Simplicial Complex

definition example

Let $V$ be a set with Power Set $\mathcal{P}(V)$. A subset $X\subseteq \mathcal{P}(V)$ is downward closed if $u\in X$ and $u^{\prime }\subseteq u$ imply $u^{\prime }\in X$, and contains all atoms if $\{v\}\in X$ for every $v\in V$. A simplicial complex is a pair $(V,X)$ with $X\subseteq \mathcal{P}(V)$ downward closed and containing all atoms. Elements of $X$ are simplices; those of cardinality $n+1$ form the set $X_n$ of $n$-simplices — vertices ($X_0\cong V$), edges ($X_1$), triangles ($X_2$), tetrahedra ($X_3$), …; the index is the dimension. (Strictly, $X_0=\{\{v\}\mid v\in V\}$, which CTfS calls “just pedantry”.)

Sources: CTfS §2.7.4.3 (Definition 2.7.4.4, Example 2.7.4.5, Exercises 2.7.4.6–2.7.4.7), §5.2.3.11 (a sheaf of worldviews on a simplicial complex), Example 4.6.1.6 (simplicial sets).

Examples

• The $n$-simplex ${\Delta }^n$ (CTfS Example 2.7.4.5): $V=\underline{n+1}$ and $X=\mathcal{P}(V)$ (all nonempty subsets; the empty set is harmless). ${\Delta }^0$ is a point, ${\Delta }^1$ an edge, ${\Delta }^2$ a filled triangle, ${\Delta }^3$ a solid tetrahedron.
• A tree: vertices $\{1,2,3,4\}$, edges $\{1,2\},\{2,3\},\{2,4\}$, nothing higher — a star with centre 2.
• The boundary of a triangle $\partial {\Delta }^2$ (CTfS Exercise 2.7.4.7): $X_0=\{\{1\},\{2\},\{3\}\}$, $X_1=\{\{1,2\},\{1,3\},\{2,3\}\}$ and $X_2=X_3=\cdots =\varnothing$ — an empty triangle, drawn with edges but not filled in.
• Shared worldviews (CTfS §5.2.3.11): vertices are people, and a simplex is a group of people with a common ground. Assigning to each simplex the Olog of concepts its members share, with a schema morphism for each face inclusion, gives a functor from the (opposite of the) poset of simplices to schemas; via the Alexandrov topology this becomes a Sheaf of categories.

Drawing and the geometry it stands for

Draw a dot per vertex, a segment per edge, a filled triangle per 2-simplex, and so on; downward closure guarantees that the boundary of every drawn simplex is drawn too. A simplicial complex is a combinatorial description of a space glued from simplices. The simplices ordered by inclusion form a Preorder (indeed a poset), and the whole complex is determined by its face poset.

Simplicial complexes have poor formal properties (products and quotients misbehave); the categorical replacement is the simplicial set, a functor ${\Delta }^{\mathrm{op}}\to \mathbf{Set}$ on the Simplex Category, which “has excellent formal properties that simplicial complexes do not” (CTfS Example 4.6.1.6). Every simplicial complex with a chosen total order on its vertices gives a simplicial set.

JuliaLeanHaskell

Docs: plain Julia — Catlab has no dedicated API for this; related: Catlab v0.16 docs · GATlab standard library

# a simplicial complex as a downward-closed family of vertex sets (CTfS Definition 2.7.4.4)
function faces(σ)                          # all nonempty subsets of σ, via bitmasks
  v = sort(collect(σ))
  [Set(v[i] for i in eachindex(v) if isodd(mask >> (i - 1))) for mask in 1:(2^length(v) - 1)]
end
closure(generators) = unique(reduce(vcat, faces.(generators)))
simplices_of_dim(X, n) = filter(s -> length(s) == n + 1, X)
∂Δ2 = closure([Set([1, 2]), Set([1, 3]), Set([2, 3])])      # the empty triangle
length.(simplices_of_dim.(Ref(∂Δ2), 0:2))                   # [3, 3, 0]
Δ3 = closure([Set(1:4)])                                     # the solid tetrahedron
length.(simplices_of_dim.(Ref(Δ3), 0:3))                     # [4, 6, 4, 1]

import Mathlib
-- Mathlib has geometric simplicial complexes (in a real vector space) and simplicial sets;
-- an abstract complex is easily written as a downward-closed family of finsets:
structure AbstractSimplicialComplex (V : Type*) where
  faces : Set (Finset V)
  down_closed : ∀ {s t}, s ∈ faces → t ⊆ s → t.Nonempty → t ∈ faces
  atoms : ∀ v, ({v} : Finset V) ∈ faces
#check @Geometry.SimplicialComplex   -- the geometric version
#check SSet                          -- simplicial sets: functors SimplexCategoryᵒᵖ ⥤ Type

import Data.List (subsequences, nub, sort)
 
type Simplex = [Int]                       -- a sorted list of vertices
 
closure :: [Simplex] -> [Simplex]          -- all nonempty faces of the given simplices
closure gens = nub [ f | g <- gens, f <- subsequences (sort g), not (null f) ]
 
dim :: Int -> [Simplex] -> [Simplex]
dim n = filter ((== n + 1) . length)
 
boundaryTriangle :: [Simplex]
boundaryTriangle = closure [[1,2], [1,3], [2,3]]
-- map (\n -> length (dim n boundaryTriangle)) [0,1,2] == [3,3,0]