0)[/latex],
    [latex]0\mapsto 0,1\mapsto 2[/latex] by a path [latex]\gamma_1:[0,1]\to \mathbb{R}^3[/latex] taking [latex](1,0,0)[/latex] to [latex](0,0,1)[/latex],
    [latex]0\mapsto 1,1\mapsto 2[/latex] by a path [latex]\gamma_2:[0,1]\to \mathbb{R}^3[/latex] taking [latex](0,1,0)[/latex] to [latex](0,0,1)[/latex].
  • For [latex]\Delta^2_2[/latex], label the unique element by any continuous map [latex]F:\mathbb{D}^2\to \mathbb{R}^3[/latex] taking [latex]\partial\mathbb{D}^2[/latex] to [latex]\cup_{i} \gamma_i[/latex].

Using these identifications, [latex]\Delta^2[/latex] is noneother than the geometric [latex]2[/latex]-simplex in [latex]\mathbb{R}^3[/latex].

As a final note, in motivation for Kan complexes, we define the notion of a horn.

Definition. Let [latex]0\leq j\leq n[/latex]. The jth horn [latex]\Lambda_j^n[/latex] of a [latex]n[/latex]-simplex [latex]\Delta^n[/latex] is defined by

(\Delta_j^n)_i=\begin{cases}\Delta^n_i, &\mbox{if } i<n-1,\\ \Delta^n_i\setminus \{f\}, &\mbox{if }i=n-1, \\ \emptyset, &\mbox{if }i>n-1,\end{cases}

where [latex]f[/latex] is the unique map with [latex]f^{-1}(j)=\emptyset[/latex].

Geometrically, the unique map [latex]f[/latex] corresponds to the [latex]j[/latex]th face (i.e. the [latex]j[/latex]th element in the [latex](n-1)[/latex]-skeleton of [latex]\Delta^n[/latex] using lexicographic ordering). Thus the [latex]j[/latex]th horn is just the [latex]n[/latex]-simplex with this face and the interior (the unique element in [latex]\Delta_n^n[/latex]) removed.

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