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 is the colored symmetric operad whose
objects are two elements, to be denoted and ;
multimorphisms form
if and for all then: the set of linear orders on elements, equivalently the elements of the symmetric group ;
if and exactly one of the then: the set of linear order such that ;
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 for the associative operad, regarded as a symmetric operad.
There is a canonical inclusion morphism
given by labeling the unique object/color of with . For 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.