TeXipedia

simeq

Denotes a relation that is both similar and equivalent, commonly used in mathematics and theoretical sciences.

Overview

Represents a stronger relationship than mere similarity but distinct from strict equality, making it particularly valuable in abstract algebra, topology, and category theory.

  • Often used to indicate isomorphism between mathematical structures
  • Appears frequently in proofs where objects share essential properties
  • Common in situations involving canonical isomorphisms or natural equivalences
  • Useful when describing relationships between algebraic objects that are structurally the same but not identical

Examples

Showing that two topological spaces are homotopy equivalent.

S1≃RP1S^1 \simeq \mathbb{R}P^1 
S^1 \simeq \mathbb{R}P^1

Indicating that two groups are isomorphic up to natural equivalence.

G/H≃KG/H \simeq K 
G/H \simeq K

Expressing asymptotic equivalence of functions.

f(x)≃x2 as x→∞f(x) \simeq x^2 \text{ as } x \to \infty 
f(x) \simeq x^2 \text{ as } x \to \infty