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

Definition

In set theory

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

A sequence is a net whose index set is the natural numbers$\mathbb{N}$.

In homotopy type theory

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.

A sequence is a net whose index type is the natural numbers$\mathbb{N}$.

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$\mathbb{Q}_+$.