category of operators



Every operad defines and is defined by a category – its category of operators – whose

  • objects are sequences consisting of colors of the operad;

  • morphisms are tuples consisting of maps of sets between these sequences of colors, and a kk-ary operation of the operad for each collection of kk source colors that are mapped to the same target color.

This is a universal construction: the category of operators is the free semicartesian monoidal category on the free semicartesian operad on the given operad. It also generalizes in a straightforward way to “colored operads”, i.e. multicategories.


For symmetric colored operads

Let FinSet *FinSet_{*} be the category of finite pointed sets. Write n={*,1,2,,n}\langle n \rangle = \{*, 1,2, \cdots, n\} for the pointed set with n+1n+1 elements.

Let AA be a colored symmetric operad over Set ( hence a symmetric multicategory). Its category of operators is the category

  • whose objects are finite sequences (c 1,,c n)(c_1, \cdots, c_n) of colors of AA;

  • whose morphisms F:(c 1,,c n)(d 1,,d m)F : (c_1, \cdots, c_n) \to (d_1, \cdots, d_m) are given by a collection consisting of

    • a morphism ϕ:nm\phi : \langle n \rangle \to \langle m\rangle in FinSet *FinSet_*;

    • for each 1im1 \leq i \leq m an operation

      f iHom A((c k) kϕ 1(i),d i) f_i \in Hom_A((c_k)_{k \in \phi^{-1}(i)}, d_i)

      from the objects whose indices are mapped to ii to the object d id_i;

  • composition is given componentwise by the composition in FinSet *FinSet_* and in AA.

(,1)(\infty,1)-Category of operators

The above definition has been categorified to a notion of (∞,1)-category of operators. See at (∞,1)-operad for more.


By construction, the category of operators C AC_A of a symmetric colored operad is canonically equipped with a functor p:C AFinSet *p : C_A \to FinSet_*.

From this functor, the original operad may be recovered up to canonical equivalence.

Reconstruction of the operad

Given a functor p:C AFinSet *p : C_A \to FinSet_* from a category of operators of a symmetric colored operad, we reconstruct the operad AA as follows:

For 1in1 \leq i \leq n let

ρ i:n1 \rho^i : \langle n \rangle \to \langle 1 \rangle

be the map that sends all elements to the point, except the element ii.

Write A n:=p 1(n)A_{n} := p^{-1}(\langle n\rangle) for the fiber of pp over n\langle n\rangle. A 1A_1 is the category underlying the operad AA: the category whose morphisms are the unary operations of the operad.

The morphisms ρ i\rho^i in FinSet *FinSet_* induce a functor

iρ * i:A n(A 1) n \prod_i \rho^i_* : A_n \to (A_1)^n

which is an isomorphism that identifies A nA_n with the nn-fold cartesian product of the category A 1A_1 with itself.

The morphisms hHom A((c 1,,c n),d)h \in Hom_A((c_1, \cdots, c_n), d) of AA are recovered as the collection of morphisms in C AC_A from (c 1,,c n)(c_1, \cdots, c_n) to (d)(d) that cover the morphism n1\langle n\rangle \to \langle 1\rangle in FinSet *FinSet_* whose preimage of the point contains just the point.

Relation to underlying multicategories

Forming categories of operators is left 2-adjoint to forming the underlying multicategory of a semi-cartesian monidal category. (For a left adjoint to the underlying multicategory of an arbitrary monoidal category, see instead props.) For the moment, see there for more details.


The notion originates in

A discussion of the general logic behind the notion is at

This summarizes aspects of

See example 11.20 there.

Revised on April 3, 2017 14:15:42 by Mike Shulman (