[[!redirects nets]] #Contents# * table of contents {:toc} Whenever editing is allowed on the [[nLab:HomePage|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]] $\mathbb{Q}_+$. ## See also ## * [[directed type]] * [[sequence]] * [[Cauchy structure]] * [[Cauchy net]] * [[limit of a net]]