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• CommentRowNumber1.
• CommentAuthormrmuon
• CommentTimeOct 26th 2019

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.

• CommentRowNumber2.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 26th 2019
The symmetric multicategory MultiC is simply the symmetric multicategory associated to the monoidal category given by the freely adding coproducts to C.

The second operad is similar in spirit to the operad of trees by Moerdijk and Weiss, but with rather different details.
• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeOct 27th 2019

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.

• CommentRowNumber4.
• CommentAuthorRuneHaugseng
• CommentTimeOct 27th 2019

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.

• CommentRowNumber5.
• CommentAuthormrmuon
• CommentTimeOct 27th 2019

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).

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeOct 28th 2019

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).