Homotopy Type Theory
equivalence

Idea

A function f:ABf : A \to B is an equivalence if it has inverses whose composition with ff is homotopic to the corresponding identity map.

Definition

Let A,BA,B be types, and f:ABf : A \to B a function. We define the property of ff being an equivalence as follows:

isequiv(f)( g:BAfgid B)×( h:BAhfid A)isequiv(f) \equiv \left( \sum_{g : B \to A} f \circ g \sim id_B \right) \times \left( \sum_{h : B \to A} h \circ f \sim id_A \right)

We define the type of equivalences from AA to BB as

(AB) f:ABisequiv(f)(A \simeq B) \equiv \sum_{f : A \to B} isequiv(f)

or, phrased differently, the type of witnesses to AA and BB being equivalent types.

Properties

See also

References

Created on October 10, 2018 at 19:32:23. See the history of this page for a list of all contributions to it.