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Operads and Related Concepts

The notion of operads was introduced by Peter May [ May72 ] in order to study iterated loop spaces , although the idea is based the work [ BV73 ] of Boardman and Vogt, which in turn is based on the study of “higher homotopies” appeared in problems in algebraic topology. Huebschmann’s [ Hue ] ranges from the study of higher homotopies in the pre-operad era to the modern applications.

Nowadays, the notion of operads are used in algebra and physics . The study of operads is much more active in these areas than in algebraic topology. According to Maninand Borisov [ BM ] , there are three interpretations of operads.

  • a tool to study algebraic structures on a set together with operations satisfying certain conditions
  • an example of a categorification of graph theory
  • abstraction of computational processes in tensor networks and quantum computations

There used to be only two choices when one wanted to study operads, i.e. the book by Boardman and Vog [ BV73 ] and the book [ May72 ] by Peter May. Now there are many choices, thanks to the popularity of operads.

The book [ MSS02 ] by Markl, Shnider, and Stasheff contains a lot of topics. It might be a good idea to browse this book first in order to grasp the whole subject. Markl’s [ Mar ] serves as a shorter introduction. It contains a good list of references. Another choice is the book by Kriz and May [ KM95 ] . One of the most concise expositions is Stasheff’s “WHAT Operad” in the Notices of the A.M.S. vol. 51 No. 6. Ionescu’s [ Ion ] contains PROPs .

It might be interesting to take a look at the proceedings of the workshop at Osnabrk in 1998, which can be available from the preprint server SFB 343 in Bielefeld University .