nLab category of operators

Idea

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 $k$-ary operation of the operad for each collection of $k$ 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.

Definition

For symmetric colored operads

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

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

• whose objects are finite sequences $(c_1, \cdots, c_n)$ of colors of $A$;

• whose morphisms $F : (c_1, \cdots, c_n) \to (d_1, \cdots, d_m)$ are given by a collection consisting of

  • a morphism $\phi : \langle n \rangle \to \langle m\rangle$ in $FinSet_*$;

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

    $$f_i \in Hom_A((c_k)_{k \in \phi^{-1}(i)}, d_i)$$

    from the objects whose indices are mapped to $i$ to the object $d_i$;

• composition is given componentwise by the composition in $FinSet_*$ and in $A$.

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

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

Properties

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

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

Reconstruction of the operad

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

For $1 \leq i \leq n$ let

$$\rho^i : \langle n \rangle \to \langle 1 \rangle$$

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

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

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

$$\prod_i \rho^i_* : A_n \to (A_1)^n$$

which is an isomorphism that identifies $A_n$ with the $n$-fold cartesian product of the category $A_1$ with itself.

The morphisms $h \in Hom_A((c_1, \cdots, c_n), d)$ of $A$ are recovered as the collection of morphisms in $C_A$ from $(c_1, \cdots, c_n)$ to $(d)$ that cover the morphism $\langle n\rangle \to \langle 1\rangle$ in $FinSet_*$ 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 monoidal 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.

Related concepts

• The (syntactic category of) a (homogeneous) globular theory is the category of operators of a globular operad.

References

The notion originates in

• Peter May, Robert Thomason, The uniqueness of infinite loop space machines , Topology 17(3) (pdf)

A discussion of the general logic behind the notion is at

• Mike Shulman, Generalized operads in classical algebraic topology

  (blog)

This summarizes aspects of

• G. Cruttwell, Mike Shulman, A unified framework for generalized multicategories (arXiv:0907.2460)

See example 11.20 there (note that this example is missing from the final version of the paper).