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The Lumenary · First-year graduate

Kan extensions

Name: ______________________

Date: ______________________

You will learn to

Define left and right Kan extensions by their universal properties, and recognise the adjunction $\mathrm{Lan}_K \dashv K^* \dashv \mathrm{Ran}_K$
Prove the pointwise formula $(\mathrm{Lan}_K F)(d) = \operatorname{colim}_{(K \downarrow d)} F$ and state when it guarantees existence
Express limits, colimits and adjoint functors as Kan extensions, and prove Mac Lane's criterion for adjoints
Use left Kan extension along the Yoneda embedding to extend a functor from representables to all presheaves
Compute Kan extensions between posets and of finite set-valued functors, including induced and coinduced group actions

1.
Let $C$ and $D$ be posets, $E$ a complete lattice, $k : C \to D$ monotone and $f : C \to E$ monotone, all viewed as thin categories and functors. Which formula gives the pointwise left Kan extension $(\mathrm{Lan}_k f)(d)$ ?
- A. $\bigvee \{\, f(c) \mid k(c) \le d \,\}$
- B. $\bigwedge \{\, f(c) \mid d \le k(c) \,\}$
- C. $\bigvee \{\, f(c) \mid d \le k(c) \,\}$
- D. $f(c)$ for the unique $c$ with $k(c) = d$
2.
Let $K : \{a, b\} \to \mathbf{1}$ be the functor from the discrete category with two objects to the terminal category, and let $F : \{a, b\} \to \mathbf{Set}$ have $|F(a)| = 3$ and $|F(b)| = 4$ . How many elements does $(\mathrm{Lan}_K F)(\ast)$ have?
3.
Let $H$ be a subgroup of index $3$ in a finite group $G$ , let $i : \mathbf{B}H \to \mathbf{B}G$ be the inclusion of one-object categories, and let $X$ be an $H$ -set (a functor $\mathbf{B}H \to \mathbf{Set}$ ) with $|X| = 2$ . How many elements does the right Kan extension $(\mathrm{Ran}_i X)(\ast) = \mathrm{Hom}_H(G, X)$ have?
4.
In the proof that the pointwise formula gives a left Kan extension, given $G : \mathcal{D} \to \mathcal{E}$ and $\alpha : F \Rightarrow G K$ , we define $\sigma_d : (\mathrm{Lan}_K F)(d) \to G(d)$ by $\sigma_d \circ \iota_{(c,f)} = G(f) \circ \alpha_c$ . For this to make sense, the maps $G(f) \circ \alpha_c$ must form a cocone over $(K \downarrow d)$ : for $u : (c, f) \to (c', f')$ we need $G(f') \circ \alpha_{c'} \circ F(u) = G(f) \circ \alpha_c$ . Which fact gives this?
- A.Naturality of $\alpha$ , together with $f' \circ K(u) = f$
- B.The universal property of the colimit $(\mathrm{Lan}_K F)(d)$
- C.The triangle identities of an adjunction
- D. $K$ is fully faithful
5.
Which situation gives a functor $F$ with no left Kan extension along $K$ ?
- A. $K$ is the inclusion of the discrete poset $\{a, b\}$ into $\{a, b, \top\}$ (with $a, b \le \top$ ), and $F$ sends $a, b$ to the two elements of the discrete poset $\{x, y\}$
- B. $\mathcal{C}$ small, $\mathcal{D}$ locally small, $\mathcal{E} = \mathbf{Set}$
- C. $C, D$ posets and $E$ a complete lattice
- D. $K = \mathrm{id}_{\mathcal{C}}$
6.
Which of these are Kan extensions? (Choose all that apply.)
- A.The limit of $F : J \to \mathcal{C}$ , as a functor $\mathbf{1} \to \mathcal{C}$
- B.A left adjoint $F$ of $G : \mathcal{A} \to \mathcal{X}$
- C.The floor function $\mathbb{R} \to \mathbb{Z} \cup \{\pm\infty\}$
- D.Every functor $G : \mathcal{D} \to \mathcal{E}$ is the left Kan extension of $G \circ K$ along $K$ , for every $K$
7.
Suppose $F \dashv G$ with $G : \mathcal{A} \to \mathcal{X}$ , unit $\eta$ and counit $\varepsilon$ . Which statement is correct?
- A. $(F, \varepsilon)$ is a right Kan extension of $\mathrm{id}_{\mathcal{A}}$ along $G$ , and it is preserved by every functor out of $\mathcal{A}$
- B. $(F, \eta)$ is a left Kan extension of $\mathrm{id}_{\mathcal{X}}$ along $G$
- C. $(F, \varepsilon)$ is a right Kan extension of $\mathrm{id}_{\mathcal{A}}$ along $G$ , but only if $\mathcal{A}$ is complete
- D.Left adjoints are left Kan extensions and right adjoints are right Kan extensions
8.
In the lesson's Lean file, lan k f d = ⨆ (c : C) (_ : k c ≤ d), f c. Which Lean statement says that $\mathrm{Lan}_k$ is left adjoint to restriction along $k$ (for monotone $k : C \to D$ , on monotone maps into a complete lattice $E$ )?
- A.GaloisConnection (fun f : C →o E => (⟨lan k f, lan_monotone k f⟩ : D →o E)) (fun g : D →o E => g.comp k)
- B.GaloisConnection (fun g : D →o E => g.comp k) (fun f : C →o E => (⟨lan k f, lan_monotone k f⟩ : D →o E))
- C.∀ f : C → E, ∀ c, lan k f (k c) = f c
- D.∀ f : C → E, lan k f ≤ ran k f
9.
Let $C$ be a finite set of sample points in $\mathbb{R}$ , $k : C \hookrightarrow \mathbb{R}$ the inclusion and $f : C \to \mathbb{R} \cup \{\pm\infty\}$ any function, given as a dictionary {point: value}. Write lan_value(f, d) returning $(\mathrm{Lan}_k f)(d) = \sup\{ f(c) \mid c \le d \}$ (use float('-inf') for an empty supremum) and ran_value(f, d) returning $(\mathrm{Ran}_k f)(d) = \inf\{ f(c) \mid c \ge d \}$ (float('inf') if empty). Then check: when $f$ is not monotone, $\mathrm{Lan}_k f$ is its running maximum, the smallest monotone function above $f$ on the sample points.
Computational — Practice

