Homotopy Type Theory directed type > history (Rev #4, changes)

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Definition

Examples

See also

Definition

A directed type or filtered (0,1)-precategory is a preorder or (0,1)-precategory $(P, \leq)$ with

a term $\iota:\Vert P \Vert$
a function $(-)\oplus(-):P \times P \to P$
a family of dependent terms

$a:P, b:P \vdash d(a, b):(a \leq a \oplus b) \times (b \leq a \oplus b)$

A codirected type or cofiltered (0,1)-precategory is a preorder or (0,1)-precategory $(P, \leq)$ with

a term $\iota:\Vert P \Vert$
a function $(-)\otimes(-):P \times P \to P$
a family of dependent terms

$a:P, b:P \vdash d(a, b):(a \otimes b \leq a) \times (a \otimes b \leq b)$

If the directed type or codirected type is 0-truncated it is called a directed set or codirected set, and if the directed type or codirected type is a poset, then it is called a directed poset or codirected poset, or a filtered (0,1)-category or cofiltered (0,1)-category.

Examples

The natural numbers $\mathbb{N}$ with addition $+$ and less than or equal to $\leq$ is a directed type.
The positive rational numbers $\mathbb{Q}_{+}$ with addition $+$ and less than or equal to $\leq$ is a directed type.
The negative rational numbers $\mathbb{Q}_{-}$ with addition $+$ and greater than or equal to $\geq$ is a directed type.
The positive integers $\mathbb{Z}_{+}$ with multiplicaiton $\cdot$ and less than or equal to $\leq$ is a directed type.

Homotopy Type Theory directed type > history (Rev #4, changes)

Contents

Definition

Examples

See also