nLab
symmetric monoidal (infinity,1)-category

Contents

Context

(,1)(\infty,1)-Category theory

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Idea

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

This means that it is

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

Equivalently, a symmetric monoidal (,1)(\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 (,1)(\infty,1)-categories (particularly, all of those that are locally presentable) can be presented by model categories, many symmetric monoidal (,1)(\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 N(Δ) opC^\otimes \to N(\Delta)^{op} of simplicial sets over the (opposite of) the nerve N(Δ) opN(\Delta)^{op} of the simplex category satisfying a certain property.

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

The following definition encodes the commutativity of all these operations by replacing Δ\Delta with the category FinSet *FinSet_* of pointed finite sets.

Definition

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

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

such that

  • for each n0n \geq 0 the associated functors C [n] C [1] C^\otimes_{[n]} \to C^\otimes_{[1]} determine an equivalence of (,1)(\infty,1)-categories C [n] C [1] nC^\otimes_{[n]} \stackrel{\simeq}{\to} C_{[1]}^n.
Remark

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

𝒪=Com \mathcal{O} = Com

the commutative (∞,1)-operad.

See (Lurie, def. 2.0.0.7).

Proposition

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

Remark

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

Examples

Classes of examples

Specific examples

Properties

Model category structure

A presentation of the (∞,1)-category of all symmetric monoidal (,1)(\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.

References

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

and definition 2.0.0.7 in

A concise treatment is also available in

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

Last revised on December 6, 2017 at 13:11:17. See the history of this page for a list of all contributions to it.