Operad terminology - Operads with and without O(0).

Question

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.

Answer 1

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.

Answer 2

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 $V$.

Answer 3

A 2020 paper Operads, monoids, monads, and bar constructions by Ruozi Zhang, Foling Zou, and J.P. May revisits the foundations of operad theory and discusses several possible choices for what $\mathcal{C}(0)$ might be in an operad $\mathcal{C}$ defined in a symmetric monoidal category $\mathcal{M}$ with unit object $\mathcal{I}$. It might be an initial object (so empty in some examples, $0$ in others). It might be a terminal object (when the operad is often called reduced), or it might be the unit $\mathcal{I}$ (in which case the cited authors and others say that $\mathcal{C}$ is unital), so unital = reduced when $\mathcal{M}$ is cartesian monoidal, but they are quite different otherwise. So whatever May may have told his ex-student Shulman over a decade ago, he firmly disagrees now. David, much as you and May usually agree, this means that May also firmly disagrees with your sentence "An operad where $O(0)$ is the monoidal unit of $\mathcal{M}$ is called reduced.'' In examples where $\mathcal{M}$ is not cartesian monoidal, unital is much more accurately informative than reduced. The cited paper shows that unital operads admit an interesting variant of Kelley's interpretation of operads as monoids in a certain monoidal category, and that reinterpretation is seriously interesting in applications.

Saying that there is no $\mathcal{C}(0)$ says that $\mathcal{C}$ is not actually an operad, or so says the pigheaded person who first defined operads. The horrid if accurate term ``constantless operad'' should be viewed as a noun. That is, constantless should not be viewed as an adjectival modifier of operad.

Answer 4

EDIT:

I did not realize there was still controversy, and I didn't know about the 2020 paper Peter May linked to today, but I like it a lot. I'm always happy to defer to Peter, so the terminology I'll try to stick to from now on is that $O$ is:

constant-free if $O(0) = \emptyset$ is the initial object of $M$

reduced if $O(0) = \ast$ is the terminal object

unital if $O(0) = I$ is the monoidal unit of $M$

I wasn't aware of Fresse's terminology for unitary (meaning unital above) and non-unitary (meaning reduced above), but I'm not a big fan of it because being "not unitary" is not the same as being "non-unitary."

It is worth pointing out that evaluation gives an $O$-algebra morphism $O(0) \to A$ for any $O$-algebra $A$, so unital operads encode flavors of unital algebras, which is nice.

ORIGINAL POST:

Twelve years later, the terminology has converged. Let $M$ be a closed symmetric monoidal category so I can consider operads valued in $M$.

An operad with no arity zero part $O(0)$ is called constant-free. This means there are objects $O(1), O(2), \dots$, with actions of the symmetric group $\Sigma_n$ on $O(n)$ if you want to work with symmetric operads. Equivalently, one can work with operads where $O(0)$ is the initial object of $M$.

An operad where $O(0)$ is the monoidal unit of $M$ is called reduced. This terminology is used in the seminal work of Berger and Moerdijk Axiomatic homotopy theory for operads and on the nLab page.

When we just say "operad", we assume we have $O(0), O(1), O(2), \dots$ and a priori $O(0)$ can be any object of $M$. Note that in the case $M = Set$, being reduced means $O(0)$ is a point.

The OP asked for "examples where these distinctions are interesting." One important example is in homotopy theory. If $M$ is a monoidal model category, the category of constant-free symmetric operads has a transferred model structure, which is left proper if $M$ is. The same is true of the category of reduced symmetric operads but is FALSE for symmetric operads. An example where the category of symmetric operads has only a transferred semi-model structure is given in Example 2.9 of the paper Bousfield Localization and Eilenberg-Moore Categories (published in HHA).

A reference for everything in this answer, as well as pointers to the literature where these terms have become standard, is Homotopy theory for algebras over polynomial monads, published in TAC.