FINITE CATEGORIES

NOTES FOR THE REU: JULY 27 SCRIBE: JUSTIN NOEL

1. Simplicial Spaces

Informally, we can think of simplicial spaces as spaces built out of oriented lines, triangles, tetrahedra...

2. Homology of simplicial complexes.

Let K_{n} be the set of n + 1 point simplices. Recall that the free abelian group ℤ[K_{n}] consists of formal sums of elements of K_{n} (i.e. ∑_{i=1}^{n} k_{i}σ_{i} ∈ ℤ[K_{n}] where σ_{i} ∈ K_{n} and k_{i} ∈ ℤ for each i). We also recall that we were given maps d_{i} : K_{n} → K_{n−1} and s_{i} : K_{n} → K_{n+1} for i = 0 … n, that satisfy certain relations. Using these maps we define d : ℤ[K_{n}] → ℤ[K_{n−1}] as d(σ) = ∑_{i=0}^{n}(−1)^{i}d_{i}(σ).

We compute

(dd)(σ) = d(∑_{i=0}^{n}(−1)^{i}d_{i}(σ))
= ∑_{0≤j<n} ∑_{0≤i≤n}(−1^{i+j})d_{j}d_{i}(σ)
= ∑_{0≤j<i≤n}(−1)^{i+j}d_{j}d_{i}(σ) + ∑_{0≤i≤j<n}(−1)^{i+j}d_{j}d_{i}(σ)
= − ∑_{0≤j<i≤n}(−1)^{i+j}d_{i−1}d_{j}(σ) + ∑_{0≤i≤j<n}(−1)^{i+j}d_{j}d_{i}(σ)
= − ∑_{0≤j≤i<n}(−1)^{i+j}d_{i}d_{j}(σ) + ∑_{0≤i≤j<n}(−1)^{i+j}d_{j}d_{i}(σ)
= 0

Using this we define B_{n}(K) = d(ℤ[K_{n+1}]) and Z_{n}(K) = {σ ∈ ℤ[K_{n}] | d(σ) = 0}. The nth homology group is defined to be H_{n}(K) = Z_{n}(K)/B_{n}(K).

Example 2.1. If we let K be the triangle with ordered points a < b < c and edges f, connecting a to b, g connecting b to c and h connecting a to c. We can compute H_{0}(K) = H_{1}(K) = ℤ. Since d(ℤ[K_{0}]) is by definition 0 we see that Z_{0}(K) = ℤ^{3} = ℤa ⊕ ℤb ⊕ ℤc. We have that d(f) = b − a, d(g) = c − b and d(h) = c − a = d(f) + d(h). So B_{0}(K) = d(ℤ[K_{1}]) = ℤ(b − a) ⊕ ℤ(c − b). Rewriting Z_{0}(K) as ℤ(b − a) ⊕ ℤ(c − b) ⊕ ℤb we see that H_{0}(K) = ℤ. Since there are no 2simplices we have B_{1}(K) = 0 which implies H_{1}(K) = Z_{1}(K) = ℤ(h − (f + g)) = ℤ.

3. The functoriality of homology.

Suppose we have a map of chain complexes f : C_{∗} → C′_{∗}. That is we have that df_{n} = f_{n}d. We see that if x ∈ Z_{n}(C) then df_{n}(x) = f_{n}(dx) = f_{n}(0) = 0 so f(x) ∈ Z_{n}(C′). And we see that if x ∈ B_{n}(C), so x = dy for some y ∈ C_{n+1}, then f(x) = f(dy) = d(f(y)) and we have f(x) ∈ B_{n}(C′). These two facts imply that we have a well defined map H_{n}(C) → H_{n}(C′).

4. The homotopy invariance of homology

We return to one of our favorite simplicial complexes I. Recall I has two 0simplices, I_{0} = {[0], [1]} , and one 1-simplex, I_{1} = {[I]} . Let K be a simplicial space, we can check that ℤ[(K × I)_{n}] = ℤ[K_{n} × I_{n}] = ℤ[K_{n}] ⊗ ℤ[I_{n}]. The n-chains C_{n}(K ×I) then are given by ℤ[K_{n}]⊗[0]⊕ℤ[K_{n}]⊗[1]⊕ℤ[K_{n−1}]⊗[I]. For x ∈ C_{n}(K) we set

d(x ⊗ [0]) = d(x) ⊗ [0]
d(x ⊗ [1]) = d(x) ⊗ [1]
d(x ⊗ [I]) = d(x) ⊗ [I] + (−1)^{n}x ⊗ [1] − (−1)^{n}x ⊗ [0].

Now we see two chain maps f, g : C → C′ are homotopic if there exists a map h : C ⊗ I → C′ such that h(x ⊗ [0]) = f(x ⊗ [0]) and h(x ⊗ [1]) = g(x ⊗ [1]) for all x ∈ C.

Theorem 4.1. If f and g are homotopic then f_{∗} = g_{∗}.