Homotopy Type Theory
monoid > history (Rev #12)
Contents
Definition
A monoid consists of
- A type ,
- A basepoint
- A binary operation
- A contractible left unit identity
- A contractible right unit identity
- A contractible associative identity
- A 0-truncator
Properties
Monoid homomorphisms
A monoid homomorphism between two monoids and consists of
- A function such that
- The basepoint is preserved
- The binary operation is preserved
For any function
the contractible left unit identity is preserved:
because
is contractible. Likewise, for any function
the contractible right unit identity is preserved:
because
is contractible, and for any function
the contractible associative identity is preserved:
because
is contractible. Finally, the 0-truncator is always preserved in a function between two sets.
Monoid isomorphisms
A monoid isomorphism between two monoids and consists of
- a monoid homomorphism
- a monoid homomorphism that is an inverse function of .
Examples
-
Every contractible magma is a monoid.
-
The integers are a monoid.
-
Given a set , the type of endofunctions has the structure of an monoid, with basepoint , operation function composition.
See also
References
Revision on June 13, 2022 at 06:22:13 by
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