Abstract
Given a type 
of algebras there is a notion of “free” algebra over a generic vector space V. Let us denote it by 
. Viewed as a functor from the category Vect of vector spaces to itself, 
is equipped with a monoid structure, that is a transformation of functors 
, which is associative, and another one 
which is a unit. The existence of this structure follows readily from the universal properties of free algebras. Such a data 
is called an algebraic operad.
This notion admits another equivalent definitions: classical, partial, and combinatorial.
The name ‘operad’ is a word that I coined myself, spending a week thinking about nothing else.
J.P. May in “Operads, algebras and modules”
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Notes
- 1.
If the grandfather J wants to make a picture of his family, then he has two choices. He can put his children E, H and S on his right side, and then the grandchildren Y, B, A further right. Or, he can put the grandchildren on the right side of their parent: E, Y, B and H, A, and then put these subfamilies on his right. That gives two different pictures since JEHSYBA≠JEYBHAS. If J had only one child, the pictures would have been the same.
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Loday, JL., Vallette, B. (2012). Algebraic Operad. In: Algebraic Operads. Grundlehren der mathematischen Wissenschaften, vol 346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30362-3_5
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