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The Lumenary · Advanced undergraduate

Adjunctions

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You will learn to

Define an adjunction $F \dashv G$ by natural bijections $\Hom(FA, B) \cong \Hom(A, GB)$ , and verify adjunctions such as free monoid $\dashv$ forgetful, $- \times B \dashv (-)^B$ in $\cat{Set}$ and the tensor–hom adjunction
Pass between the hom-set and the unit–counit definitions of an adjunction using the triangle identities
Recognise Galois connections as adjunctions between posets and use them (floor and ceiling, image and preimage)
Prove that adjoints are unique up to unique isomorphism and that right adjoints preserve limits and left adjoints preserve colimits
Use preservation of limits and colimits to show that some functors have no adjoint, and explain why an adjunction need not be an equivalence

1.
For the free group functor $F : \cat{Set} \to \cat{Grp}$ and the forgetful functor $U : \cat{Grp} \to \cat{Set}$ , which statement is correct?
- A. $F \dashv U$ : $\Hom_{\cat{Grp}}(F S, G) \cong \Hom_{\cat{Set}}(S, U G)$
- B. $U \dashv F$ : $\Hom_{\cat{Set}}(U G, S) \cong \Hom_{\cat{Grp}}(G, F S)$
- C. $F$ and $U$ are inverse equivalences
- D.Neither is adjoint to the other
2.
Let $|A| = 2$ , $|B| = 3$ and $|C| = 2$ . How many functions $A \to C^B$ are there (where $C^B$ is the set of functions $B \to C$ )? Check your answer against $|\Hom(A \times B, C)|$ .
3.
On $(\Z, \le)$ let $l(n) = 2n$ . Its right adjoint $u$ satisfies $2n \le m \iff n \le u(m)$ for all integers $n, m$ . What is $u(7)$ ? (And what is $u(-7)$ , for yourself?)
4.
Given an adjunction $\varphi : \Hom(FA, B) \cong \Hom(A, GB)$ , the unit is $\eta_A = \varphi(\mathrm{id}_{FA})$ . Which formula gives the counit $\varepsilon_B : FGB \to B$ ?
- A. $\varepsilon_B = \varphi^{-1}(\mathrm{id}_{GB})$
- B. $\varepsilon_B = \varphi(\mathrm{id}_{B})$
- C. $\varepsilon_B = G(\eta_B)$
- D. $\varepsilon_B = \eta_{GB}^{-1}$
5.
Why does the forgetful functor $U : \cat{Field} \to \cat{Set}$ have no left adjoint?
- A.A left adjoint would preserve the initial object, so it would send $\varnothing$ to an initial field, and $\cat{Field}$ has no initial object
- B.Because $U$ is not faithful
- C.Because fields are infinite
- D.Because $U$ does not reflect isomorphisms
6.
Abelianisation $G \mapsto G^{\mathrm{ab}}$ is left adjoint to the inclusion $\iota : \cat{Ab} \hookrightarrow \cat{Grp}$ . Which statements follow from “right adjoints preserve limits, left adjoints preserve colimits”? Select all that apply.
- A.A product of abelian groups, computed in $\cat{Grp}$ , is abelian (and is the product in $\cat{Ab}$ ).
- B. $(G * H)^{\mathrm{ab}} \cong G^{\mathrm{ab}} \oplus H^{\mathrm{ab}}$ .
- C.The inclusion $\iota$ preserves coproducts: $\Z \oplus \Z$ is the coproduct of $\Z$ and $\Z$ in $\cat{Grp}$ .
- D.Abelianisation preserves all products: $(\prod_i G_i)^{\mathrm{ab}} \cong \prod_i G_i^{\mathrm{ab}}$ for every family of groups.
7.
In proving that unit and counit with the triangle identities give an adjunction, we show $\varepsilon_B \circ F(Gk \circ \eta_A) = k$ for $k : FA \to B$ . The computation is $\varepsilon_B \circ FGk \circ F\eta_A = k \circ \varepsilon_{FA} \circ F\eta_A = k$ . Which facts justify the two equalities, in order?
- A.Naturality of $\varepsilon$ at $k$ ; then the first triangle identity
- B.Naturality of $\eta$ ; then the second triangle identity
- C.Functoriality of $F$ twice
- D.The hom-set bijection, applied twice
8.
Mathlib defines GaloisConnection l u for maps l : α → β and u : β → α between preorders. Which proposition is it?
- A.∀ a b, l a ≤ b ↔ a ≤ u b
- B.∀ a, u (l a) = a
- C.∀ a b, a ≤ b ↔ l a ≤ l b
- D.∀ a b, l a ≤ b → a ≤ u b
9.
Write right_adjoint(l, P, Q, le_p, le_q) for a monotone map l (a dict) from a finite poset P to a finite poset Q. Return the dict u with l[a] ≤ b ⟺ a ≤ u[b] if it exists, using u[b] = max{a : l[a] ≤ b}, and None otherwise (including when that maximum does not exist).
Computational — Practice

