symmetric monoidal (infinity,1)-category


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

Monoidal categories



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.


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.

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.


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


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.


Classes of examples

Specific examples


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.


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

and definition 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

Revised on December 6, 2017 13:11:17 by Mike Shulman (