Homotopy Type Theory equivalence > history (Rev #2)

Idea

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

Definition

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

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)

The above type could also be defined as

isequiv(f) \equiv \prod_{b:B} isContr(fiber(f))

where $fiber(f, b)$ is the fiber of $f$ over a term $b:B$

We define the type of equivalences from $A$ to $B$ as

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

or, phrased differently, the type of witnesses to $A$ and $B$ being equivalent types.

Properties

Every equivalence is a $0$ -monic function.

See also

References

HoTT book

Revision on March 12, 2022 at 19:36:21 by Anonymous?. See the history of this page for a list of all contributions to it.