ID	063b34e7-073e-4990-8321-f4b3101ece4f
DeertopiaVisibility	public
ROAM_REFS	[cite:@maclane2000categories]

Categories for the Working Mathematician

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Exercises

Exercise 1.4.1

Let $s$ be a fixed set, and $t^{s}$ the set of all functions $s \to t$ . Show that $t \mapsto t^{s}$ is the object restriction of a functor $𝐒 𝐞 𝐭 \to 𝐒 𝐞 𝐭$ , and that application $e_{t} : t^{s} \times s \Rightarrow t$ , defined by $e f x = f x$ , defines a natural transformation.

Let $F : 𝐒 𝐞 𝐭 \to 𝐒 𝐞 𝐭$ be the endofunctor defined on objects by $F x = x^{s}$ , and on morphisms by $F f = g \mapsto g f$ .

Functorality of $F$ comes easy: identity is preserved:

\begin{align*} F \id{x} &= g \mapsto g\id{x} \\ &= g \mapsto g, \end{align*} ParseError: Expected 'EOF', got '&' at position 25: …ign*} F \id{x} &̲= g \mapsto g\i…

and composition is respected:

\begin{matrix} a a u u u} : 3 \end{matrix}

Exercise 1.4.2

If $H$ is a fixed group, show that $G \mapsto H \times G$ defines a functor $H \times - : \Grp \to \Grp$ and that each morphism $f : H \to K$ defines a natural transformation $H \times - \Rightarrow K \times -$ .

The functor $H \times -$ is defined on objects by $G \mapsto H \times g$ and on morphisms by \phi \mapsto \id{H} \times \phi ParseError: Function "\id" is not trusted at position 14: \phi \mapsto \̲i̲d̲{H} \times \phi, lifting homomorphisms. into the right-hand side of the direct product and leaving $H$ unaffected. Identity and composition are preserved/respected:

\begin{tabular}{p{4cm}p{3cm}} {\begin{align*} & \left(H \times -\right) \id{G} \\ = &\id{H} \times \id{G} \\ = &\id{H \times G} \end{align*}} & {\begin{align*} & \left[\left(H\times -\right) \phi\right]\left[\left(H\times -\right) \psi \right] \\ = & \left(\id{H} \times \phi\right) \left(\id{H} \times \psi \right) \\ = & \left(\id{H} \times \phi\psi \right) \\ = & \left(H\times -\right)\left(\phi\psi\right). \end{align*}} \end{tabular} ParseError: No such environment: tabular at position 7: \begin{̲t̲a̲b̲u̲l̲a̲r̲}̲{p{4cm}p{3cm}} …

Given a group homomorphism $f : H \to K$ , the arrows \eta_G = f \times \id{G} ParseError: Unexpected end of input in a macro argument, expected '}' at end of input: …f \times \id{G} define a natural transformation.

Exercise 1.5.6

Show that all idempotents in $𝐒 𝐞 𝐭$ split.

Let $f : A \to A$ be an idempotent function. We will factor $f$ into two new functions whose composite is $f$ , but with a detour through $f$ 's image.

Restrict $f$ 's codomain to its image, yielding $g : A ↠ Image f$ . By definition of the image, every element has a fiber in $A$ , making $g$ a surjection.

Define $h : Image f ↣ A$ as the inclusion of $f$ 's image back into its codomain. Trivially an injection!!!!!!!

Now, $g$ , being a restriction of $f$ , behaves exactly like $f$ and differs only in signature. Similarly, the inclusion $h$ behaves exactly like the identity function, again differing only in signature. Composing them under this new light, we get $g h = f 1 = f$ .

Finally, we can substitute $g h = f$ into the equation for $f$ 's idempotence:

$g h = g h g h .$

As $g$ is surjective, it is also right-cancellable:

$h = h g h,$

and $h$ is a surjection, making it left-cancellable:

$1 = h g .$

quetzal ed deez or whatever.