Background
Basic concepts
equivalences in/of $(\infty,1)$-categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
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 $E_\infty$, in the sense of E-infinity operad.
A commutative monoid in a symmetric monoidal (infinity,1)-category $C$ is a lax symmetric monoidal $(\infty,1)$-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 $C$ is a section
such that $A$ carries collapsing morphisms in $FinSet_*$ to coCartesian morphisms in $C^\otimes$.
For $C$ a symmetric monoidal (∞,1)-category write $CMon(C)$ for the $(\infty,1)$-category of commutative monoids in $C$.
$CMon(C)$ has all (∞,1)-coproducts and these are computed as tensor products in $C$.
For $K$ a sifted (infinity,1)-category , (∞,1)-colimits of shape $K$ exist in $CMon(C)$ and are computed in $C$ if $K$-colimits exist in $C$ are preserved by tensor product with any object.
$CMon(C)$ has all (∞,1)-limits and these are computed in $C$.
This is (Lurie DAG III, section 4) or (Lurie HA, sections 3.2.2 and 3.2.3).
$(\infty,1)$-Colimits over simplicial diagrams exists in $CMon(C)$ and are computed in $C$ if they exist in $C$ 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 October 24, 2023 at 05:10:41. See the history of this page for a list of all contributions to it.