From Vector Spaces to Categories. Part 6.

We began by thinking of categories as “posets with extra arrows”. This analogy gives excellent intuition for the general facts about adjoint functors. However, our intuition from posets is insufficient to actually prove anything about adjoint functors.

To complete the proofs we will switch to a new analogy between categories and vector spaces. Let V be a vector space over a field K and let V ∗ be the dual space consisting of K-linear functions V → K. Now consider any K-bilinear function ⟨−,−⟩ ∶ V × V → K. We say that the function ⟨−,−⟩ is non-degenerate in both coordinates if we have

⟨u₁,v⟩ = ⟨u₂,v⟩ ∀ v ∈ V ⇒ u₁ = u₂, ⟨u,v₁⟩ = ⟨u,v₂⟩ ∀ u ∈ V ⇒ v₁ = v₂

We say that two K-linear operators L ∶ V ⇄ V ∶ R define an adjunction with respect to ⟨−, −⟩ if, ∀ vectors u,v ∈ V, we have

⟨u, R(v)⟩ = ⟨L(u), v⟩

Uniqueness of Adjoint Operators. Let L ⊣ R be an adjoint pair of operators with respect to a non-degenerate bilinear function ⟨−, −⟩ ∶ V × V → K. Then each of L and R determines

the other uniquely.

Proof: To show that R determines L, suppose that L′ ⊣ R is another adjoint pair. Thus, ∀ vectors u,v ∈ V we have

⟨L(u), v⟩ = ⟨u, R(v)⟩ = ⟨L′(u), v⟩

Now consider any vector u ∈ V. The non-degeneracy of ⟨−, −⟩ tells us that

⟨L(u), v⟩ = ⟨L′(u), v⟩ ∀ v ∈ V ⇒ L(u) = L′(u)

and since this is true ∀ u ∈ V we conclude that L = L′

RAPL for Operators:

Suppose that the function ⟨−, −⟩ ∶ V × V → K is non-degenerate and continuous. Now let T ∶ V → V be any linear operator. If T has a left or a right adjoint, then T is continuous.

Proof:

Suppose that T ∶ V → V has a left adjoint L ⊣ T, and suppose that the sequence of vectors v_i ∈ V has a limit lim_iv_i ∈ V. Furthermore, suppose that the limit lim_iT(v_i) ∈ V exists. Then for each u ∈ V, the continuity of ⟨−, −⟩ in the second coordinate tells us that

⟨u, T (lim_iv_i)⟩ = ⟨L(u), lim_iv_i⟩

= lim_i⟨L(u), v_i⟩

= lim_i⟨u,T(v_i)⟩

= ⟨u, lim_iT (v_i)⟩

Since this is true for all u ∈ V, non-degeneracy gives

T (lim_iv_i) = lim_iT (v_i)

The theorem can be made rigorous if we work with topological vector spaces. If (V, ∥ − ∥) is a normed (real or complex) vector space, then an operator T ∶ V → V is bounded if and only if it is continuous. Furthermore, if (V,⟨−,−⟩) is a Hilbert space then an operator T ∶ V → V having an adjoint is necessarily bounded, hence continuous. Many theorems of category have direct analogues in functional analysis. After all, Grothendieck began as a functional analyst.

We can summarize these two results as follows. Let ⟨−,−⟩ ∶ V ×V → K be a K-bilinear function. Then for each vector v ∈ V we have two elements of the dual space H^v, H_v ∈ V^∗ defined by

H^v ∶= ⟨v,−⟩ ∶ V → K,

H_v ∶= ⟨−,v⟩ ∶ V → K

The mappings v ↦ H^v and v ↦ H_v thus define two K-linear functions from V to V^∗ : H(−) ∶V → V^∗ and H(−) ∶ V → V^∗

Furthermore, if the function is ⟨−,−⟩ is non-degenerate and continuous then the functions H(−), H(−) ∶ V → V^∗ are both injective and continuous.

the hom bifunctor

Hom_C(−,−) ∶ C^op × C → Set behaves like a “non-degenerate and continuous bilinear function”……

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