# How now to study operads in homotopy theory?

There is a great introduction by May, "The Geometry of Iterated Loop Spaces". I really enjoy reading it, but it was written 50 years ago and contains outdated technical details related to the language of topological spaces. Now, as far as I understand, there is no doubt that this should be stated in the language of simplicial sets. What are the best references nowadays? I found, for example, Simplicial and Operad Methods in Algebraic Topology. Is this a good introductory text?

In this question, I am interested in operads specifically in homotopy theory, in particular, the connection with iterated loop spaces (so the question about algebraic operads is not relevant to the current topic, as far as I understand, although the algebraic theory of operads also fascinates me).

• I do not think it is true that the technical details involving topological spaces are outdated because such details ought to be handled within simplicial sets. If you consider operads as constructions entirely internal to simplicial sets, you miss out on some insights and intuitions that are available if you are willing to think in top. spaces. The very fruitful relationship between the little n-cubes operad and the configuration spaces of points in R^n, for example, is obscured if you are insist that you must view the spaces in the operad as simplicial sets and not as topological spaces.
– user164898
Jul 31, 2022 at 16:08
• @A.S. Additionally, there is a strong relationship between the $E_n$ operad and manifolds, see factorization homology and embedding calculus. As you say, all of this would be obscured if one insisted on using simplicial sets. I'd say there is a reason the book is entitled "Geometry of iterated loop spaces". Jul 31, 2022 at 16:11
• Great, thanks a lot for your comments! I really like geometric intuition, so I'm happy with just such an answer. I thought simplicial sets were certainly better because they already form a topos (this seems to be exactly what the technical requirements like CGWH are aiming for). My friends and I are planning a seminar on operads this fall, and I wanted to avoid (what I thought) outdated math crutch in the program. Aug 2, 2022 at 9:27
• @AivazianArshak: A book that must be mentioned is Markl–Shnider–Stasheff. Chapter 5 of Lurie's Higher Algebra has some interesting material. You can also look at the list of references for the seminar I organized some time ago. Aug 4, 2022 at 21:37
• I do not agree AT ALL with the sentence "there is no doubt that this should be stated in the language of simplicial sets". A great motivation for operads in topological spaces (in addition to little n-cubes, already mentioned) is equivariant orthogonal spectra, $N_\infty$-operads, etc. It's a thriving research area. I agree with A.S. that the technical details about Top are relevant. Sep 10, 2022 at 11:55

One of the most comprehensive references today is certainly:

B. Fresse, Homotopy of Operads and Grothendieck–Teichmüller Groups, Mathematical Surveys and Monographs 217. https://bookstore.ams.org/surv-217/

However, the idea that topological spaces are obsolete and that we should only use simplicial sets is a bit misguided. How do you define a manifold using just simplicial sets? Or the little disks/cubes operads? Simplicial sets of course have their place, but there are times when you just can't avoid topological spaces.

Najib's answer pointing to Fresse's book is a solid choice. I actually learned even more from Fresse's other book Modules over Operads and Functors. And, my favorite book about operads using modern language is Donald Yau's book Colored Operads, which is explicitly written for grad students (published in 2016). This might be a good choice for the seminar the OP is organizing.

I agree the Fresse's book is fantastic, as are the others mentioned by David White.

For (one perspective on) modern interactions of operads and their role in homotopy theory, I would like to add a recommendation for the recent book by Heuts and Moerdijk: Simplicial and Dendroidal Homotopy Theory. In fact, the original papers by Cisinski-Moerdijk-Weiss are very readable, and make nice companions.

These present the foundations of a theory of homotopy coherent operads defined intrinsically to modern homotopy theory. The book gives a friendly and detailed introduction to the theory of dendroidal sets and the Cisinski-Heuts-Moerdijk-Weiss model of infinity operads. It's also an excellent book on simplicial homotopy theory if you are after a complement to Goerss-Jardine's Simplicial Homotopy Theory

• Thank you very much! Apr 2, 2023 at 14:05