Algebras and modules
Model category presentations
Geometry on formal duals of algebras
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 -ary operation of the operad for each collection of 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.
For symmetric colored operads
Let be the category of finite pointed sets. Write for the pointed set with elements.
Let be a colored symmetric operad over Set ( hence a symmetric multicategory). Its category of operators is the category
whose objects are finite sequences of colors of ;
whose morphisms are given by a collection consisting of
a morphism in ;
for each an operation
from the objects whose indices are mapped to to the object ;
composition is given componentwise by the composition in and in .
-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 of a symmetric colored operad is canonically equipped with a functor .
From this functor, the original operad may be recovered up to canonical equivalence.
Reconstruction of the operad
Given a functor from a category of operators of a symmetric colored operad, we reconstruct the operad as follows:
be the map that sends all elements to the point, except the element .
Write for the fiber of over . is the category underlying the operad : the category whose morphisms are the unary operations of the operad.
The morphisms in induce a functor
which is an isomorphism that identifies with the -fold cartesian product of the category with itself.
The morphisms of are recovered as the collection of morphisms in from to that cover the morphism in whose preimage of the point contains just the point.
Relation to endomorphism operads
Forming categories of operators is left 2-adjoint to forming endomorphism operads. For the moment, see there for more details.
The notion originates in
- Peter May, R. Thomason, The uniqueness of infinite loop space machines , Topology 17(3) (pdf)
A discussion of the general logic behind the notion is at
This summarizes aspects of
See example 11.20 there.