FINITE CATEGORIES

NOTES FOR THE REU: JULY 28 SCRIBE: WEI REN

1. Simplicial Sets and Chain Complexes

Recall that if we have a chain complex

\xymatrix{⋯ \ar[r] & C_{n+1} \ar[r]^{d_{n+1}} & C_{n} \ar[r]^{d_{n}} & C_{n−1} \ar[r] & ⋯} ,

we define the homology groups to be

H_{n} = \frac{Z_{n}}{B_{n}} ,

where the Z_{n} := ker(d_{n}) consists of cycles, and B_{n} := Im(d_{n+1}) is the group of boundaries. The condition d^{2} = 0, i.e. d_{n} ∘ d_{n+1} = 0, guarantees B_{n} ⊂ Z_{n} and this is well-defined.

Now suppose we have a simplicial set, {K_{n}}, with maps d_{i} and s_{i}. We define A_{n}(K) to be the free abelian group generated by K_{n}. In other words, the elements of A_{n}(K) are finite formal ℤ−linear combinations of elements of K_{n}:

A_{n}(K) = { ∑_{k_{i}∈K_{n}} n_{i}k_{i}, n_{i} ∈ ℤ}.

The maps d_{i} : K_{n} → K_{n−1} naturally extend to A_{n}(K) by linearity (that is, we define d_{i}(∑ n_{i}k_{i}) = ∑ n_{i}d(k_{i})). Now we can define d : A_{n}(K) → A_{n−1}(K) by

d := ∑_{i=0}^{n}(−1)^{i}d_{i}.

This map makes the sequence {A_{n}(K)} into a chain complex, because

d ∘ d = ∑_{i=0}^{n−1} ∑_{j=0}^{n}(−1)^{i+j}d_{i} ∘ d_{j},

and the identity d_{i} ∘ d_{j} = d_{j−1} ∘ d_{i}, i < j ensures that d ∘ d = 0.

We can now define C_{n}(A) := A_{n}(K)/D_{n}(K), where D_{n}(K) is the degenerate subgroup that consists of all the degenerate elements, which are finite linear combinations of elements of the form s_{i}(y). Note that commutation relations between d_{i} and s_{i} ensure that the map d : A_{n}(K) → A_{n−1}(K) descends to the quotient, and we get a map d : C_{n}(K) → C_{n−1}(K). This is equivalent to saying that the maps that we obtain by restriction, d : D_{n}(K) → A_{n−1}(K), is in fact a map into the degenerate subgroup D_{n−1}(K), and therefore the induced map on the quotient groups C_{n}(K) → C_{n−1}(K) is well-defined.

Yet another way of stating this fact is to say that the groups D_{n}(K) form a subchain complex, with maps d obtained by restriction.

Exercise 1.1. Verify the claims made above. Also check that H_{n}(D_{∗}(K)) is trivial for all n. We will see later that this implies we have an isomorphism: H_{∗}(A_{∗}(K)) ≅ H_{∗}(C_{∗}(K)).

Now recall that we have defined the singular complex. For any topology space X, S_{n} is the set of all continuous maps {f : Δ_{n} → X}, where Δ_{n} is the standard n−simplex. The sets {S_{n}} naturally come with the operators d_{i} and s_{i}, by precomposing with the maps δ_{i} : Δ_{n−1} → Δ_{n} and σ_{i} : Δ_{n+1} → Δ_{n}, which we have defined previously. We define d_{i} : S_{n} → S_{n−1} by f ↦ f ∘ δ_{i}, and s_{i} : S_{n} → S_{n+1} by f ↦ f ∘ σ_{i}. We can check that the commutation relations are satisfied, and thus the sets {S_{n}} form a simplicial set, and we have in fact defined a functor from the category of topological spaces into the category of simplicial sets.

Now we have a sequence of functors, all of which can be shown to be homotopyinvariant:

\xymatrix{Spaces \ar[r]^{S_{∗}} & Simplicial Sets \ar[r] & Simplicial Abelian Groups \ar[r] & Chain Complexes} .

This sequence shows homotopic maps induce induce identical maps on homology.

2. Homology

To better understand homology, we consider the following sequnece of abelian groups:

\xymatrix{0 \ar[r] & A \ar[r]^{f} & B \ar[r]^{g} & C \ar[r] & 0}.

If we require g ∘ f = 0, i.e., Im(f) ⊂ ker(g), then we have a chain complex and homology groups are defined.

We say the sequence is exact (at B) if we have ker(g) = Im(f). This condition is to equivalent to saying the homology group at B is trivial, and thus homology groups of a chain complex measures the “non-exactness” of the complex.

Exercise 2.1. Check that the condition that the above complex is exact at A is equivalent to f being injective, and the complex being exact at C is equivalent to g being surjective.

If a sequence of abelian groups is exact at every group, then we say it is an exact sequence. One particularly useful sequence is the one of the form presented above, and if it is exact then it is known as a short exact sequence.

Recall that the chain complexes form a category, where a map f : C_{∗} → D_{∗} between two complexes is defined to be a sequence of maps {f_{i} : C_{i} → D_{i}}, such that the following diagrams commute:

\xymatrix{C_{i} \ar[r]^{d} \ar[d]^{f_{i}} & C_{i−1} \ar[d]^{f_{i−1}} \\ D_{i} \ar[r]^{d} & D_{i−1}.}

We say the sequence of chain complexes

\xymatrix{0 \ar[r] & C′_{∗} \ar[r]^{f} & C_{∗} \ar[r]^{g} & C′′_{∗} \ar[r] & 0}

is exact if in the following (big) commutative diagram, each column is a short exact sequence (of abelian groups):

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ ⋯ \ar[r] & C′_{n+1} \ar[r] \ar[d]^{f_{n+1}} & C′_{n} \ar[r] \ar[d]^{f_{n}} & C′_{n−1} \ar[r] \ar[d]^{f_{n−1}} & ⋯ \\ ⋯ \ar[r] & C_{n+1} \ar[r] \ar[d]^{g_{n+1}} & C_{n} \ar[r] \ar[d]^{g_{n}} & C_{n−1} \ar[r] \ar[d]^{g_{n−1}} & ⋯ \\ ⋯ \ar[r] & C′′_{n+1} \ar[r] \ar[d] & C′′_{n} \ar[r] \ar[d] & C′′_{n−1} \ar[r] \ar[d] & ⋯ \\  & 0 & 0 & 0}

The most fundamental result of homological algebra tells us there is a nice relation between the homology groups in this case.

Theorem 2.2. Suppose we have a short exact sequence of chain complexes,

\xymatrix{0 \ar[r] & C′_{∗} \ar[r]^{f} & C_{∗} \ar[r]^{g} & C′′_{∗} \ar[r] & 0}.

Then the following sequence is exact:

\xymatrix{⋯ \ar[r] & H_{n+1}(C′′) \ar[r]^{δ} & H_{n}(C′) \ar[r]^{f_{∗}} & H_{n}(C) \ar[r]^{g_{∗}} & H_{n}(C′′) \ar[r]^{δ} & H_{n−1}(C′) \ar[r] & ⋯,}

where f_{∗}, g_{∗} are the induced maps, and δ is a mysterious (yet well-defined) map known as the connecting homomorphism.

This sequence of homology groups is a long exact sequence. As we will see later, this long exact sequence will enable us to compute the homology groups of various familiar topological spaces, for example the n−sphere.