#Contents# * table of contents {:toc} ## Definition ## ### Propositions ### A **proposition** is a type $A$ with an identification $p:a =_A b$ for all elements $a:A$ and $b:A$. ### Sets ### A **set** is a type $A$ such that for all elements $a:A$ and $b:A$, the identity type $a =_A b$ is a proposition. ### Pointed abelian groups ### A **pointed abelian group** is a set $A$ with an element $1:A$ called **one** and a binary operation $(-)-(-):A \times A \to A$ called **subtraction** such that * for all elements $a:A$, $a - a = 1 - 1$ * for all elements $a:A$, $(1 - 1) - ((1 - 1) - a) = a$ * for all elements $a:A$ and $b:A$, $a - ((1 - 1) - b) = b - ((1 - 1) - a)$ * for all elements $a:A$, $b:A$, and $c:A$, $a - (b - c) = (a - ((1 - 1) - c)) - b$ ### Pointed halving groups ### A **pointed halving group** is a pointed abelian group *G* with a function $(-)/2:G \to G$ called **halving** or **dividing by two** such that for all elements $g:G$, $g/2 = g - g/2$. ### Totally ordered pointed halving groups ### A pointed halving group $R$ is a totally ordered pointed halving group if it comes with a function $\max:R \times R \to R$ such that * for all elements $a:R$, $\max(a, a) = a$ * for all elements $a:R$ and $b:R$, $\max(a, b) = \max(b, a)$ * for all elements $a:R$, $b:R$, and $c:R$, $\max(a, \max(b, c)) = \max(\max(a, b), c)$ * for all elements $a:R$ and $b:R$, $\max(a, b) = a$ or $\max(a, b) = b$ * for all elements $a:R$ and $b:R$, $\max(a, b) = b$ implies that for all elements $c:R$, $\max(a - c, b - c) = b - c$ ### Strictly ordered pointed halving groups ### A totally ordered pointed halving group $R$ is a strictly ordered pointed abelian group if it comes with a type family $\lt$ such that * for all elements $a:R$ and $b:R$, $a \lt b$ is a proposition * for all elements $a:R$, $a \lt a$ is false * for all elements $a:R$, $b:R$, and $c:R$, if $a \lt c$, then $a \lt b$ or $b \lt c$ * for all elements $a:R$ and $b:R$, if $a \lt b$ is false and $b \lt a$ is false, then $a = b$ * for all elements $a:R$ and $b:R$, if $a \lt b$, then $b \lt a$ is false. * $1 - 1 \lt 1$ * for all elements $a:R$ and $b:R$, if $1 - 1 \lt a$ and $1 - 1 \lt b$, then $(1 - 1) \lt a - ((1 - 1) - b)$ The homotopy initial strictly ordered pointed halving group is the dyadic rational numbers $\mathbb{D}$. ### Archimedean ordered pointed halving groups ### A strictly ordered pointed halving group $A$ is an Archimedean ordered pointed halving group if for all elements $a:A$ and $b:A$, if $a \lt b$, then there merely exists a dyadic rational number $d:\mathbb{D}$ such that $a \lt h(d)$ and $h(d) \lt b$. ### Sequentially Cauchy complete Archimedean ordered pointed halving groups ### Let $A$ be an Archimedean ordered pointed halving group and let $$A_{+} \coloneqq \sum_{a:A} 1 - 1 \lt a$$ be the positive elements in $A$. A **sequence** in $A$ is a function $x:\mathbb{N} \to A$. A sequence $x:\mathbb{N} \to A$ is a **Cauchy sequence** if for all positive elements $\epsilon:A_{+}$, there merely exists a natural number $N:\mathbb{N}$ such that for all natural numbers $i:\mathbb{N}$ and $j:\mathbb{N}$ such that $N \leq i$ and $N \leq j$, $\max(x_i - x_j, x_j - x_i) \lt \epsilon$. An element $l:A$ is said to be a **limit** of the sequence $x:\mathbb{N} \to A$ if for all positive elements $\epsilon:A_{+}$, there merely exists a natural number $N:\mathbb{N}$ such that for all natural numbers $i:\mathbb{N}$ such that $N \leq i$, $\max(x_i - l, l - x_i) \lt \epsilon$ $A$ is **sequentially Cauchy complete** if every Cauchy sequence in $A$ merely has a limit. category: not redirected to nlab yet