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---
title: Exercises in Category Theory — 1.1
published: 2016-01-29
tags: Category Theory
---
<div class="corollary">
Sets and functions do form a category $\ca{Set}$.
<div class="proof">
The objects of $\ca{Set}$ are sets and its arrows are functions.
For any set $A$ we construct $\idarr{A}$ by restricting $\id$ to $A$:
$$\idarr{A} = \id \upharpoonright A$$
$\circ$ is indeed associative.
</div>
</div>
<div class="corollary">
Each [poset](https://en.wikipedia.org/wiki/Partially_ordered_set) is a category.
<div class="proof">
The objects of each poset are its elements and we construct an arrow $\begin{tikzcd} a \arrow[r, "\leq"] & b \end{tikzcd}$ 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.
</div>
</div>
<div class="corollary">
Each [monoid](https://en.wikipedia.org/wiki/Monoid) is a category.
<div class="proof">
The objects of each monoid are its elements and we construct for every pair of elements $(a, b)$ an arrow $\begin{tikzcd} a \arrow[r, "\cdot b"] & a \cdot b \end{tikzcd}$.
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 $\circ$.
</div>
</div>
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