nLab
symmetric monoidal (infinity,1)-category

Idea

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

• an (∞,1)-category

• which is “$\infty$-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…)

Equivalently, a symmetric monoidal $(\infty,1)$-category is a commutative algebra in an (infinity,1)-category in the (infinity,1)-category of (infinity,1)-categories.

Just as many ordinary $(\infty,1)$-categories (particularly, all of those that are locally presentable) can be presented by model categories, many symmetric monoidal $(\infty,1)$-categories can be presented by symmetric monoidal model categories. See for instance NikolausSagave15.

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 $C^\otimes \to N(\Delta)^{op}$ of simplicial sets over the (opposite of) the nerve $N(\Delta)^{op}$ 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 $FinSet_*$ of pointed finite sets.

Definition

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

$$p : C^\otimes \to N(FinSet_*)$$

such that

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

Remark

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

$$\mathcal{O} = Com$$

the commutative (∞,1)-operad.

See (Lurie, def. 2.0.0.7).

Proposition

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

Remark

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

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 $(\infty,1)$-categories is provided by the model structure for dendroidal coCartesian fibrations.

Commutative $\infty$-monoids

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

Related concepts

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

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

• tensor (∞,1)-category

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

• K-theory of a symmetric monoidal (∞,1)-category

• prime spectrum of a symmetric monoidal stable (∞,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

A concise treatment is also available in

• Moritz Groth, A short course on infinity-categories, pdf.

Relation to monoidal model categories (in particular, that every locally presentable symmetric monoidal $(\infty,1)$-category arises from a symmetric monoidal model category) is discussed in

• Thomas Nikolaus, Steffen Sagave, Presentably symmetric monoidal infinity-categories are represented by symmetric monoidal model categories. Algebr. Geom. Topol. 17 (2017), no. 5, 3189–3212. (arXiv:1506.01475)