nLab monoidal (infinity,1)-category

Idea

The notion of monoidal $(\infty,1)$-category is the analogue of the notion of monoidal category in the context of (∞,1)-categories. Monoidal $(\infty,1)$-categories form an (∞,2)-category, Mon(∞,1)Cat.

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

• Jacob Lurie, Noncommutative algebra (arXiv)

Another is in terms of (∞,1)-operads (see there). This approach has been taken in (Francis). Both are described below.

Just as many ordinary $(\infty,1)$-categories (particularly, all of those that are locally presentable) can be presented by model categories, many monoidal $(\infty,1)$-categories can be presented by monoidal model categories. See for instance NikolausSagave15.

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

$$* \to \mathbf{B} Cat$$

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 $lim^{\Delta} const_{\mathbf{B}Cat}$, namely a diagram

$$\array{ F \Delta^4 &\stackrel{=}{\to}& \mathbf{B} Cat \\ \uparrow && \uparrow^{Id} \\ F \Delta^3 &\stackrel{\alpha}{\to}& \mathbf{B} Cat \\ \uparrow && \uparrow^{Id} \\ F \Delta^2 &\stackrel{\otimes}{\to}& \mathbf{B} Cat \\ \uparrow && \uparrow^{Id} \\ F \Delta^1 &\stackrel{C}{\to}& \mathbf{B}Cat \\ \uparrow && \uparrow^{Id} \\ F\Delta^0 &\stackrel{\bullet}{\to}& \mathbf{B}Cat }$$

where $F \Delta^n$ is the free 3-category of the $n$-simplex (an oriental), where the horizontal morphisms are the chosen data – the category $C$, product $\otimes$, associator $\alpha$ and, in degree 4, the respect for the pentagon identity – and the condition is that this commutes for all vertical morphisms $F(\Delta^n \to \Delta^m)$.

So this is a 4-functor

$$F(\Delta^4) \to \mathbf{B}Cat$$

subject to a certain constraint.

Using the general mechanism of generalized universal bundles, this classifies a Cat-bundle

$$\array{ C^\otimes \to F(\Delta^4) } \,.$$

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.

Definition

Plain monoidal $(\infty,1)$-category

Definition

A monoidal ($\infty,1$)-category $(C, \otimes)$ is

• a simplicial set $C^\otimes$;

• and a coCartesian fibration of simplicial sets $p_\otimes : C^\otimes \to N(\Delta)^{op}$

• such that for each $n \in \mathbb{N}$ the induced (infinity,1)-functor $C^\otimes_{[n]} \to C^\otimes_{\{i,i+1\}}$ determines an equivalence of (infinity,1)-categories

$$C^\otimes_{[n]} \to C^\otimes_{\{0,1\}} \times \cdots \times C^\otimes_{\{n-1,n\}} \simeq (C^\otimes_{[1]})^n \,.$$

Here $\Delta$ is the simplex category and $N(\Delta)$ its nerve.

$\mathcal{O}$-monoidal $(\infty,1)$-category

The following defines symmetric monoidal (∞,1)-categories and their variants, where the commutative operad is replaced by any other (∞,1)-operad.

Definition

Let $\mathcal{O}^\otimes$ be an (∞,1)-operad. A coCartesian fibration of (∞,1)-operads is

1. a coCartesian fibration $p : \mathcal{C}^\otimes \to \mathcal{O}^\otimes$ of the underlying quasi-categories;

2. such that the composite

   $$\mathcal{C}^\otimes \stackrel{p}{\to} \mathcal{O}^\otimes \to Fin_* = Comm$$

   exhibits $\mathcal{C}^\otimes$ as an (∞,1)-operad.

In this case we say that the underlying (∞,1)-category

$$\mathcal{C} = \mathcal{C}^\otimes \times_{\mathcal{O}^\otimes} \mathcal{O}$$

is equipped by $p$ with the structure of an $\mathcal{O}$-monoidal $(\infty,1)$-category.

This is (Lurie, def. 2.1.2.13).

Definition

For $\mathcal{O}$ = Comm, an $\mathcal{O}$-monoidal $(\infty,1)$-category is a symmetric monoidal (∞,1)-category.

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 $k$. The lowest value of $k= 1$ (since for $k = 0$ there is no monoidal structure at all) corresponds to monoidal product which is $\infty$-associative, i.e. associative up to higher coherent homotopies, but need not have any degree of commutativity.

One says that an $n$-category is symmetric monoiodal if it is “as monoidal as possible”, i.e. $\infty$-tuply monoidal. In particular, in Noncommutative algebra and Commutative algebra we have

• the 1-fold monoidal (∞,1)-categories described here;

• ∞-tuply monoidal (∞,1)-categories.

Operadic/algebraic definition of monoidal structure

For each $1 \leq n \leq \infty$ let $E_n$ denote the little n-disk operad whose topological space of $E_n^k$ of $k$-ary operations is the space of embeddings of $k$ $n$-dimensional disks (balls) in one $n$-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 $E_n$-operad is established, so that

• $(\infty,1)$-categories with an $E_1$-action are precisely monoidal (∞,1)-categories – 1-fold monoidal $(\infty,1)$-categories;

• $(\infty,1)$-categories with an $E_\infty$-action are precisely symmetric monoidal (∞,1)-categories – $\infty$-tuply monoidal $(\infty,1)$-categories;

• $(\infty,1)$-categories with an $E_n$-action for $1 \lt n \lt \infty$ are the corresponding $n$-tuply monoidal $(\infty,1)$-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 $E_n$-operadic structures in the definition of $k$-tuply monoidal $(\infty,1)$-categories.

Related concepts

• monoidal category, monoidal (2,1)-category, monoidal $(\infty,1)$-category

  • monoidal (∞,1)-functor

  • Picard ∞-group

• braided monoidal category, braided monoidal (∞,1)-category

  • braided ∞-group

• symmetric monoidal category, symmetric monoidal (2,1)-category, symmetric monoidal (∞,1)-category

  • abelian ∞-group

• closed monoidal category , closed monoidal (∞,1)-category

• prime spectrum of a monoidal stable (∞,1)-category

References

The simplicial definition for plain monoidal $(\infty,1)$-categories is definition 1.1.2 in

• Jacob Lurie, Noncommutative Algebra (arXiv)

John Francis’ work on little cubes operad-actions on $(\infty,1)$-categories is in

• John Francis, PhD thesis (web)

The general notion of an $\mathcal{O}$-monoidal $(\infty,1)$-category is around definition 2.1.2.13 of

• Jacob Lurie, Higher Algebra .

An introductory survey is in

• Moritz Groth, A short course on $\infty$-categories (pdf)

A relation to monoidal model categories (in particular, that every locally presentable symmetric monoidal $(\infty,1)$-category arises from a symmetric monoidal model category) is described in

• Thomas Nikolaus, Steffen Sagave, Presentably symmetric monoidal infinity-categories are represented by symmetric monoidal model categories. Algebr. Geom. Topol. 17 (2017), no. 5, 3189–3212. (arXiv:1506.01475)