Answer key — Kan extensions

1.
In a thin category an arrow $k(c) \to d$ exists exactly when $k(c) \le d$ , so the comma category $(k \downarrow d)$ is $\{c \mid k(c) \le d\}$ with the order of $C$ . A colimit in a complete lattice is the supremum of the values, so

$(\mathrm{Lan}_k f)(d) = \bigvee \{\, f(c) \mid k(c) \le d \,\}.$

Dually $(\mathrm{Ran}_k f)(d) = \bigwedge \{ f(c) \mid d \le k(c) \}$ .
2.
The comma category $(K \downarrow \ast)$ is the whole discrete category $\{a, b\}$ , so $(\mathrm{Lan}_K F)(\ast) = \operatorname{colim} F = F(a) \sqcup F(b)$ , which has $3 + 4 = 7$ elements. The right Kan extension is the product $F(a) \times F(b)$ , with $12$ elements.
3.
A compatible family satisfies $x_{hg} = h \cdot x_g$ , so it is determined by its value at one representative of each right coset $Hg$ , and these values can be chosen freely in $X$ . With $3$ cosets and $|X| = 2$ there are $2^3 = 8$ families. Compare the left Kan extension $G \times_H X$ , with $[G:H]\,|X| = 6$ elements. (For linear representations of a finite-index subgroup, induction and coinduction are isomorphic; for sets they are not.)
4.
Since $u$ is a morphism of $(K \downarrow d)$ , $f' \circ K(u) = f$ . By naturality of $\alpha$ , $\alpha_{c'} \circ F(u) = G(K(u)) \circ \alpha_c$ . Hence

$G(f') \circ \alpha_{c'} \circ F(u) = G(f') \circ G(K(u)) \circ \alpha_c = G(f) \circ \alpha_c.$

Only then does the universal property of the colimit give a unique $\sigma_d$ .
5.
In option (a), a functor $G : \{a, b, \top\} \to \{x, y\}$ with $F \Rightarrow G K$ needs $x \le G(a) \le G(\top)$ and $y \le G(b) \le G(\top)$ ; in the discrete poset $\{x, y\}$ that forces $G(\top) = x = y$ , impossible. So no extension exists and in particular no Kan extension. The pointwise formula fails because the join $x \vee y$ does not exist. In (b), (c) and (d) Kan extensions exist.
6.
Limits are right Kan extensions along $! : J \to \mathbf{1}$ ; a left adjoint $F$ of $G$ is $\mathrm{Ran}_G \mathrm{id}$ (Mac Lane's theorem, proved in the lesson); the floor function is $\mathrm{Lan}$ of the inclusion along $\mathbb{Z} \hookrightarrow \mathbb{R}$ by the poset formula. The last statement is false: for $K : \{a, b\} \to \mathbf{1}$ and $G(\ast) = \{0\}$ we get $\mathrm{Lan}_K(GK)(\ast) = \{0\} \sqcup \{0\}$ , with two elements.
7.
A right Kan extension of $\mathrm{id}_{\mathcal{A}}$ along $G$ is a functor $R : \mathcal{X} \to \mathcal{A}$ with a transformation $RG \Rightarrow \mathrm{id}_{\mathcal{A}}$ ; the counit $\varepsilon : FG \Rightarrow \mathrm{id}$ has exactly this shape. Given $H$ and $\beta : HG \Rightarrow P$ , the unique factorisation is $\sigma = \beta F \circ H\eta$ , and the triangle identities give existence and uniqueness for every $P$ (so the extension is absolute). Thus left adjoints are right Kan extensions — e.g. $\lceil x \rceil = \inf \{ n \mid x \le n \}$ .
8.
An adjunction between preorders is a Galois connection l a ≤ b ↔ a ≤ u b with l the left adjoint. Taking l = Lan_k and u = (· ∘ k) gives Lan_k f ≤ g ↔ f ≤ g ∘ k, which is the universal property lan_le_iff. So (a) is right: Lumenary.KanExtensions.lan_gc. Option (b) describes the dual property of ran, proved as le_ran_iff.
9.
```
def lan_value(f, d):
    return max((v for c, v in f.items() if c <= d), default=float('-inf'))


def ran_value(f, d):
    return min((v for c, v in f.items() if c >= d), default=float('inf'))
```
These are the pointwise formulas: the comma category $(k \downarrow d)$ is $\{ c \mid c \le d \}$ , and colimits in $\mathbb{R} \cup \{\pm\infty\}$ are suprema. For $f = \{0: 1, 2: 5, 5: 3\}$ , $\mathrm{Lan}_k f$ at $5$ is $5$ , not $f(5) = 3$ : the extension does not restrict back to $f$ because $f$ is not monotone (not a functor). For monotone $f$ , both extensions agree with $f$ at the sample points, because $k$ is an order embedding.