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Let ( S , ∗ ) {\displaystyle (S,*)} and ( T , ⋅ ) {\displaystyle (T,\cdot )} be magmas (i.e., S {\displaystyle S} and T {\displaystyle T} are sets with binary operations ∗ {\displaystyle *} and ⋅ {\displaystyle \cdot } ). A homotopy of magmas from ( S , ∗ ) {\displaystyle (S,*)} to ( T , ⋅ ) {\displaystyle (T,\cdot )} is a triple ( α , β , γ ) {\displaystyle (\alpha ,\beta ,\gamma )} of maps from S {\displaystyle S} to T {\displaystyle T} such that for all g , h ∈ S {\displaystyle g,h\in S} , we have:
α ( g ) ⋅ β ( h ) = γ ( g ∗ h ) {\displaystyle \alpha (g)\cdot \beta (h)=\gamma (g*h)}
and 
be magmas (i.e., 
and 
are sets with binary operations 
and 
). A homotopy of magmas from to is a triple 
of maps from to such that for all 
, we have:
