The objects of $\ca{Set}$ are sets and its arrows are functions.
For any set $A$ we identify $\idarr{A}$ with the identity function on $A$, $\id : A \to A$.
$\circ$ is indeed associative.

Each [poset](https://en.wikipedia.org/wiki/Partially_ordered_set) is a category.

The objects of each poset are its elements and we construct an arrow $\arr{a}{\leq}{b}$ for every pair of objects $(a, b)$ iff $a \leq b$.
Reflexivity ($a \leq a$) and transitivity ($a \leq b \land b \leq c \implies a \leq c$) provide a construction for $\idarr{a}$ and $\circ$, respectively.

Each [monoid](https://en.wikipedia.org/wiki/Monoid) is a category.

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The objects of each monoid are its elements and we construct for every pair of elements $(a, b)$ an arrow $\arr{a}{\cdot b}{a \cdot b}$.
The existence of an identity element and associativity, as required by the monoid structure, immediately provide us with a construction for $\id$ and associativity of $\circ$.

$\id$ on $A$ is indeed a function which preserves the structure ($\id \circ \alpha = \alpha = \alpha \circ \id$, $\id(a) = a$) and thus a morphism

2. There exists an associative partial binary operation $\circ$ on the arrows of $\ca{Pno}$
Given three objects $(A, \alpha, a), (B, \beta, b), (C, \gamma, c)$ and two functions $g: A \to B$ and $f: B \to C$ the function $f \circ g: A \to C$ is an arrow in $\ca{Pno}$, that is to say, it preserves structure:
$$(f \circ g) \circ \alpha = f \circ \beta \circ g = \gamma \circ (f \circ g)$$
and
$$(f \circ g)(a) = f(b) = c$$

b) Show that $(\N, \textrm{succ}, 0)$ is a $\ca{Pno}$-object
$\N$ is indeed a Set, $\textrm{succ}$ is indeed an unary function and $0$ is indeed an element of $\N$.

c) Show that for each $\ca{Pno}$-object $(A, \alpha, a)$ there is an unique arrow
$$\arr{(\N, \textrm{succ}, 0)}{}{(A, \alpha, a)}$$
and describe the behaviour of the carrying function.
We construct a carrying function recursively:
$$\begin{aligned}
f : \N & \to A \\
0 & \mapsto a \\
\textrm{succ}(x) & \mapsto \alpha(f(x))
\end{aligned}$$
$f : \N \to A$ is indeed a morphism and thus an arrow.
Given two morphisms $f : \N \to A$ and $g : \N \to A$ we show that they are pointwise identical by induction over $\N$:
* $f(0) = a = g(0)$
* Given $n \in \N$:
$$(f \circ \mathrm{succ})(n) = (\alpha \circ f)(n) \overset{\text{ind.}}{=} (\alpha \circ g)(n) = (g \circ \mathrm{succ})(n)$$
The morphism maps $\N$ to the transitive closure of $a$ under $\alpha$.

Consider objects of form $(A, a)$ where $A$ is a set and $a \subseteq A$.
For two such objects a morphism $\arr{(A, a)}{f}{(B, b)}$ is a function $f : A \to B$ that respects the selected subsets:
$$\forall \alpha \in a \ldotp f(\alpha) \in b$$
Show that such objects and morphisms form a category $\ca{SetD}$
1. For every $(A, a) \in \ca{SetD}$ there exists $\idarr{(A, a)}$

$\id$ on $A$ is indeed a function which respects the distinguished subset ($\forall \alpha \in a \ldotp \id(\alpha) \in a$) and thus a morphism

2. There exists an associative partial binary operation $\circ$ on the arrows of $\ca{SetD}$
Given three objects $(A, a), (B, b), (C, c)$ and two functions $g: A \to B$ and $f: B \to C$ the function $f \circ g: A \to C$ is an arrow in $\ca{SetD}$, that is to say, $\forall \alpha \in a$:
$$(f \circ g)(\alpha) = f(b) = c$$

Consider pairs $(A, \odot)$ where $A$ is a set and $\odot \subseteq A \times A$ is a binary relation on $A$.
Show that these pairs are objects of a category finding a sensible notion of morphism.

For two such objects $(A, \odot), (B, \oplus)$ a morphism shall be a function $f : A \to B$ that respects the binary relation:
$$\forall a \odot a^\prime \ldotp f(a) \oplus f(a^\prime)$$
We call the category comprised of such objects and arrows $\ca{RelH}$.
1. For every $(A, \odot) \in \ca{RelH}$ there exists $\idarr{(A, \odot)}$
$\id$ on $A$ is indeed a function which respects the binary relation ($\forall a \odot a^\prime \ldotp \id(a) \odot \id(a^\prime)$) and thus a morphism
2. There exists an associative partial binary operation $\circ$ on the arrows of $\ca{RelH}$
Given three objects $(A, \odot), (B, \oplus), (C, \otimes)$ and two functions $g: A \to B$ and $f: B \to C$ the function $f \circ g: A \to C$ is an arrow in $\ca{RelH}$, that is to say, $\forall a \odot a^\prime$:
$$a \odot a^\prime \implies g(a) \oplus g(a^\prime) \implies (g \circ f)(a) \otimes (g \circ f)(a^\prime)$$

Topological spaces $(S, \mathcal{O}_S)$, where $S$ is a Set and $\mathcal{O}_S \subseteq \powerset{S}$, and continuous maps $f : S \to P$, that is $\forall V \in \mathcal{O}_T \ldotp f^\leftarrow(V) \in \mathcal{O}_S$, form a Category $\ca{Top}$.
1. For every $(S, \mathcal{O}_S) \in \ca{Top}$ there exists $\idarr{(S, \mathcal{O}_S)}$

