Practice: Kan extensions

Kan extensions

Warm-upHuman

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 $(\mathrm{Lan}_k f)(d)$ ?

Choose one

\bigvee \{\, f(c) \mid k(c) \le d \,\}

\bigwedge \{\, f(c) \mid d \le k(c) \,\}

\bigvee \{\, f(c) \mid d \le k(c) \,\}

f(c)

for the unique

c

with

k(c) = d

Kan extensions

Warm-upHuman

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?

Kan extensions

CoreHuman

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?

Kan extensions

CoreHuman

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?

Choose one

Naturality of

\alpha

, together with

f' \circ K(u) = f

The universal property of the colimit

(\mathrm{Lan}_K F)(d)

The triangle identities of an adjunction

K

is fully faithful

Kan extensions

CoreHuman

Which situation gives a functor $F$ with no left Kan extension along $K$ ?

Choose one

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\}

\mathcal{C}

small,

\mathcal{D}

locally small,

\mathcal{E} = \mathbf{Set}

C, D

posets and

E

a complete lattice

K = \mathrm{id}_{\mathcal{C}}

Kan extensions

CoreHuman

Which of these are Kan extensions? (Choose all that apply.)

Choose all that apply

The limit of

F : J \to \mathcal{C}

, as a functor

\mathbf{1} \to \mathcal{C}

A left adjoint

F

of

G : \mathcal{A} \to \mathcal{X}

The floor function

\mathbb{R} \to \mathbb{Z} \cup \{\pm\infty\}

Every functor

G : \mathcal{D} \to \mathcal{E}

is the left Kan extension of

G \circ K

along

K

, for every

K

Kan extensions

StretchHuman

Suppose $F \dashv G$ with $G : \mathcal{A} \to \mathcal{X}$ , unit $\eta$ and counit $\varepsilon$ . Which statement is correct?

Choose one

(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}

(F, \eta)

is a left Kan extension of

\mathrm{id}_{\mathcal{X}}

along

G

(F, \varepsilon)

is a right Kan extension of

\mathrm{id}_{\mathcal{A}}

along

G

, but only if

\mathcal{A}

is completeLeft adjoints are left Kan extensions and right adjoints are right Kan extensions

Kan extensions

CoreFormal

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 $E$ )?

Choose one

GaloisConnection (fun f : C →o E => (⟨lan k f, lan_monotone k f⟩ : D →o E)) (fun g : D →o E => g.comp k)GaloisConnection (fun g : D →o E => g.comp k) (fun f : C →o E => (⟨lan k f, lan_monotone k f⟩ : D →o E))∀ f : C → E, ∀ c, lan k f (k c) = f c∀ f : C → E, lan k f ≤ ran k f

Kan extensions

CoreComputational

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.

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