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Idea
The commutative version of the A3-space up to homotopy, without any higher commutative coherences.
Definition
A commutative -space or commutative -algebra in homotopy types or commutative H-monoid consists of
A type ,
A basepoint
A binary operation
A left unitor
A right unitor
An asssociator
A commutator
Homomorphisms of commutative -spaces
One could also speak of commutative -spaces where commutativity is mere property rather than structure, which is a commutative -space as defined above with additional structure specifying that the type is contractible:
A homomorphism of commutative -spaces between two commutative -spaces and consists of
A function such that
The basepoint is preserved
The binary operation is preserved
A function
Homomorphisms of commutative -spaces
A homomorphism of commutative -spaces between two commutative -spaces and consists of