colored symmetric sequence




A symmetric sequence in a symmetric monoidal category CC can be thought of as a functor ΣC\Sigma\to C, where Σ\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 Σ\Sigma. The category of colored symmetric sequences in a symmetric monoidal category CC, as defined below, admits a composition product?, with respect to which the algebras are precisely the colored operads in CC.

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 CC might be better referred to as a CC-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.


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()P(\mathfrak{C}), to be the groupoid whose objects are finite lists of elements of \mathfrak{C} and in which there is an isomorphism c={c 1,,c n}d={d 1,,d n}\vec{c}=\{c_1,\ldots,c_n\}\to \vec{d}=\{d_1,\ldots,d_n\} if d\vec{d} is a permutation of c\vec{c}. Note that if =1\mathfrak{C}=1 then we recover the category Σ\Sigma of finite ordinals and bijections.

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

The idea of course is that the target of the pair (c,d)=({c 1,,c n},d)(\vec{c},d)=(\{c_1,\ldots,c_n\},d) (where some of the c ic_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 ic_i for each c ic_i composing c\vec{c} and have output of color dd.

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 CC (more generally than for a mere set CC), the construct Set Σ(C) opSet^{\Sigma(C)^{op}} is the free symmetric monoidally cocomplete category generated by CC, in the sense that given any cocomplete DD with a symmetric monoidal structure whose tensor product is cocontinuous in each variable, any functor CDC \to D extends (uniquely up to coherent symmetric monoidal isomorphism) to a cocontinuous symmetric monoidal functor Set Σ(C) opSet^{\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 Σ(C)\Sigma(C).

It follows that the hom-category of functors and natural transformations Hom(C,D)Hom(C, D) is equivalent to SymMonCocont(Set Σ(C) op,D)SymMonCocont(Set^{\Sigma(C)^{op}}, D) (the category of symmetric monoidal cocontinuous functors and symmetric monoidal natural transformations between them). Hence, taking D=Set Σ(C) opD = Set^{\Sigma(C)^{op}}, the evident endofunctor composition on SymMonCocont(Set Σ(C) op,Set Σ(C) op)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 Σ(C) op)Set Σ(C) op×CHom(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 Σ(C)×C\Sigma(C) \times C for a set of colors CC being the category of CC-profiles, and presheaves on that category the category of CC-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 CC, called a Σ S()\Sigma_{S(\mathfrak{C})}-bimodule in HRY, is a functor P()×P() opCP(\mathfrak{C})\times P(\mathfrak{C})^{op}\to C.


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

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

Last revised on July 23, 2016 at 12:40:26. See the history of this page for a list of all contributions to it.