I came across this Math Overflow question pertaining to the usage of sheaves in understanding CA. Piquing my curiosity, I did some quick searches and discovered that this particular application of sheaves is rather deep. Unfortunately, I couldn't find nearly as many applications of operads to studying specifically cellular automata, which is unfortunate as it would seem to admit a natural language in which to express/recover all kinds of results about cellular automata.
My question is simple: have there been serious investigations into the extent to which operads (or, at least, ideas from that side of category theory) aid in working with any abstract structure that can be called a cellular automaton?