Simplicial Complexes, Part 1

The Abstract Simplicial Complex Category

Pre-requisites: none

This entry is the start of my foray into algebraic topology. It is the first of a series of notes I hope to write that focus on standard homology and cohomology theories of topological spaces (such as simplicial, singular, and cellular (co)homology). From there, I will give some examples of how these theories can be used to distinguish between topological spaces, as well as prove celebrated results like the Brower fixed point theorem.

These flashy results are still far out on the horizon. After all, topological spaces are pretty complicated objects, and understanding them will take some time. I am instead going to start with a much more simplified model: that of abstract simplicial complexes.

An abstract simplicial complex is a set $X$ with a collection $\Delta$ of finite, non-empty subsets such that if $A \in \Delta$ and $B \subset A$ is non-empty, then $B \in \Delta$.

We call elements of $\Delta$ simplices, and they will later be replaced with real geometric simplices when we turn our gaze to topology. The dimension of a simplex is one less than its size. We refer to a simplex of dimension $n$ as an $n$-simplex.

As an example, consider $X = \{a, b\}$ and $\Delta = \mathcal{P}(X) \setminus \{\varnothing\}$, the set of all non-empty subsets of $X$. For reasons that will become later on, we will call this complex the interval. The interval has three simplices: it has two 0-simplices and one 1-simplex.

We often want to visualize what is happening in an abstract simplicial complex. To do this, we can draw a kind of graph of the space. Start with a point for every 0-simplex. For every 1-simplex $\{a, b\}$, we draw a line between he 0-simplices $\{a\}$ and $\{b\}$. This now forms a graph. Below is this construction applied to the interval.

For a 2-simplex $\{a, b, c\}$, we shade in a triangle connecting the 0-simplices $\{a\}$, $\{b\}$, and $\{c\}$. For higher-dimensional simplices, we continue this process, but it becomes harder to make out the different simplices at play. As a result, in this page, I will stick to drawing complexes whose simplices are at most 2-dimensional.

As an example, the following is a drawing of an abstract simplicial complex.

It's important to note that, while every simplex is finite, there may be no global bound on the size of simplices in a complex. For example, take $X$ to be any infinite set, and let $\Delta = \mathcal{P}(X) \setminus \{\varnothing\}$. The complex $(X, \Delta)$ has simplices of arbitrarily large size. These are hard to visualize, but they will be very important to us as well.

Now that we understand abstract simplicial complexes, we proceed in true mathematical fashion and define maps between abstract simplicial complexes.

Let $K = (X, \Delta_X)$ and $L = (Y, \Delta_Y)$ be abstract simplicial complexes. A map from $K$ to $L$ is defined as a set function $f: X \rightarrow Y$ such that, for every simplex $\sigma \in \Delta_X$, the set $f(\sigma)$ is a simplex in $\Delta_Y$. In other words, a map between simplicial complexes is one that sends simplices to simplices.

As an example, consider the following example. $f$ is a function from $X$ to $Y$ shown by downward arrows, and it represents a map on abstract simplicial complexes because each simplex of $K$ gets sent to a simplex of $L$.

Given maps of abstract simplicial complexes $f: K \rightarrow L$ and $g: L \rightarrow E$, we can compose these to get a third map $g\circ f: K \rightarrow E$. For every abstract simplicia complex $K$, we have a unique map called the identity map from $K$ to $K$, denoted $i_K$. These notions of composition and identity give us a category whose objects are abstract simplicial complexes and whose arrows are maps between abstract simplicial complexes. We call this category the abstract simplicial complex category, $\mathbf{SCpx}$.

* For those unfamiliar with categories, these are basically a generalization of many different mathematical theories. A category consists of a collection of objects and a collection of arrows between those objects, as well as identity arrows and a rule for composing arrows. We also require that, if $f: X \rightarrow Y$ is an arrow and $i_A$ denotes the identity on $A$, then $f \circ i_X = f = i_Y \circ f$. Prototypical examples of categories include: $\mathbf{Set}$, whose objects are sets and whose arrows are functions between sets; $\mathbf{Grp}$, whose objects are groups and whose arrows are group homomorphisms; and $\mathbf{Top}$, whose objects are topological spaces and whose arrows are continuous functions.

In any category, we can define an isomorphism. This is an invertible map. In the language we have been using, this is a map $f: K \rightarrow L$ such that for some $g: L \rightarrow K$, $f \circ g = i_L$ and $g \circ f = i_K$. We call such a $g$ an inverse of $f$. If there is an isomorphism from $K$ to $L$, we say that $K$ and $L$ are isomorphic. As far as we will be concerned, isomorphic abstract simplicial complexes are basically the same.

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Exercise: prove that the identity map is an isomorphism.

Exercise: prove that the composition of two isomorphisms is an isomorphism.

Exercise: prove that the composition of an isomorphism and a non-isomorphism is not an isomorphism.

Exercise: prove that isomorphism is an equivalence relation.

Exercise+: find an alternative characterization of isomorphisms specifically for abstract simplicial complexes.

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More Reading: see Wikipedia on abstract simplicial complexes.