Background
Basic concepts
equivalences in/of $(\infty,1)$-categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
A symmetric monoidal $(\infty,1)$-category is
which is “$\infty$-tuply monoidal”, or “stably monoidal”.
This means that it is
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.
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.
A symmetric monoidal $(\infty,1)$-category is a coCartesian fibration of simplicial sets
such that
In other words, a symmetric monoidal $(\infty,1)$-category is an $\mathcal{O}$-monoidal (∞,1)-category for
the commutative (∞,1)-operad.
See (Lurie, def. 2.0.0.7).
The homotopy category of a symmetric monoidal $(\infty,1)$-category is an ordinary symmetric monoidal category.
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 $\varphi$.
A presentation of the (∞,1)-category of all symmetric monoidal $(\infty,1)$-categories is provided by the model structure for dendroidal coCartesian fibrations.
See commutative monoid in a symmetric monoidal (∞,1)-category.
symmetric monoidal category, symmetric monoidal $(\infty,1)$-category, symmetric monoidal (∞,n)-category
prime spectrum of a symmetric monoidal stable (∞,1)-category
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 $(\infty,1)$-category arises from a symmetric monoidal model category) is discussed in
Last revised on December 6, 2017 at 18:11:17. See the history of this page for a list of all contributions to it.