Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
symmetric monoidal (∞,1)-category of spectra
The notion of commutative monoid (or commutative monoid object, commutative algebra, commutative algebra object) in a symmetric monoidal (infinity,1)-category is the (infinity,1)-categorical generalization of the notion of commutative monoid in a symmetric monoidal category. It is the commutative version of monoid in a monoidal (infinity,1)-category.
Note that commutative here really means , in the sense of E-infinity operad.
A commutative monoid in a symmetric monoidal (infinity,1)-category is a lax symmetric monoidal -functor
In more detail, this means the following:
Given a symmetric monoidal (infinity,1)-category in its quasi-categorical incarnation as a coCartesian fibration of simplicial sets
a commutative monoid in is a section
such that carries collapsing morphisms in to coCartesian morphisms in .
For a symmetric monoidal (∞,1)-category write for the -category of commutative monoids in .
has all (∞,1)-coproducts and these are computed as tensor products in .
For a sifted (infinity,1)-category , (∞,1)-colimits of shape exist in and are computed in if -colimits exist in are preserved by tensor product with any object.
has all (∞,1)-limits and these are computed in .
This is (Lurie DAG III, section 4) or (Lurie HA, sections 3.2.2 and 3.2.3).
-Colimits over simplicial diagrams exists in and are computed in if they exist in and a preserved by tensor products.
Because the simplex category is a sifted (infinity,1)-category (as discussed there).
A commutative monoid in the stable (infinity,1)-category of spectra is a commutative ring spectrum or E-infinity ring.
An equivalent reformulation of commutative monoids in terms (∞,1)-algebraic theories is in
Last revised on March 9, 2015 at 12:35:40. See the history of this page for a list of all contributions to it.