Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A symmetric monoidal -category is
which is “-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 -category is a commutative algebra in an (infinity,1)-category in the (infinity,1)-category of (infinity,1)-categories.
Just as many ordinary -categories (particularly, all of those that are locally presentable) can be presented by model categories, many symmetric monoidal -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 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
such that
In other words, a symmetric monoidal -category is an -monoidal (∞,1)-category for
the commutative (∞,1)-operad.
See (Lurie, def. 2.0.0.7).
The homotopy category of a symmetric monoidal -category is an ordinary symmetric monoidal category.
There is a functor such that the monoidal (infinity,1)-category underlying a symmetric monoidal -category is the (infinity,1)-pullback of along .
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.
symmetric monoidal category, symmetric monoidal -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 -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.