# Contents

Whenever editing is allowed on the nLab again, this article should be ported over there.

## Definition

A net is a function $a: I \to A$ from a directed type $I$ to a type $A$. $I$ is called the index type, the terms of $I$ are called indices (singular index), and $A$ is called the indexed type.

## Subnets

Let $I$ be a preordered type. Given a term $i:I$, the positive cone of $I$ with respect to $i$ is defined as the type

$I^+_i \coloneqq \sum_{j:I} i \leq j$

with monic function $f:I^+_i \to I$ such that for all terms $j:I$, $i \leq f(j)$.

Given a net $a: I \to A$ and a net $b:J \to A$, we say that $b$ is a subnet of $a$ if $b$ comes with a function $f:I \to J$ and a dependent function $g:\prod_{i:I} J^+_{f(i)} \to I$ such that for every dependent term $j(i):J^+_{f(i)}$, there is a dependent identification $p(i, j(i)): a_{j(i)} = b_{g(i)(j(i))}$.

$b \subseteq a \coloneqq \prod_{i:I} \prod_{k:J} (f(i) \leq k) \times \left[\sum_{l:I} (i \leq l) \times (a_k = b_l)\right]$

## Examples

• The Cauchy approximations used to define the HoTT book real numbers are nets indexed by a dense subsemiring $R_{+}$ of the positive rational numbers $\mathbb{Q}_+$.