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Jacques Carette (Feb 08 2022 at 14:44):If I was interested in the question What are Operads, I would easily find many, many resources. I'm instead interested in the following (related) questions:
I get the feeling that there are well-known answers in the community to both of these, and that this information may be conveyed in lectures on the topic, but somehow this part seems to be skipped in the papers that I've seen.
I would be just as happy with post-facto rationalizations, in the same way that various items in category theory have had their understanding revisited.
Reid Barton (Feb 08 2022 at 15:11):I think operads were introduced more than once, but one answer is in Peter May's "The geometry of iterated loop spaces". In algebraic topology, when you want to put some kind of algebraic structure on a space, it's usually too restrictive to ask that the equations hold on the nose, but you want more than just knowing that they hold up to some unspecified homotopy. (Same situation as for monoidal categories.) One way to deal with this is to "blow up" the sets of operations into spaces, i.e., an operad, in such a way that the operad acts strictly on the space but now there is a whole space of multiplication operations (for instance) rather than just a single one.
Reid Barton (Feb 08 2022 at 15:12):There are also interesting topological operads that don't arise from ordinary algebra in this way (e.g. operads for ).
Todd Trimble (Feb 08 2022 at 16:44):I would add that the first real appearance of an operad, although not formalized under that name, was what is now called the Stasheff operad or associahedral operad; see the papers by Stasheff, Homotopy Associativity of H-Spaces I, II (1963). The interest there was to uncover structures and conditions under which a space admits a delooping, i.e., is homotopy equivalent to a loop space. This is also a motivation for May, where now one was interested in multiple deloopings and infinite deloopings. Anyway, Stasheff proved that a space that admits a structure of algebra over his operad and which is grouplike (i.e., the homotopy multiplication on coming from the binary component induces a group structure on ) can be delooped. I believe this is an if and only if. Meanwhile, deloopings are closely connected with being able to construct classifying bundles, and this was a prime motivation for Stasheff.
Mike Shulman (Feb 08 2022 at 18:23):GOILS was certainly the origin of the word "operad", and as far as I know the first general definition in the form used nowadays.
John Baez (Feb 08 2022 at 19:51):The other people already said this, but here's how I'd try to say it without getting into details.
By the early 1970s topologists had realized the immense importance of infinite loop spaces - if you've heard the buzzwords "spectra" and "stable homotopy theory", these are closely connected. People had many different approaches to building infinite loop spaces, colloquially called infinite loop space machines, but the subject was a bit disorganized.
At around this point Peter May entered the scene and emphasized that an infinite loop space could be thought of as a topological space equipped with a bunch of -ary operations obeying equations that have no repeated or omitted variables. E.g. if you have a binary operation , the equation
is allowed, but
is not allowed (because the is omitted at right), nor is
(because the is duplicated at left).
He called a gadget that describes a bunch of -ary operations obeying equations of this limited kind an 'operad', and he developed the theory of operads to understand and work with the operad that describes infinite loop spaces.
John Baez (Feb 08 2022 at 19:57):Here of course I'm summarizing a long history and ignoring a lot of important work. For example Getzler argues that Boardman and Vogt explained the idea of operad before May in their important book Homotopy Invariant Algebraic Structures. I'm not trying to take sides on this, just trying to emphasize that the full history of operads would involve Stasheff, Boardman, Vogt, Mac Lane and many others - and various structures related to operads, such as PROPs, PACTs, etc.
John Baez (Feb 08 2022 at 19:59):By now operads are used for many other things. To take a random example, the team I'm working with at the Topos Institute are using operads in software that assembles complicated models of epidemic disease from smaller models. But I think it's good to know the origin story of operads and how they were used to tackle important problems in topology.
Jacques Carette (Feb 09 2022 at 00:25):So what motivates the question is this vague understanding of mine that operads algebra-ify "sets of operations", and their composition. They may also help with issues of 'plumbing' (i.e. getting values from where they are provided to where they are used), with a notion of value that isn't strictly 1-dimensional, as is usually the case. That's all extremely vague, I know, but it still seemed like operads intersected those thoughts in a non-trivial way.
John Baez (Feb 09 2022 at 00:36):Yes, an operad consists of a set of -ary operations for each , together with all the ways of composing them.
John Baez (Feb 09 2022 at 00:39):If you're not quite sure how operads work, this is a quite painless explanation:
Jacques Carette (Feb 09 2022 at 14:11):I understand the abstract definition of operad, what I'm 'missing' is what it is abstracting. The blog post by Tai-Danae Bradley is indeed extremely clear, thanks.
The "Axiom 2: Permutations behave nicely" seems to basically say "your planar diagrams are an illusion, all that matters is the labels for the tips, and these labels have identity and not order, so using positive integers is an especially perverse and misleading choice".
In fact, the whole thing reminds me strongly of how awkward natural-number indexed multicategories are (and how most of universal algebra doesn't need natural number arities either). If instead one uses type-valued arities, the axioms ought to simplify a lot (they did for multicategories, and had no effect on universal algebra).
This is part of a larger quest of mine to move beyond 0-dimensional notions of "plugging in" (in operads, you plug in at a label, i.e. a point; this 'point' can actually represent higher dimensional things, as shown in part 2, but these are still regarded as having no shape of their own. The obvious thing this doesn't seem to model is plugging in 2 cylinders along their circle 'interface' but having each twist being a different 'plugging in' without going to the silly extreme of having uncountably many operations. We all know that faking multi-sorted algebra in single-sorted algebra by having "large infinities" of operations is very inelegant.)
Reid Barton (Feb 09 2022 at 14:21):Operads and multicategories are the same thing up to differences in focus and "default settings" (e.g. symmetric/non-symmetric, or single sorted vs multisorted).
Jacques Carette (Feb 09 2022 at 14:22):Thanks for confirming - I was starting to suspect so, I just had not worked through the details.
Reid Barton (Feb 09 2022 at 14:35):At least in algebra/topology, the term "operad" suggests you're viewing the "elements" of the operad as operations, i.e., you're really mostly interested in the algebras of your operad. In the multicategory language these would be symmetric multifunctors from your multicategory to or another symmetric monoidal category , viewed as a representable multicategory .