FINITE CATEGORIES: JULY 26

NOTES FOR THE REU

In the previous lecture, abstract simplicial complexes were introduced. Since they are essential to this lecture as well, the definition is repeated.

Definition 1.1. An abstract simplicial complex K is a set of vertices V (K) and a set of nonempty subsets of V (K), called simplices, such that

(1) Every vertex is in some simplex.

(2) Every subset of a simplex is a simplex..

1.1. Background on Topological Spaces. An earlier handout gives an overview of topological spaces, but a few concepts are reiterated here for review.

The spaces about which we have the most intuition, and therefore think about the most, are Euclidean spaces ℝ^{n} and subsets of Euclidean spaces. The Euclidean space ℝ^{n} has a distance function

\begin{matrix}d : ℝ^{n} × ℝ^{n} → ℝ \\ d(x, y) = \sqrt{∑(x_{i} − y_{i})^{2}}. \end{matrix}

Then define U_{x,ϵ}, the ϵ-ball centered at x, to be {y ∈ ℝ^{n} : d(x, y) < ϵ}. The open sets of ℝ^{n} are all the finite intersections and arbitrary unions of these sets for any choice of x and ϵ.

The concept of a topological space generalizes Euclidean spaces:

Definition 1.2. A topological space is a set X and a set 𝒰 of subsets of X, called the open sets of X, such that ∅ and X are in 𝒰 and finite intersections and arbitrary unions of elements of 𝒰 are in 𝒰 .

This definition is much more general than Euclidean space. For example, an arbitrary topological space does not necessarily satisfy the following familiar separation property of ℝ^{n}: for all distinct x, y ∈ X, there exist open sets U and V such that x ∈ U, y ∈ V, and U ∩ V = ∅.

A set function f : X → Y from one topological space X to another Y may not take nearby points in X to nearby points in Y, so we need a restriction on the types of maps between topological spaces. These maps are called continuous, and in the case of metric spaces they are the ones which satisfy the usual ϵ-δ definition: f is continous if for all x ∈ X and all ϵ > 0, there is a δ > 0 such that f(U_{x,δ}) ⊆ U_{f(x),ϵ}. The generalized definition for arbitrary topological spaces, which is equivalent to the ϵ-δ definition in the case of metric spaces, is

Definition 1.3. A function f : X → Y from a topological space X to a topological space Y is continuous if f^{−1}(V) is open in X for all open V in Y.

1.2. Simplicial Sets. Even though a topological space is essentially a geometric object, we can study them by assigning algebraic objects to them. Our next step is to develop the tools to investigate topological spaces algebraically.

Suppose we totally order the set of vertices [n] so that it makes sense to refer to the ith vertex. Let K_{n} be the set of n + 1 point simplices allowing redundancies. Define functions for 0 ≤ i ≤ n

d_{i} : K_{n} → K_{n−1}  delete ith entry
s_{i} : K_{n} → K_{n+1}  repeat ith entry.

Recall that a category 𝒞 defines a sequence of sets and functions by 𝒞_{0} = Obj(𝒞) and 𝒞_{n} = {(f_{n}, …, f_{1}) | the f_{i} are composable }. Then define

d_{0}(f_{n}, …, f_{1}) = (f_{n}, …, f_{2})
d_{i}(f_{n}, …, f_{1}) = (f_{n}, …, f_{i+1} ∘ f_{i}, …, f_{1})
d_{n}(f_{n}, …, f_{1}) = (f_{n−1}, …, f_{1})
s_{i}(f_{n}, …, f_{1}) = (f_{n}, …, f_{i+1}, id, f_{i}, …, f_{1})

Note that in both cases, these functions satisfy the following relations:

\begin{aligned}d_{i}d_{j} & = d_{j−1}d_{i} i < j \\ s_{i}s_{j} & = s_{j+1}s_{i} i ≤ j \\ d_{i}s_{j} & = \{ \begin{matrix}s_{j−1}d_{i} & i < j \\ id & i = j, j + 1 \\ s_{j}d_{i−1} & i > j + 1. \end{matrix} \end{aligned}

Define a category Δ whose objects are the sets \underline{n} = \{0, 1, …, n\} and whose morphisms τ : \underline{m} → \underline{n} are monotonic nondecreasing functions. That is, if i < j, then τ(i) ≤ τ(j). For example, there are morphisms δ_{i} : \underline{n} → \underline{n + 1} defined by

δ_{i}(j) = \{ \begin{matrix}j & j < i \\ j + 1 & j ≥ i. \end{matrix}

The δ_{i} increment every element greater than or equal to i by one. There are also morphisms σ_{i} : \underline{n + 1} → \underline{n} defined by

σ_{i}(j) = \{ \begin{matrix}j & j ≤ i \\ j − 1 & j > i. \end{matrix}

Exercise 1.4. Every morphism in Δ is a composite of δ_{i}s and σ_{i}s.

In these terms, a simplicial set is a contravariant functor from Δ to Sets. Note the contravariance switches the relations around. Moreover, by the exercise, one need not define the functor on all morphisms of Δ, but only on the δ_{i} and σ_{i}, which we have already done for abstract simplicial complexes and for categories. That is, we have already defined a functor from abstract simpicial complexes to simplicial sets, K → K_{∗}, and from (small) categories to simplicial sets, 𝒞 → 𝒞_{∗}. Now we would like to define a functor from topological spaces to simplicial sets, X → S_{∗}X.

Let S_{n}(X) be the set of continous functions Δ_{n} → X. If τ : \underline{m} → \underline{n} is a morphism in Δ, then τ also determines a continuous map Δ_{m} → Δ_{n}. Thus for each continous f : Δ_{n} → X, define t(f) = f ∘ τ : Δ_{m} → X, which is also a continous map. Then t maps S_{n}(X) to S_{m}(X). Next time we will see how the sets K_{∗}, 𝒞_{∗}, or S_{∗}(X) lead to a sequence of abelian groups, which will be algebraic objects associated to the simplicial complex K, category 𝒞, or topological space X.