symmetric monoidal (∞,1)-category of spectra
An operad whose algebras over an operad are triples consistsing of two associative algebras , and an --bimodule .
Write for the colored symmetric operad whose
objects are three elements, to be denoted and ;
multimorphisms form
if and all then: the set of linear orders of elements;
if and all then again: the set of linear orders of elements;
if : the set of linear orders such that there is exactly one index with and for all and for all .
composition is given by the composition of linear orders as for the associative operad.
There are two canonical inclusions of the associative operad given by labelling its unique color/object with either or , respectively. For
a morphism to a symmetric monoidal category, there compositions pick the left and the right algebra
There is also a morphism given by forgetting the labels and just remembering the linear orders.
This is a fibration of (∞,1)-operads.
In (Lurie) this appears as remark 4.3.1.8.
This is such that for
an algebra, the composite
exhibits canonically as a bimodule over itself.
Similarly, there is an inclusion of the operad for modules over an algebra
etc.
A coCartesian fibration of (∞,1)-operads , hence the structure of a -monoidal (∞,1)-category is a bitensoring of
over the (ordinary) monoidal (∞,1)-categories
By the microcosm principle, bitensored -categories are the right context into which to internalize bimodules. See Relation to the category of bimodules below.
For a fibration of (∞,1)-operads the corresponding (∞,1)-category of (∞,1)-algebras over an (∞,1)-operad
is the (∞,1)-category of (∞,1)-bimodules in .
Section 4.3.1 in
Last revised on September 20, 2024 at 19:56:36. See the history of this page for a list of all contributions to it.