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monoidal (infinity,1)-category

higher category theory - contents

Idea

The notion of monoidal $(\mathrm{\infty }\mathrm{,1})$-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

• Jacob Lurie, Noncommutative algebra (arXiv)

Another is in terms of operads. This approach has been taken by John Francis in his thesis.

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

$$*\to B\mathrm{Cat}$$* \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 ${\lim }^{\Delta }{\mathrm{const}}_{B\mathrm{Cat}}$, namely a diagram

$$\begin{array}{ccc}F{\Delta }^4& \stackrel{=}{\to }& B\mathrm{Cat}\\ \uparrow & & {\uparrow }^{\mathrm{Id}}\\ F{\Delta }^3& \stackrel{\alpha }{\to }& B\mathrm{Cat}\\ \uparrow & & {\uparrow }^{\mathrm{Id}}\\ F{\Delta }^2& \stackrel{\otimes }{\to }& B\mathrm{Cat}\\ \uparrow & & {\uparrow }^{\mathrm{Id}}\\ F{\Delta }^1& \stackrel{C}{\to }& B\mathrm{Cat}\\ \uparrow & & {\uparrow }^{\mathrm{Id}}\\ F{\Delta }^0& \stackrel{•}{\to }& B\mathrm{Cat}\end{array}$$\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

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

$$\begin{array}{c}{C}^{\otimes }\to F({\Delta }^4)\end{array}.$$\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 equivlent to monoidal categoriesy. Alternatively, one can read pages 5 and 6 of LurieNonCom cited below.

In any case, this motivates the following definition.

Definition

simplicial/geometric definition

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

• a simplicial set $C^{\otimes }$;

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

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

$$C_{[n]}^{\otimes }\to C_{\{\mathrm{0,1}\}}^{\otimes }\times \cdots \times C_{\{n-1,n\}}^{\otimes }\simeq (C_{[1]}^{\otimes }{)}^n.$$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.

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 $\mathrm{\infty }$-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. $\mathrm{\infty }$-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 $1\le n\le \mathrm{\infty }$ 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

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

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

• $(\mathrm{\infty }\mathrm{,1})$-categories with an $E_n$-action for $1<n<\mathrm{\infty }$ are the corresponding $n$-tuply monoidal $(\mathrm{\infty }\mathrm{,1})$-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 $(\mathrm{\infty }\mathrm{,1})$-categories.

References

the simplicial definition is definition 1.1.2 in

• Jacob Lurie, Noncommutative algebra (arXiv)

John Francis’ work on $E_n$-actions on $(\mathrm{\infty }\mathrm{,1})$-categories is here

• EnAction John Francis, PhD thesis (web)