TeXipedia

coloncolon

Represents a double colon operator commonly used in category theory and advanced mathematics to denote specific relationships or mappings.

Overview

Serves as a specialized notation primarily found in advanced mathematical contexts, particularly in category theory and type theory where precise relationship notation is crucial.

  • Frequently used to denote natural transformations between functors
  • Appears in formal mathematical proofs and theoretical computer science
  • Provides a distinct alternative to single colons when more specific notation is needed
  • Common in papers and texts dealing with categorical logic and higher-order abstractions

Examples

Type theory notation showing a variable's type declaration

x∷Nx \coloncolon \mathbb{N} 
x \coloncolon \mathbb{N}

Category theory notation for natural transformation

F∷C→DF \coloncolon C \to D 
F \coloncolon C \to D

Programming language type annotation syntax

map∷(a→b)→[a]→[b]\text{map} \coloncolon (a \to b) \to [a] \to [b] 
\text{map} \coloncolon (a \to b) \to [a] \to [b]