Marching From Galois Connections to Adjunctions. Part 4.

To make the transition from Galois connections to adjoint functors we make a slight change of notation. The change is only cosmetic but it is very important for our intuition.

Definition of Poset Adjunction. Let (P, ≤P) and (Q, ≤Q) be posets. A pair of functions L ∶ P ⇄ Q ∶ R is called an adjunction if ∀ p ∈ P and q ∈ Q we have

p ≤P R(q) ⇐⇒ L(p) ≤Q q

In this case we write L ⊣ R and call this an adjoint pair of functions. The function L is the left adjoint and R is the right adjoint.

The only difference between Galois connections and poset adjunctions is that we have reversed the partial order on Q. To be precise, we define the opposite poset Qop with the same underlying set Q, such that for all q1 , q2 ∈ Q we have

q1Qop q2 ⇐⇒ q2Q q1

Then an adjunction P ⇄ Q is just the same thing as a Galois connection P ⇄ Qop.

However, this difference is important because it breaks the symmetry. It also prepares us for the notation of an adjunction between categories, where it is more common to use an “asymmetric pair of covariant functors” as opposed to a “symmetric pair of contravariant functors”.

Uniqueness of Adjoints for Posets: Let P and Q be posets and let L ∶ P ⇄ Q ∶ R be an adjunction. Then each of the adjoint functions L ⊣ R uniquely determines the other.

Proof: To prove that R determines L, suppose that L′ ∶ P ⇄ Q ∶ R is another adjunction. Then by definition of adjunction we have for all q ∈ Q that

L(p) ≤Q q ⇐⇒ p ≤P R(q) ⇐⇒ L′(p) ≤Q q

In particular, setting q = L(p) gives

L(p) ≤Q L(p) ⇒ L′(p) ≤Q L′(p)

and setting q = L′(p) gives

L′(p) ≤Q L(p) ⇒ L(p) ≤Q L′(p)

Then by the antisymmetry of Q we have L(p) = L′(p). Since this holds for all p ∈ P we conclude that L = L′, as desired.

RAPL Theorem for Posets. Let L ∶ P ⇄ Q ∶ R be an adjunction of posets. Then for all subsets S ⊆ P and T ⊆ Q we have

L (∨P S) = ∨Q L(S) and R (∧Q T) = ∧P R(T).

In words, this could be said as “left adjoints preserve join” and “right adjoints preserve meet”.

Proof: We just have to observe that sending Q to its opposite Qop switches the definitions of join and meet: Qop = ∧Q and Qop = ∨Q.

It seems worthwhile to emphasize the new terminology with a picture. Suppose that the posets P and Q have top and bottom elements: 1P , 0P ∈ P and 1Q, 0Q ∈ Q. Then a poset adjunction L ∶ P ⇄ Q ∶ R looks like this:

img_20170204_163208

In this case RL ∶ P → P is a closure operator as before, but now LR ∶ Q → Q is called an interior operator. From the case of Galois connections we also know that LRL = L and RLR = R. Since bottom elements are colmits and top elements are limits, the identities L(0P ) = 0Q and R(1Q) = 1P are special cases of the RAPL Theorem.

Just as with Galois connections, adjunctions between the Boolean lattices 2U and 2V are in bijection with relations ∼ ⊆ U × V, but this time we will view the relation as a function f ∼ ∶ U → 2V that sends each to the set f ∼ (u)∶= {v∈V ∶ u∼v}. We can also think off as a “multi-valued function” from U to V.

Adjunctions of Boolean Lattices: Let U,V be sets and consider an arbitrary function f ∶ U → 2V. Then subsets S ∈ 2U and T ∈ 2V we define

L(S) ∶= ∪s∈S f(s) ∈ 2V,

R(T) ∶= {u∈U ∶ f(u) ⊆ T} ∈ 2U

The pair of functions Lf ∶ 2U ⇄ 2V ∶ Rf is an adjunction of Boolean lattices. To see this, note  S ∈ 2U and T ∈ 2V

S ⊆ Rf (T) ⇐⇒ ∀ s∈S, s ∈ R(T)

⇐⇒ ∀ s∈S, f(s) ⊆ T

⇐⇒ ∪s∈S f(s) ⊆ T

⇐⇒ L(S) ⊆ T

Functions : Let f ∶ U → V be any function. We can extend this to a function f ∶ U → 2V by defining f(u) ∶= {f(u)} ∀ u ∈ U. In this case we denote the corresponding left and right adjoint functions by f ∶= Lf ∶ 2U → 2V and f−1 ∶= Rf ∶ 2V → 2U, so that ∀ S ∈ 2U and T ∈ 2V we have

f(S) = {f(s) ∶ s ∈ S}, f−1(T)={u∈U ∶ f(s) ∈ T}

The resulting adjunction f ∶ 2U ⇄ 2V ∶ f−1 is called the image and preimage of the function. It follows from RAPL that image preserves unions and preimage preserves intersections.

But now something surprising happens. We can restrict the preimage f−1 ∶ 2V → 2U to a function f−1 ∶ V → 2U by defining f−1(v) ∶= f−1({v}) for each v ∈ V. Then since f−1 = Lf−1 we obtain another adjunction

f−1 ∶ 2V ⇄ 2U ∶ Rf−1,
where this time f−1 is the left adjoint. The new right adjoint is defined for each S ∈ 2U by

R f−1(S) = {v∈V ∶ f−1(v) ⊆ S}

There seems to be no standard notation for this function, but people call it f! ∶= Rf−1 (the “!” is pronounced “shriek”). In summary, each function f ∶ U → V determines a triple of

adjoints f ⊣ f−1 ⊣ f! where f preserves unions, f! preserves intersections, and f−1 preserves both unions and intersections. Logicians will tell you that the functions f and f! are closely related to the existential (∃) and universal (∀) quantifiers, in the sense that for all S ∈ 2U we have

f∗ (S) = {v∈V ∶ ∃ u ∈ f−1 (v), u ∈ S}, f(S)={v ∈ V ∶ ∀ u ∈ f−1(v), u ∈ S}

Group Homomorphisms: Given a group G we let (L (G), ⊆) denote its poset of subgroups. Since the intersection of subgroups is again a subgroup, we have ∧ = ∩. Then since L (G) has arbitrary meets it also has arbitrary joins. In particular, the join of two subgroups A, B ∈ L (G) is given by

A ∨ B = ⋂ {C ∈ L(G) ∶ A ⊆ C and B ⊆ C},

which is the smallest subgroup containing the union A ∪ B. Thus L (G) is a lattice, but since A ∨ B ≠ A ∪ B (in general) it is not a sublattice of 2G.

Now let φ ∶ G → H be an arbitrary group homomorphism. One can check that the image and preimage φ ∶ 2G ⇄ 2H ∶ φ−1 send subgroups to subgroups, hence they restrict to an adjunction between subgroup lattices:

φ ∶L(G) ⇄ L(H)∶ φ−1.

The function φ! ∶ 2G → 2H does not send subgroups to subgroups, and in general the function φ−1 ∶ L(H) → L(G) does not have a right adjoint. For all subgroups A ∈ L (G) and B ∈ L (H) one can check that

φ−1φ(A)=A ∨ ker φ and φφ−1(B) = B ∧ im φ

Thus the φ−1φ-fixed subgroups of G are precisely those that contain the kernel and the φφ−1-fixed subgroups of H are precisely those contained in the image. Finally, the Fundamental Theorem gives us an order-preserving bijection as in the following picture:

img_20170204_173156

…..

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