I have just started to read about operads, so this question might be silly.


So it seems to me that any "reasonable" class of algebras can actually be defined as a class of all algebras over a certain operad. For example, associative algebras are algebras over the associative operad $\mathcal{As}$, commutative algebras - over the commutative operad $\mathcal{Com}$. The same is true for Lie, Poisson, Gerstenhaber, $A\_{\infty}$-algebras.


So my question is: is there a (interesting) class of algebras that can't be viewed as a class of algebras over a certain operad? I mean, how universal is the operadic approach to studying algebras?


If there are algebras that are not algebras over an operad, then what are the obstructions?


Thank you very much!



How universal is operadic approach to studying algebras?
One example is a bialgebra. An operad only has operations taking several inputs and one output, so there is no room for comultiplication. On the other hand this is not a serious obstacle. A small modification of the defining axioms will accommodate also this example: bialgebras are algebras over a properad, rather than an operad.


A better example is maybe then a field. There is no possibility of encoding the existence of the inverse in the language of operads. More generally, operads and their relatives are all about encoding operations and relations which are multilinear, and inversion is just the simplest example of something nonlinear.


