symmetric monoidal (∞,1)-category of spectra
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.
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
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 $A$.
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 $C_A$ 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.
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
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
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.
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.
Last revised on April 3, 2017 at 14:15:42. See the history of this page for a list of all contributions to it.