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Added a page about a colored generalization of the notion of a symmetric sequence at symmetric colored sequence. I’m happy to merge this (or some heavily edited and corrected version of it) with the page on symmetric sequences. Also open to massive edits or whatever. Just feel like something like this should be on here.
I would have been more inclined to say “colored symmetric sequence” than “symmetric colored sequence”. Not sure exactly why – maybe because a “colored sequence” is not as obvious of a thing that I would apply the adjective “symmetric” to. In fact the whole name seems wrong – the only reason a symmetric sequence can be thought of as a sequence is because it has only one color so that the objects are in bijection with $\mathbb{N}$ – but at least with “colored symmetric sequence” I can think “okay, a symmetric sequence, I know what that is and it’s associated to one-colored operads, now obviously what we mean is an analogous thing for colored operads”.
However… I don’t think this is correct! In the underlying data of a colored operad (whatever we call it), the objects have output types as well as input types. So it should not be just a functor $P(\mathfrak{C}) \to C$, but a functor $P(\mathfrak{C})\times \mathfrak{C} \to C$, and that seems even less deserving of either of these names. I would call $\mathfrak{C}$ together with such a functor a “$C$-enriched symmetric multigraph” (going along with the fact that I prefer the terminology “multicategory” to “colored operad”).
Yes you’re right! I agree with you on all counts! I was sort of working backwards from the properads case, in which you need $P(\mathfrak{C})^{op}\times P(\mathfrak{C})\to C$ but I forgot that the outputs still have colors! Thanks! Will change the name, and this error, soon.
Added a little bit about colored bisymmetric sequences (although I’m sure this also has a better name), which are useful in defining properads.
The concept of colored symmetric sequence has a long history which is liable to be forgotten unless it’s pointed out.
Baez and Dolan made use of the same concepts in HDA III, almost 20 years ago now (the category $\Sigma(C) \times C$ for a set of colors $C$ being called the category of $C$-profiles, and presheaves on that being the category of $C$-signatures in their terminology).
I don’t have a ready reference right now, but the idea is even older and might even be in Kelly’s original paper on operads (early 70’s). The conceptual point is that for a category $C$, the construct $Set^{\Sigma(C)^{op}}$ is the free symmetric monoidally cocomplete category generated by $C$, in the sense that given any cocomplete $D$ with a symmetric monoidal structure whose tensor product is cocontinuous in each variable, any functor $C \to D$ extends (uniquely up to coherent isomorphism) to a cocontinuous symmetric monoidal functor $Set^{\Sigma(C)^{op}}$, where the symmetric monoidal structure there is given by the Day convolution product induced by the symmetric monoidal structure on the free symmetric monoidal category $\Sigma(C)$.
It follows that the hom-category of functors and natural transformations $Hom(C, D)$ is equivalent to $SymMonCocont(Set^{\Sigma(C)^{op}}, D)$. Hence, taking $D = Set^{\Sigma(C)^{op}}$, the evident endofunctor composition on $SymMonCocont(Set^{\Sigma(C)^{op}}, Set^{\Sigma(C)^{op}})$ gives a monoidal structure which may be transferred across the equivalence to a monoidal structure on $Hom(C, Set^{\Sigma(C)^{op}}) \cong Set^{\Sigma(C)^{op} \times C}$, and monoids in the latter monoidal structure are of course colored operads.
I may add this material to the article later.
Please do! I didn’t know any of this other stuff.
Okay, I have added a somewhat ad hoc subsection titled “Conceptual Significance”, which could stand some further development.
I also corrected two errors in the Idea section. The more serious correction is that an operad or colored operad is not a commutative monoid in the relevant monoidal category. For one thing, that monoidal category is not even symmetric (or braided) monoidal, so there is no direct way to define the notion of commutative monoid therein. Now there is a notion of commutative monad, but still most operads or their associated monads don’t have this feature either.
Some additional light on this material may be seen in the theory of species (another name for symmetric sequence), where the relevant monoidal product is called “substitution product” or “plethystic (monoidal) product”, which is highly noncommutative and is really a categorification of functional composition of generating functions. More on this at Schur functor.
Another way to look at it is that $\Sigma(C)$ alone is the free symmetric (strict) monoidal category on $C$, so that $\Sigma$ is a monad that extends to the bicategory Prof. A functor $\Sigma(C)^{op} \times C\to Set$ is then just a profunctor from $C$ to $\Sigma(C)$, and the monoidal structure on such functors is just composition in the Kleisli bicategory of $\Sigma$.
This is the perspective of generalized multicategories. One advantage of it is that it generalizes easily to other monads on other bicategories. Another is that by replacing bicategories with double categories, we immediately obtain the right notion of functors between symmetric multicategories (a term I prefer over “colored operad”) that don’t necessarily have the same set of objects.
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