Homotopy Type Theory
abelian group > history (Rev #7, changes)
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Definition
An abelian group consists of
As a group
- A type ,
- A basepoint
- A binary operation
- A unary operation
- A contractible left unit identity
- A contractible right unit identity
- A contractible associative identity
- A contractible left inverse identity
- A contractible right inverse identity
- A contractible commutative identity
- A 0-truncator
An abelian group or -module consists of
- A type ,
- A basepoint
- A binary operation
- A unary operation
- A contractible left unit identity
- A contractible right unit identity
- A contractible associative identity
- A contractible left inverse identity
- A contractible right inverse identity
- A contractible commutative identity
- A 0-truncator
As a module
An abelian group or -module is a set with a term and a binary function , and a left multiplicative -action , such that
We define the functions and to be
and is an abelian group and a -bimodule
Examples
-
Every contractible magma with a function is an abelian group.
-
The integers are an abelian group.
See also
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