I may be confused somehow. Feel free to ignore. But: * At first I thought you meant the input alphabet to be the colors, not the operations. * Instead, am I correct that “the free operad generated by the input alphabet of the tree automaton” is an operad with just one color, and the “operations” are basically all the labeled trees where labels of the nodes are the elements of the alphabet, such that the number of children of a node is always equal to the arity of that label in the input alphabet? * That would make sense, as the algebra would then I guess assign the state space of the tree automaton to the single color of the operad, and each arity n operation would be mapped to the mathematical function from Q^n to Q. * That would make sense I think, but then why do you talk about a “colored” operad in: “we can now define a deterministic automaton over a (colored) operad O to be an O-algebra”?
You now understand correctly. The reason I switch to colored operads is to add even more generality. My key use case is when the operad consists of terms-with-holes in a programming language, in which case the colors are the types of the terms/holes.
I may be confused somehow. Feel free to ignore. But:
* At first I thought you meant the input alphabet to be the colors, not the operations.
* Instead, am I correct that “the free operad generated by the input alphabet of the tree automaton” is an operad with just one color, and the “operations” are basically all the labeled trees where labels of the nodes are the elements of the alphabet, such that the number of children of a node is always equal to the arity of that label in the input alphabet?
* That would make sense, as the algebra would then I guess assign the state space of the tree automaton to the single color of the operad, and each arity n operation would be mapped to the mathematical function from Q^n to Q.
* That would make sense I think, but then why do you talk about a “colored” operad in: “we can now define a deterministic automaton over a (colored) operad O to be an O-algebra”?
You now understand correctly. The reason I switch to colored operads is to add even more generality. My key use case is when the operad consists of terms-with-holes in a programming language, in which case the colors are the types of the terms/holes.