nLab
symmetric monoidal (infinity,1)-category

Idea

A symmetric monoidal $(\mathrm{\infty }\mathrm{,1})$-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 $(\mathrm{\infty }\mathrm{,1})$-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 $(\mathrm{\infty }\mathrm{,1})$-categories $C_{[n]}^{\otimes }\stackrel{\simeq }{\to }{C}_{[1]}^n$.

Proposition

The homotopy category of a symmetric monoidal $(\mathrm{\infty }\mathrm{,1})$-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 $(\mathrm{\infty }\mathrm{,1})$-category $p:C^{\otimes }\to N({\mathrm{FinSet}}_{*})$ is the (infinity,1)-pullback? of $p$ along $\phi$.

Examples

• stable (infinity,1)-category of spectra

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

References

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

• Jacob Lurie, Commutative algebra