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I have 2 examples of symmetric multicategories (colored operads of sets) that are simple enough that I expect they already exist in the literature. I’m hoping that someone here can point me to suitable references. I haven’t found anything in Leinster’s book.
The first example, Multi$C$, is constructed from an ordinary category, $C$. The objects are just the objects of $C$. The $n$-ary multimorphisms are $n$-tuples of morphisms of $C$ (with the same target). The symmetric group acts by permuting the components of these $n$-tuples. The $j$’th partial composition is composition with the $j$’th component.
That seems ludicrously simplistic, but I have actually found it useful!
For the second example, IndTree, the objects (colors) are the natural numbers. (I think of a rooted tree as a directed graph with the edges directed away from the root.) An $n$-ary multimorphism is a rooted tree with $n$ vertices and 1. the vertices are ordered, 2. the vertices are colored with natural numbers, 3. the edges coming from a vertex colored $r$ are each colored with a different number from $1$ to $r$. The symmetric group acts by permuting the order of the vertices.
The partial composition $S \circ_j T$ is given by replacing the $j$’th vertex of $S$ with $T$, and attaching the edges on a way determined by the colorings. I think this is easiest to describe by giving an action of IndTree.
Consider some arbitrary set, $X$. An $r$-colored function is a function $X^r\to X$. An $n$-ary multimorphism in IndTree describes a way of composing $n$ such functions. The partial composition $S \circ_j T$ corresponds to composing some functions according to $T$, and then composing the result with some more functions according to $S$.
Obviously, my names here are just placeholders.
I’m sure I’ve seen your MultiC somewhere, and I thought it was in Leinster’s book, but I could be wrong.
Your IndTree sounds kind of like the free symmetric multicategory generated by one $r$-ary morphism for all $r$, but I don’t understand how “the objects are natural numbers” squares with your description of an action of it on one set – if the objects are natural numbers, then an “action” of it should take place on a countable family of sets.
I think what Dmitri describes is rather the “symmetric monoidal envelope” of MultiC (the canonical enlargement to a symmetric monoidal category), while if C has coproducts then MultiC itself is the symmetric monoidal structure on C with coproduct as tensor product.
Mike:
IndTree acts on a set of functions. There is a base map from that set to the set of natural numbers. Specifically, any function $X^r\to X$ is mapped to $r$. I think that set is the endomorphism operad of $X$.
It is now occurring to me that IndTree is the colored operad governing non symmetric operads, i.e., a IndTree-algebra is the same thing as a ns-operad (in any given category).
Ah, yes. I was confused because I expected you to be describing a general action of IndTree, but you were just describing an example of such an action (on functions).
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