# Homotopy Type Theory net > history (Rev #11, changes)

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# 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.

## Eventuality Subnets

Let  A I be a type, let$I$preordered type . be Given a directed term type, and let a: i:I I \to A , be the a positive net. cone Given of a term i:I I , the with positive respect cone to of I 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 subtype net B a: I \to A of and a net b:J \to A , with we monic say function that g:B b \to A , is a$a$subnet is ofeventually$a$ in if B b if comes with a function I f:I \to J comes and with a term dependent function i:I g:\prod_{i:I} J^+_{f(i)} \to I and such a that function for every dependent term b:I^+_i j(i):J^+_{f(i)} \to B , such there that is for a all dependent terms identification j:I^+_i p(i, j(i)): a_{j(i)} = b_{g(i)(j(i))}, $a_{f(j)} = g(b_j)$.

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

## 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 $\mathbb{Q}_+$.