nLab colored symmetric sequence

Idea

A symmetric sequence in a symmetric monoidal category $C$ can be thought of as a functor $\Sigma\to C$, where $\Sigma$ is the category of finite ordinals with bijections between them¹. One use of this idea is to define operads as monoids in symmetric sequences (with a certain composition product²). However, operads defined in this way are necessarily operads with one color. To define colored operads as a type of “symmetric sequence”, one has to generalize the category $\Sigma$. The category of colored symmetric sequences in a symmetric monoidal category $C$, as defined below, admits a composition product?, with respect to which the algebras are precisely the colored operads in $C$.

Note that the terminology “colored symmetric sequence” doesn’t really make any sense, since we know longer have a sequence at all, much less a “colored” one. Mike Shulman has suggested that what we call a colored symmetric sequence in $C$ might be better referred to as a $C$-enriched multigraph. The only drawback of such terminology is that it obscures the relevance of this construction to symmetric sequences (which it generalizes) and its relationship to colored operads.

Definition

Let $\mathfrak{C}$ be a set that we will refer to as the set of colors. Define the category of $\mathfrak{C}$-profiles, denoted $P(\mathfrak{C})$, to be the groupoid whose objects are finite lists of elements of $\mathfrak{C}$ and in which there is an isomorphism $\vec{c}=\{c_1,\ldots,c_n\}\to \vec{d}=\{d_1,\ldots,d_n\}$ if $\vec{d}$ is a permutation of $\vec{c}$. Note that if $\mathfrak{C}=1$ then we recover the category $\Sigma$ of finite ordinals and bijections.

A symmetric $\mathfrak{C}$-colored sequence in a category $C$ is then a functor $P(\mathfrak{C})\times\mathfrak{C}\to C$.

The idea of course is that the target of the pair $(\vec{c},d)=(\{c_1,\ldots,c_n\},d)$ (where some of the $c_i$ might be the same element of $\mathfrak{C}$) is the object in the colored operad governing operations that take as input an object of color $c_i$ for each $c_i$ composing $\vec{c}$ and have output of color $d$.

Conceptual significance

The concept of colored symmetric sequence, seemingly rediscovered several times, involves concepts originally developed by Kelly (reference to be added).

The underlying conceptual point is that for a category $C$ (more generally than for a mere set $C$), 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 symmetric monoidal 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 $SymMonCocont(Set^{\Sigma(C)^{op}}, D)$ (the category of symmetric monoidal cocontinuous functors and symmetric monoidal natural transformations between them). 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.

Baez and Dolan made use of the same concepts in Higher Dimensional Algebra III (where their theory of opetopic sets was developed), calling the category $\Sigma(C) \times C$ for a set of colors $C$ being the category of $C$-profiles, and presheaves on that category the category of $C$-signatures.

Colored Bisymmetric Sequences

The above definition can be generalized to support colored properads instead of just operads. The way to do this is to include the possibility of multiple outputs as well as multiple inputs. Thus a colored bisymmetric sequence in a category $C$, called a $\Sigma_{S(\mathfrak{C})}$-bimodule in HRY, is a functor $P(\mathfrak{C})\times P(\mathfrak{C})^{op}\to C$.

References

• Philip Hackney, Marcy Robertson and Donald Yau. Infinity Properads and Infinity Wheeled Properads, Lecture Notes in Mathematics, 2147. Springer, Cham, 2015. (arxiv version)

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1. Cf. species, another name for symmetric sequence. ↩

2. Essentially synonymous with substitution product as explained at club, a concept due to Max Kelly. ↩