Answer key — Adjunctions

1.
$F \dashv U$ : a homomorphism $F(S) \to G$ is determined, freely, by the images of the generators, i.e. by a function $S \to U(G)$ . The free functor is the left adjoint; its unit $S \to UF(S)$ inserts the generators.
2.
$|C^B| = 2^3 = 8$ , so $|\Hom(A, C^B)| = 8^2 = 64$ . And $|\Hom(A \times B, C)| = 2^6 = 64$ : the exponential law $(C^B)^A = C^{A \times B}$ is the cardinality shadow of the currying adjunction.
3.
$u(m) = \max\{n : 2n \le m\} = \lfloor m/2 \rfloor$ , so $u(7) = 3$ (and $u(-7) = -4$ ). Check: $2n \le 7 \iff n \le 3.5 \iff n \le 3$ . The left adjoint of $l$ would have to satisfy $\lambda(m) \le n \iff m \le 2n$ , giving $\lambda(m) = \lceil m/2 \rceil$ .
4.
$\varepsilon_B = \varphi^{-1}(\mathrm{id}_{GB}) : FGB \to B$ , the transpose of the identity of $GB$ — just as $\eta_A$ is the transpose of the identity of $FA$ . For currying, $\varepsilon$ is evaluation $C^B \times B \to C$ , the uncurried form of $\mathrm{id}_{C^B}$ .
5.
If $F \dashv U$ , then $\Hom_{\cat{Field}}(F\varnothing, K) \cong \Hom_{\cat{Set}}(\varnothing, UK)$ is a one-element set for every field $K$ , so $F\varnothing$ would be initial. But ring homomorphisms between fields preserve the characteristic, so no field maps to both $\mathbb{F}_2$ and $\mathbb{F}_3$ . There is no free field on any set. Alternatively, use limits: $\mathbb{F}_p$ with both projections equal to the identity is a product of $\mathbb{F}_p$ with itself in $\cat{Field}$ (every field mapping to $\mathbb{F}_p$ is $\mathbb{F}_p$ , with only the identity map), but $U$ sends it to a $p$ -element set rather than $\mathbb{F}_p \times \mathbb{F}_p$ ; so $U$ does not preserve limits and cannot be a right adjoint.
6.
$\iota$ is the right adjoint, so it preserves limits: products of abelian groups in $\cat{Grp}$ are abelian. Abelianisation is the left adjoint, so it preserves colimits: $(G * H)^{\mathrm{ab}} \cong G^{\mathrm{ab}} \oplus H^{\mathrm{ab}}$ . Neither principle says anything about $\iota$ and colimits (it does not preserve coproducts) or about abelianisation and limits.
7.
The naturality square of $\varepsilon$ at $k : FA \to B$ reads $\varepsilon_B \circ FG(k) = k \circ \varepsilon_{FA}$ ; the first triangle identity $\varepsilon_{FA} \circ F\eta_A = \mathrm{id}_{FA}$ finishes. The symmetric computation uses naturality of $\eta$ and the second triangle identity.
8.
GaloisConnection l u := ∀ a b, l a ≤ b ↔ a ≤ u b. The unit and counit are the derived inequalities a ≤ u (l a) and l (u b) ≤ b (Mathlib: GaloisConnection.le_u_l, GaloisConnection.l_u_le), and monotonicity of l and u follows.

def right_adjoint(l, P, Q, le_p, le_q):
    u = {}
    for b in Q:
        below = [a for a in P if le_q(l[a], b)]
        top = [a for a in below if all(le_p(x, a) for x in below)]
        if not top:
            return None
        u[b] = top[0]
    if all(le_q(l[a], b) == le_p(a, u[b]) for a in P for b in Q):
        return u
    return None

For l(a) = 2a the right adjoint is $u(b) = \lfloor b/2 \rfloor$ (capped at the top of $P$ ). The constant map $3$ has none: $\{a : 3 \le b\}$ is empty for $b < 3$ , reflecting that $l$ does not preserve the least element — left adjoints preserve all joins, including the empty one.