nLab
symmetric monoidal (infinity,1)-category

Idea

A symmetric monoidal $(\mathrm{\infty },1)$-category is

• an (∞,1)-category

• which is ”$\mathrm{\infty }$-tuply monoidal”, or “stably monoidal”.

This means that it is

• a monoidal (∞,1)-category;

• for which the tensor product is commutative up to infinite coherent homotopy.

This can be understood as a special case of an (∞,1)-operad (…to be expanded on…)

Definition in terms of quasi-categories

Recall that in terms of quasi-categories a general monoidal (infinity,1)-category is conceived as a coCartesian fibration $C^{\otimes }\to N(\Delta {)}^{\mathrm{op}}$ of simplicial sets over the (opposite of) the nerve $N(\Delta {)}^{\mathrm{op}}$ of the simplex category satisfying a certain property.

The fiber of this fibration over the 1-simplex $[1]$ is the monoidal (infinity,1)-category $C$ itself, its value over a map $[n]\to [1]$ encodes the tensor product of $n$ factors of $C$ with itself.

The following definition encodes the commutativity of all these operations by replacing $\Delta$ with the category ${\mathrm{FinSet}}_{*}$ of pointed finite sets.

Definition

A symmetric monoidal $(\mathrm{\infty },1)$-category is a coCartesian fibration of simplicial sets

$$p:C^{\otimes }\to N({\mathrm{FinSet}}_{*})$$p : C^\otimes \to N(FinSet_*)

such that

• for each $n\ge 0$ the associated functors $C_{[n]}^{\otimes }\to C_{[1]}^{\otimes }$ determine an equivalence of $(\mathrm{\infty },1)$-categories $C_{[n]}^{\otimes }\stackrel{\simeq }{\to }{C}_{[1]}^n$.

Remark

In other words, a symmetric monoidal $(\mathrm{\infty },1)$-category is an $𝒪$-monoidal (∞,1)-category for

$$𝒪=\mathrm{Com}$$\mathcal{O} = Com

the commutative (∞,1)-operad.

See (Lurie, def. 2.0.0.7).

Proposition

The homotopy category of a symmetric monoidal $(\mathrm{\infty },1)$-category is an ordinary symmetric monoidal category.

Remark

There is a functor $\phi :{\Delta }^{\mathrm{op}}\to {\mathrm{FinSet}}_{*}$ such that the monoidal (infinity,1)-category underlying a symmetric monoidal $(\mathrm{\infty },1)$-category $p:C^{\otimes }\to N({\mathrm{FinSet}}_{*})$ is the (infinity,1)-pullback of $p$ along $\phi$.

Examples

Classes of examples

• cartesian monoidal (∞,1)-category

Specific examples

• stable (infinity,1)-category of spectra

• symmetric monoidal (infinity,1)-category of presentable (infinity,1)-categories

Properties

Model category structure

A presentation of the (∞,1)-category of all symmetric monoidal $(\mathrm{\infty },1)$-categories is provided by the model structure for dendroidal coCartesian fibrations.

Commutative $\mathrm{\infty }$-monoids

See commutative monoid in a symmetric monoidal (∞,1)-category.

Related concepts

• monoidal category, monoidal (∞,1)-category

• symmetric monoidal category, symmetric monoidal $(\mathrm{\infty },1)$-category, symmetric monoidal (∞,n)-category

• closed monoidal category , closed monoidal (∞,1)-category

References

The defintion of symmetric monoidal quasi-category is definition 1.2 in

• Jacob Lurie, Commutative Algebra

and definition 2.0.0.7 in

• Jacob Lurie, Higher Algebra