The Category of Simplicial Complexes

Abstract

Let us recall that a subset C ⊂ ℝ ⁿ is convex if x, y ∈ C, t ∈ [0, 1] ${\Rightarrow} tx + (1 - t)y \in C$. The convex hull of a subset X ⊂ ℝ ⁿ is the smallest convex subset of ℝ ⁿ, which contains X. We say that d + 1 points x ₀, x ₁, …, x _d belonging to the Euclidean space ℝ ⁿ are linearly independent (from the affine point of view) if the vectors x ₁ − x ₀, x ₂ − x ₀, $\ldots$, x _d − x ₀ are linearly independent. A vector x − x ₀ of the vector space generated by these vectors can be written as a sum $x - {x}_{0} ={ \sum \nolimits }_{i=1}^{d}{r}_{i}({x}_{i} - {x}_{0})$ with real coefficients r _i ; notice that if we write x as $x ={ \sum \nolimits }_{i=0}^{d}{\alpha }_{i}{x}_{i}$, then ${\sum \nolimits }_{i=0}^{d}{\alpha }_{i} = 1$. If $\{{x}_{0},\ldots, {x}_{d}\} \in X$ are affinely independent, the convex hull of X is said to be an (Euclidean) simplex of dimension d contained in ℝ ⁿ; its points x can be written in a unique fashion as linear combinations

$$x ={ \sum \nolimits }_{i=1}^{d}{\lambda }_{ i}{x}_{i},$$

with real coefficients λ _i . The coefficients λ _i are called barycentric coordinates of x; they are nonnegative real numbers and satisfy the equality ${\sum \nolimits }_{i=0}^{d}{\lambda }_{i} = 1$. The points x _i are the vertices of the simplex. The standard n-simplex is the simplex obtained by taking the convex hull of the n + 1 points of the standard basis of ℝ ^{n + 1} (see Figs. II.1 and II.2 for dimensions n = 1 and n = 2, respectively). The faces of a simplex s ⊂ ℝ ⁿ are the convex hulls of the subsets of its vertices; the faces which do not coincide with s are the proper faces. We can define the interior of a simplex s as the set of all points of s with positive barycentric coordinates λ _i > 0. We indicate the interior of s with ˚s. If the dimension of s is at least 1, ˚s coincides with the topological interior. At any rate, it is not hard to prove that we obtain the interior of a simplex by removing all of its proper faces.