What is the free product in the category of operads?

Question

Is the Hadamard product (5.3.3 in Loday-Vallette's book on Algebraic operads) the free product in the category of operads? Is there a free product defined in the category of 1/2props or props?

Answer 1

The Hadamard product is not the free product. The free product $O_1 \ast O_2$ is the operad generated by all the operations in $O_1$ and the relations they satisfy there, and all the operations in $O_2$ and the relations they satisfy there, with no additional relations, just like the free product of groups. Its algebras are objects equipped with both an $O_1$-algebra structure and an $O_2$-algebra structure, with no relationship between them, and it is generally going to be much bigger than the Hadamard product.

You can very concretely see the difference already for operads with only $1$-ary operations in, say, $k$-vector spaces: these correspond to $k$-algebras, and the Hadamard product is the tensor product while the free product is, well, the free product.

Also note that Loday and Vallette note that the commutative operad is the unit for the Hadamard product, whereas the unit for the free product is the initial operad; more explicitly, the operad generated by no operations.