I just discovered that something I've been working with has the structure of an operad. So I'm wondering what natural basic questions does one ask about operads? For example, if I knew I had the structure of a group, I'd ask if it is abelian or has torsion, etc. So what are these questions for operads?



What are natural questions to ask about an operad?
There's a lot of things you could ask. 


* Operads act on things, that's their point. What things does your operad act on? Presumably this is how you found your operad. Moreover, once you know it acts on something you can ask if that action is maximal, whether or not your operad fits into a bigger operad that also acts on the thing in question, etc. Similarly, you can ask what does that operad tell you about the thing its acting on.
* Operads can have sub-operads, do you have any interesting ones? That would lead to other related questions, like is your operad an extension of other operads? Take a look at the Markl and Stasheff operad book to get a sense for some of the operads out in the literature, and what they're good for.
* There are things like totalizations and bar constructions for operads. What might that look like for your operad?
* Operads induce other operads, for example, the homology of a topological operad is another operad. Does your operad have any related operads that are known or otherwise interesting?
* (Edit, idea from Jeff's cyclicity suggestion) Operads sometimes fit into even larger higher-algebraic structures. Jeff mentions cyclic operads, but there are also PROPs, for example. You might want to consider that maybe you're dealing with something that's "more than" an operad.


That'd be a start.


