Maps between operads of different arities Is there a category of operads which allows morphisms which take operations to operations with more or fewer arguments? One example should be when you fix arguments to obtain maps with fewer inputs. I'm grateful for any comments or references.
 A: I'm not certain I understand the intent of the question, but perhaps the following is the kind of thing Poisson is looking for.
A non-symmetric operad is a sequence $(P_n)_{n \geq 0}$ of sets together with an identity element and maps defining composition, all satisfying some axioms.
A symmetric operad is the same, but also comes with a map $\theta_*\colon P_n \to P_n$ for each bijection $\theta\colon \mathbf{n} \to \mathbf{n}$, again satisfying axioms.  (Here I use $\mathbf{n}$ to denote a fixed $n$-element set, say $\{1, \ldots, n\}$.)
A finitary algebraic theory is the same, but also comes with a map $\theta_* \colon P_n \to P_m$ for each function $\theta\colon \mathbf{n} \to \mathbf{m}$, again satisfying axioms.
The three types of structure have successively greater expressive power.  For example, there is no non-symmetric operad encoding the theory of commutative monoids (because expressing the equation $xy = yx$ requires the nontrivial bijection $\mathbf{2} \to \mathbf{2}$).  There is a symmetric operad encoding the theory of commutative monoids, but there is none encoding the theory of commutative monoids in which every element is idempotent (because expressing the term $x^2$ requires the surjection $\mathbf{2} \to \mathbf{1}$).
This doesn't mean that finitary algebraic theories are 'better' than operads, because there's a trade-off: the contexts in which it's possible to talk about algebras are successively narrower.  That is, given an operad $P$, you can talk about algebras for $P$ in an arbitrary monoidal category $\mathcal{E}$; but if $P$ has the structure of a symmetric operad, you need $\mathcal{E}$ to be symmetric monoidal in order to talk about $P$-algebras in $\mathcal{E}$; and if $P$ is a finitary algebraic theory, you need the monoidal structure on $\mathcal{E}$ to be actual categorical product $\times$.
There are many well-known equivalent definitions of "finitary algebraic theory": clone, Lawvere theory, finitary monad on $\mathbf{Set}$, etc, or the classical definition from universal algebra (modulo choice of presentation).  The definition I'm implicitly referring to above is perhaps not so well-known, though it's simple enough.  You can find the details in Definitions 2.3.1 and 2.3.2 of Miles Gould's thesis, and the equivalence to the other definitions is Theorem 2.3.12.
