With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
Algebras and modules
Model category presentations
Geometry on formal duals of algebras
The notion of monoidal -category is the analogue of the notion of monoidal category in the context of (∞,1)-categories.
There are various ways to state the monoidal structure. One is in terms of fibrations over the simplex category. This is the approach taken in
Another is in terms of (∞,1)-operads (see there). This approach has been taken in (Fracis)
Both are described below.
Idea of the simplicial definition
As discussed at the end of the entry on monoidal category, an ordinary monoidal category may be thought of as a lax functor
from the terminal category to the one-object 3-category whose single hom-object is the 2-category Cat of all categories and for which composition is the cartesian monoidal structure on Cat.
More concretely, as also described there, such a lax functor is a kind of descent object in a weighted limit , namely a diagram
where is the free 3-category of the -simplex (an oriental), where the horizontal morphisms are the chosen data – the category , product , associator and, in degree 4, the respect for the pentagon identity – and the condition is that this commutes for all vertical morphisms .
So this is a 4-functor
subject to a certain constraint.
Using the general mechanism of generalized universal bundles, this classifies a Cat-bundle
With a bit more time than I have on the train one can figure out that conversely suitable such fibrations are equivalent to monoidal categories. Alternatively, one can read pages 5 and 6 of LurieNonCom cited below.
In any case, this motivates the following definition.
Plain monoidal -category
A monoidal ()-category is
Here is the simplex category and its nerve.
The following defines symmetric monoidal (∞,1)-categories and their variants, where the commutative operad is replaced by any other (∞,1)-operad.
Let be an (∞,1)-operad. A coCartesian fibration of (∞,1)-operads is
a coCartesian fibration of the underlying quasi-categories;
such that the composite
exhibits as an (∞,1)-operad.
In this case we say that the underlying (∞,1)-category
is equipped by with the structure of an -monoidal -category.
This is (Lurie, def. 188.8.131.52).
Higher monoidal structure
While for an ordinary monoid there is just one notion of commutativity (either it is or it is not commutative), already a monoidal category distinguishes between being just braided monoidal or fully symmetric monoidal.
This pattern continues, as expressed by the periodic table of k-tuply monoidal categories.
A higher category may be a k-tuply monoidal n-category or more generally k-tuply monoidal (n,r)-category for different values of . The lowest value of (since for there is no monoidal structure at all) corresponds to monoidal product which is -associative, i.e. associative up to higher coherent homotopies, but need not have any degree of commutativity.
One says that an -category is symmetric monoiodal if it is “as monoidal as possible”, i.e. -tuply monoidal. In particular, in Noncommutative algebra and Commutative algebra we have
Operadic/algebraic definition of monoidal structure
For each let denote the little n-disk operad whose topological space of of -ary operations is the space of embeddings of -dimensional disks (balls) in one -dimensional disk without intersection, and whose composition operation is the obvious one obtained from gluing the big outer disks into given inner disks.
In John Francis’ PhD thesis the theory of (∞,1)-categories equipped with an action of the -operad is established, so that
-categories with an -action are precisely monoidal (∞,1)-categories – 1-fold monoidal -categories;
-categories with an -action are precisely symmetric monoidal (∞,1)-categories – -tuply monoidal -categories;
-categories with an -action for are the corresponding -tuply monoidal -categories in between.
Remark The second statement is example 2.3.8 in EnAction. The first seems to be clear but is maybe not in the literature. Jacob Lurie is currently rewriting Higher Algebra such as to build in a discussion of -operadic structures in the definition of -tuply monoidal -categories.
monoidal category, monoidal (2,1)-category, monoidal -category
braided monoidal category, braided monoidal (∞,1)-category
symmetric monoidal category, symmetric monoidal (2,1)-category, symmetric monoidal (∞,1)-category
closed monoidal category , closed monoidal (∞,1)-category
prime spectrum of a monoidal stable (∞,1)-category
The simplicial definition for plain monoidal -categories is definition 1.1.2 in
John Francis’ work on little cubes operad-actions on -categories is in
The general notion of an -monoidal -category is around definition 184.108.40.206 of
An introductory survey is in