In the Markl, Schneider and Stasheff text, topological operads are an indexed collection of spaces $O(n)$ for $n \in \{1,2,3,\cdots\}$ satisfying some axioms. In May's text, the index set is allowed to include zero.

1) Is there a standard terminology for operads with and without $O(0)$?

2) Is there standard terminology for topological operads where $O(0)$ is a point, vs. $O(0)$ not being a point?

Although it's less important I'd be curious if people have examples where these distinctions are interesting.

Since any operad acts on its $O(0)$ part perhaps the $O(0)$ part should be called something like its "base"? But then "baseless operad" would sound kind of pejorative.

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    $\begingroup$ In my experience, there are so many variations on terminology with operads that there is almost never a standard term for anything. Regarding your question specifically, Lazarev (e.g., in arXiv:0704.2561) tends to refer to O(0) as the 'vacuum part' of the operad, though this is more in the context of modular operads. His terminology comes from analogy with QFT where one would talk about vacuum diagrams (no legs) contributing to the vacuum expectation value. $\endgroup$ – Jeffrey Giansiracusa Jun 26 '10 at 20:53
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    $\begingroup$ The "n" in O(n) could be referred to as "operadic degree". I certainly don't approve of "level". $\endgroup$ – André Henriques Jun 26 '10 at 22:00
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    $\begingroup$ According to wikipedia, a synonym of "arity" is "adicity", so you could call it the "operadicity". $\endgroup$ – Harry Gindi Jun 26 '10 at 22:07
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    $\begingroup$ FWIW, I'm in the habit of referring to n as the arity. (Certainly it is standard to refer to the "arity" of operations in logic and in universal algebra.) So $O(0)$ would be the 0-ary (or nullary) component. Also in logic, 0-ary function symbols are what are usually called "constants", so an operad where $O(0)$ is empty could logically be called an "operad without constants" or a "constant-free operad". The only trouble with that is that many people won't understand what the heck you are talking about until you explain. :-) $\endgroup$ – Todd Trimble Jun 27 '10 at 2:19
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    $\begingroup$ More recently, Peter May has called operads with $O(0)$ a point reduced. I like that terminology better. $\endgroup$ – Mike Shulman Jun 27 '10 at 3:12

I can second Jeffrey's comment, reduced is used to say that O(1) is just the monoidal unit (it allows us to use the Boardman Vogt resolution in homotopy theory). It's my opinion that this terminology will probably stick.

I would also say that a $\mu$ in O(n) had arity n.

That the O(0) part of an operad is referred to as the 'constants' of the operad makes a lot of sense, every algebra for O must contain O(0) and the composition of those must behave in a certain way.

Calling O(0) the point also makes sense, because in the category of algebras O(0) will be the initial object.

Here my comment has become too long, just as I've got to the point of my comment:

The comments to the question tend to prefer terminology that relates to the behaviour of the operad (eg "reduction", because a unit lowers the arity). My personal preference (and I think the literature follows it), is that terminology should have more of a relation to the category of algebras than to the operad itself.

So my vote is that you call O(0) the initial of O. And you call an operad without O(0) initial-less or uninitiated.

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    $\begingroup$ No, reduced is taken: means the 0th object is a point, not the first. That terminology is the one that sticks. $\endgroup$ – Peter May Dec 16 '11 at 17:39
  • $\begingroup$ It's worth pointing out that in Berger and Moerdijk's "Axiomatic Homotopy Theory for Operads" an operad $P$ is reduced if $P(0)$ is the unit of the underlying category. This matches Peter May's comment more than the answer. $\endgroup$ – David White Jun 7 '12 at 17:04

It is interesting to note that the general theory of operads with constants and that of operads without constants (here I refer to $O(0)$ as $\it{constants}$ showing my personal preference for terminology) admit the following distinct difference. Just for simplicity, let's consider operads enriched in sets and let's allow all (coloured) operads instead of just the monochromatic ones. Thus, let $\mathbf{Ope}$ be the category of all small coloured operads (symmetric or not, does not matter for this example) in $\mathbf{Set}$. Let $\mathbf{cfOpe}$ be the full subcategory of $\mathbf{Ope}$ consisting of the constant-free operads (that is those operads in which no $0$-ary arrows exist).

Now consider the obvious functors $j:\mathbf{Cat}\to\mathbf{Ope}$ and $l:\mathbf{Cat}\to\mathbf{cfOpe}$. It is rather simple to show that each of these functors has a right adjoint so we get $j':\mathbf{Ope}\to\mathbf{Cat}$ and $l': \mathbf{cfOpe}\to\mathbf{Cat}$. However, $l'$ has again a right adjoint while $j'$ does not.

Not much changes if one considers operads enriched in some symmetric monoidal category $V$.


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