$\id$ on $S$ is indeed a continuous map and thus a morphism

2. There exists an associative partial binary operation $\circ$ on the arrows of $\ca{Top}$
Given three spaces $(S, \mathcal{O}_S), (T, \mathcal{O}_T), (U, \mathcal{O}_U)$ and two continuous maps $g : S \to T$ and $f : T \to U$ the map $f \circ g : S \to U$ is an arrow in $\ca{Top}$, that is to say, it is contiuous:
$$V \in \mathcal{O}_U \implies g^\leftarrow(V) \in \mathcal{O}_T \implies (f \circ g)^\leftarrow(V) \in \mathcal{O}_S$$

Show that $\End{\ca{C}}{A}$ is a monoid under composition, where $A$ is an object of a category $\ca{C}$.

$\circ$ is associative and total on $\End{\ca{C}}{A}$ by the definition of category and $\End{\ca{C}}{A}$ is obviously closed under $\circ$.
$\idarr{A}$ is required to exist by the definition of category and indeed an identity of $\circ$ on $\End{\ca{C}}{A}$.

Each monoid $(M, \cdot)$ is a category ([again](./1.1.md)).

We construct a category with exactly one object $\ast$ and associate to every element $m \in M$ an arrow $\arr{\ast}{m}{\ast}$.
We further define $m \circ n = m \cdot n$.

Show that $\circ$ in $\ca{Pfn}$ is associative.

Consider the composition $g \circ f = \rest{g}{\bar{B}} \circ \rest{f}{\bar{\bar{A}}}$ of two partial functions $f : A \to B$ and $g : B \to C$:
$$
\begin{tikzcd}
A \arrow[r, "f"] & B \arrow[r, "g"] & C \\
\bar{A} \arrow[u, hook] \arrow[ru, "\rest{f}{\bar{A}}" description] & \bar{B} \arrow[u, hook] \arrow[ru, "\rest{g}{\bar{B}}" description] & \\
\bar{\bar{A}} \arrow[u, hook] \arrow[ru, "\rest{f}{\bar{\bar{A}}}" description] & & \\
\end{tikzcd}
$$
where all restricted versions of $f$, $g$, and $h$ are total.
Extending the above to three partial functions $f : A \to B$, $g : B \to C$, and $h : C \to D$:
$$
\begin{aligned}
(h \circ g) \circ f &= \rest{\left (\rest{h}{\bar{C}} \circ \rest{g}{\bar{\bar{B}}} \right )}{\bar{B}} \circ \rest{f}{\bar{\bar{A}}} \\
&= \rest{h}{\bar{C}} \circ \rest{g}{\bar{\bar{B}}} \circ \rest{f}{\bar{\bar{A}}} \\
&= \rest{h}{\bar{C}} \circ \rest{\left (\rest{g}{\bar{\bar{B}}} \circ \rest{f}{\bar{\bar{A}}} \right)}{\bar{\bar{A}}} \\
&= h \circ (g \circ f)
\end{aligned}
$$

Consider the category $\ca{Set_\bot}$ of pointed sets.
Show that $\ca{Set_\bot}$ and $\ca{Pfn}$ are "essentially the same" category.

The idea is to identify the undefined parts of a partial functions image with $\bot$:
$$
\begin{aligned}
\Phi : \ca{Pfn} & \to \ca{Set_\bot} \\
A & \mapsto A \cup \{ \bot_A \} \\
f\ \text{s.t.} \begin{tikzcd}
A \arrow[r, "f"] & B \\
\bar{A} \arrow[u, hook] \arrow[ru, "\rest{f}{\bar{A}}" description] &
\end{tikzcd} & \mapsto \begin{aligned}
\Phi(f) : A \cup \{ \bot_A \} & \to B \cup \{ \bot_B \} \\
\bot_A & \mapsto \bot_B \\
a & \mapsto \begin{cases}
f(a) & \quad x \in \bar{A} \\
\bot & \quad \text{else}
\end{cases}
\end{aligned} \\
\Phi^\leftarrow : \ca{Set_\bot} & \to \ca{Pfn} \\
A & \mapsto A - \{ \bot \} \\
\arr{A}{f}{B} & \mapsto \rest{f}{\left ( A - \{ \bot \} \right )}
\end{aligned}
$$
where $\rest{f}{\bar{A}}$ is total.

Verify that for every monoid $R$ both $R\ca{Set}$ and $\ca{Set}R$ are categories of structured sets.

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Since the two proofs are perfectly analogous we cover only the case $R\ca{Set}$.
1. For every $R$-Set $A \in R\ca{Set}$ there exists $\idarr{A}$
$\id$ on $A$ is indeed a structure preserving function ($\forall a \in A, r \in R \ldotp \id(ra) = ra = r \id(a)$).
2. There exists an associative partial binary operation $\circ$ on the arrows of $R\ca{Set}$
Given three R-Sets $A$, $B$, and $C$ and two continuous maps $g : A \to B$ and $f : B \to C$ the map $f \circ g : A \to C$ is an arrow in $R\ca{Set}$, that is to say $\forall a \in A, r \in R$:
$$(f \circ g)(ra) = f(r g(a)) = r (f \circ g)(a)$$
The format of the proof was chosen to demonstrate that $R\ca{Set}$ is indeed a structured set.