TITLE: Operad terminology - Operads with and without O(0). 
QUESTION [8 upvotes]: 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.

REPLY [3 votes]: 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 $V$.