representation, ∞-representation?
symmetric monoidal (∞,1)-category of spectra
An operad whose algebras over an operad are pairs consisting of an associative algebra and a module over that algebra
The operad for modules over an algebra $LM$ is the colored symmetric operad whose
objects are two elements, to be denoted $\mathfrak{a}$ and $\mathfrak{n}$;
multimorphisms$(X_i)_{i = 1}^n \to Y$ form
if $Y = \mathfrak{a}$ and $X_i = \mathfrak{a}$ for all $i$ then: the set of linear orders on $n$ elements, equivalently the elements of the symmetric group $\Sigma_n$;
if $Y = \mathfrak{n}$ and exactly one of the $X_i = \mathfrak{n}$ then: the set of linear order $\{i_1 \lt \cdots \lt i_n\}$ such that $X_{i_n} = \mathfrak{n}$;
otherwise: the empty set;
composition is given by composition of linear orders as for the associative operad.
In (Lurie) this appears as def. 4.2.1.1.
Write $Assoc$ for the associative operad, regarded as a symmetric operad.
There is a canonical inclusion morphism
given by labeling the unique object/color of $Assoc$ with $\mathfrak{a}$. For $(A,N) \colon LM^\otimes \to \mathcal{C}^\otimes$ a map to a symmetric monoidal category the composite
is the corresponding associative algebra.
There is also a morphism
given by forgetting the color and just remembering the linear orders. This is such that for
and algebra, the composite
is the algebra canonically regarded as a module over itself.
Analogous relations exist to the operad for bimodules over algebras.
(…)
Section 4.2.1 of
Last revised on February 12, 2013 at 10:03:43. See the history of this page for a list of all contributions to it.