With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
A symmetric monoidal -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 -category is a commutative algebra in an (infinity,1)-category in the (infinity,1)-category of (infinity,1)-categories.
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 of simplicial sets over the (opposite of) the nerve of the simplex category satisfying a certain property.
The fiber of this fibration over the 1-simplex is the monoidal (infinity,1)-category itself, its value over a map encodes the tensor product of factors of with itself.
The following definition encodes the commutativity of all these operations by replacing with the category of pointed finite sets.
A symmetric monoidal -category is a coCartesian fibration of simplicial sets
- for each the associated functors determine an equivalence of -categories .
See (Lurie, def. 220.127.116.11).
Classes of examples
Model category structure
A presentation of the (∞,1)-category of all symmetric monoidal -categories is provided by the model structure for dendroidal coCartesian fibrations.
See commutative monoid in a symmetric monoidal (∞,1)-category.
monoidal category, monoidal (∞,1)-category
symmetric monoidal category, symmetric monoidal -category, symmetric monoidal (∞,n)-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
The defintion of symmetric monoidal quasi-category is definition 1.2 in
and definition 18.104.22.168 in
A concise treatment is also available in
Relation to monoidal model categories is discussed in