Practice: Functors · The Lumenary

Functors

Warm-upHuman

Which of these are functors? Choose all that apply.

Choose all that apply

\mathbf{Grp} \to \mathbf{Set}

: a group goes to its underlying set, a homomorphism to its underlying function.

\mathbf{Set} \to \mathbf{Set}

:

X

goes to the set of finite lists in

X

, and

f

to "apply

f

to every entry".

\mathbf{Grp} \to \mathbf{Grp}

:

G

goes to its centre

Z(G)

, and

f

to its restriction to the centre.Between two posets regarded as categories: a monotone map

p

, with

x \le y

sent to

p(x) \le p(y)

.

\mathbf{Set} \to \mathbf{Set}

:

X

goes to its power set, and

f : X \to Y

to the preimage map

T \mapsto f^{-1}(T)

.

Functors

Warm-upHuman

How many group homomorphisms are there from $\mathbb{Z}/2$ to $\mathbb{Z}/6$ ? (That is, what is $|\mathrm{Hom}(\mathbb{Z}/2, \mathbb{Z}/6)|$ ?)

Functors

CoreHuman

Which argument correctly proves that the groups $\mathbb{Z}/4$ and $\mathbb{Z}/2 \times \mathbb{Z}/2$ are not isomorphic?

Choose one

|\mathrm{Hom}(\mathbb{Z}/2, \mathbb{Z}/4)| = 2

but

|\mathrm{Hom}(\mathbb{Z}/2, \mathbb{Z}/2 \times \mathbb{Z}/2)| = 4

, and the functor

\mathrm{Hom}(\mathbb{Z}/2, -)

would send an isomorphism to a bijection.They have different orders.One is abelian and the other is not.The bijection

0 \mapsto (0,0)

,

1 \mapsto (1,0)

,

2 \mapsto (0,1)

,

3 \mapsto (1,1)

is not a homomorphism.

Functors

CoreHuman

For the contravariant power-set functor $\mathcal{P}^\ast$ (which sends $f : X \to Y$ to the preimage map $\mathcal{P}^\ast(f)(T) = f^{-1}(T)$ ), and functions $f : X \to Y$ , $g : Y \to Z$ , which equation holds for all $T \subseteq Z$ ?

Choose one

(g \circ f)^{-1}(T) = f^{-1}\big(g^{-1}(T)\big)

, i.e.

\mathcal{P}^\ast(g \circ f) = \mathcal{P}^\ast(f) \circ \mathcal{P}^\ast(g)

(g \circ f)^{-1}(T) = g^{-1}\big(f^{-1}(T)\big)

(g \circ f)^{-1}(T) = g\big(f(T)\big)

None — preimages do not compose.

Functors

CoreHuman

Which of the following are morphisms $f$ that are not isomorphisms, but whose image $F(f)$ under the indicated functor is an isomorphism? Choose all that apply.

Choose all that apply

The identity map from

\mathbb{R}

with the discrete topology to

\mathbb{R}

with the usual topology, and

F

the forgetful functor

\mathbf{Top} \to \mathbf{Set}

.The morphism

0 \to 1

in the poset

\{0 < 1\}

, and

F

the functor to

\mathbf{Set}

sending both objects to

\{\ast\}

.A bijective group homomorphism, and

F

the forgetful functor

\mathbf{Grp} \to \mathbf{Set}

.A matrix

A \in \mathrm{GL}_2(\mathbb{R})

with

\det A = 1

, regarded as a morphism of the one-object category

\mathrm{GL}_2(\mathbb{R})

, and

F = \det

.

Functors

CoreFormal

In Lean/Mathlib, which type is right for the preimage power-set functor, which sends $f : X \to Y$ to $T \mapsto f^{-1}(T)$ ?

Choose one

(Type u)ᵒᵖ ⥤ Type uType u ⥤ Type u(Type u)ᵒᵖ ⥤ (Type u)ᵒᵖType u → Type u

Functors

StretchHuman

In the proof that there is no retraction $r : D^2 \to S^1$ (with $r \circ i = \mathrm{id}_{S^1}$ ), which facts are used? Choose the most complete correct answer.

Choose one

Functoriality of

\pi_1

gives

\pi_1(r) \circ \pi_1(i) = \mathrm{id}

on

\pi_1(S^1) \cong \mathbb{Z}

; this identity would factor through

\pi_1(D^2) = 0

, which is impossible.

\pi_1

preserves isomorphisms, and

D^2

is not isomorphic to

S^1

.

D^2

is compact and

S^1

is not.

S^1

minus a point is connected, so

r

cannot be continuous.

Functors

CoreComputational

Functions between finite sets are stored as Python dicts. Write image(f, S) (the set $f(S)$ ) and preimage(f, T) (the set $f^{-1}(T)$ ), and compose(g, f) (first f, then g). The tests check the functor laws: images compose covariantly, $(g \circ f)(S) = g(f(S))$ , and preimages contravariantly, $(g \circ f)^{-1}(T) = f^{-1}(g^{-1}(T))$ .

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