What are operad automorphisms? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T03:05:41Z http://mathoverflow.net/feeds/question/40291 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40291/what-are-operad-automorphisms What are operad automorphisms? Romeo 2010-09-28T09:13:22Z 2010-12-20T13:44:16Z <p>What is the general concept of an <strong>operad automorphism</strong>$?$ Is there a "standard" definition?</p> <p>[added after comment] If an <em>operad automorphism</em> is an invertible <em>operad endomorphism</em>, how then is <strong>operad endomorphism</strong> defined? </p> <p>I'm being drawn into these ideas for a section of a paper I'm workign on and most likely this is written-up nicely somewhere,so</p> <blockquote> <p>references would be most welcome.</p> </blockquote> http://mathoverflow.net/questions/40291/what-are-operad-automorphisms/40311#40311 Answer by James Griffin for What are operad automorphisms? James Griffin 2010-09-28T13:15:59Z 2010-09-29T13:18:42Z <p>I'm not sure what the question you're <strong>trying</strong> to ask is, but the answer to the question that you have asked is that an operad automorphism is an invertible operad endomorphism.</p> <p>[EDIT] (just restating Ryan's comment on the original post) An operad endomorphism is an operad morphism where the source and target operads are the same. An operad morphism is a collection of maps $\mathcal{O}(n)\rightarrow\mathcal{P}(n)$, one for each arity, such that the obvious squares involving the operad structure maps commute. [/EDIT]</p> <p>Perhaps you want to know about operad automorphisms in a homotopy category. In that case you want to understand what an operad quasi-isomorphism is. This is an operad morphism which is a quasi-isomorphism on each underlying space of operations.</p> <p>Would you like to refine the question?</p> http://mathoverflow.net/questions/40291/what-are-operad-automorphisms/40423#40423 Answer by Sean Tilson for What are operad automorphisms? Sean Tilson 2010-09-29T05:37:39Z 2010-09-29T05:37:39Z <p>Suppose we only care about operads in chain complexes, although i think this can all work more generally. Then an operad is a monoid in the category of symmetric sequences with respect to a particular monoidal product. so a morphism of operads would then be a morphism of monoids. I think this framework might help clarify things. The reference I have in mind, although i am sure there are earlier ones, is Kathryn Hess's lecture notes on the cobar construction. You want to look at the second lecture, page 9 specifically.</p> <p>This different monoidal product is just what you want it to be in order for a monoid to be an operad! I would explain more, but I can't do any better than Hess: <a href="http://sma.epfl.ch/~hessbell/Minicourse_Louvain_Notes.pdf" rel="nofollow">http://sma.epfl.ch/~hessbell/Minicourse_Louvain_Notes.pdf</a></p> <p>PS: a symmetric sequence is a functor from the groupoid $\Sigma$ (where the objects are sets {1}, {1,2}, ... {1,...,n},... and the morphisms are bijections) to chain complexes.</p>