nLab
k-tuply monoidal n-category

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higher category theory

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Contents

Idea

A kk-tuply monoidal nn-category is an nn-category in which objects can be multiplied in kk different ways, all of which interchange with each other up to equivalence. By the Eckmann–Hilton argument, this implies that these kk ways all end up being equivalent, but that the single resulting operation is more and more commutative as kk increases. The stabilization hypothesis states that by the time we reach k=n+2k = n + 2, the multiplication has become “maximally commutative.”

While there is maybe no generally accepted definition of kk-tuply monoidal nn-category yet, it seems that defining it to be an nn-category with an action of the little k-cubes operad makes good sense, as discussed further below. With this definition the stabilization hypothesis is a theorem at least for (n,1)(n,1)-categories (see below).

By the delooping hypothesis a kk-tuply monoidal nn-category can be interpreted as a special kind of (n+k)(n+k)-category. One may take this hypothesis as a definition, but it has been verified in many low-dimensional cases; see below.

Definition

For purposes of this page, a kk-tuply monoidal nn-category is a pointed (n+k)(n+k)-category such that any two parallel jj-morphisms are equivalent for j<kj \lt k. One usually relabels the jj-morphisms as (jk)(j-k)-morphisms. You may interpret this definition as weakly or strictly as you like, by starting with weak or strict notions of (n+k)(n+k)-category.

The given point serves as an equivalence between (1)(-1)-morphisms? (for now, see (n,r)(n,r)-category for these), so there is nothing to say if k0k \leq 0 except that the category is pointed. Thus we may as well assume that k0k \geq 0. Also, according to the stabilisation hypothesis, every kk-tuply monoidal nn-category for k>n+2k \gt n + 2 may be reinterpreted as an (n+2)(n+2)-tuply monoidal nn-category. Unlike the restriction k0k\ge 0, this one is not trivial.

Special cases

The periodic table

There is a periodic table of kk-tuply monoidal nn-categories:

k k ↓\ n n 1 -1 $0$ 1 1 2 2 ...
0 0 trivialpointed setpointed category?pointed 2-category?...
1 1 trivialmonoidmonoidal categorymonoidal 2-category...
2 2 "abelian monoidbraided monoidal categorybraided monoidal 2-category...
3 3 ""symmetric monoidal categorysylleptic monoidal 2-category...
4 4 """symmetric monoidal 2-category...
""""

Historical notes

Originally the importance of pointedness was not fully appreciated, so any nn-category was accepted as 00-tuply monoidal, and kk-tuply monoidal nn-categories were identified simply with (k1)(k-1)-simply connected (n+k)(n+k)-categories (those in which any two parallel jj-morphisms are equivalent for j<kj \lt k). See periodic table for this original.

Low dimensions

k=0k=0

As remarked above, a 00-tuply monoidal nn-category is just a pointed one, and functors and transformations between such are required to preserve the chosen object, at least up to specified coherent isomorphism. (In other words, the (n+1)(n+1)-category of 00-tuply monoidal nn-categories is the co-slice (n+1)(n+1)-category 1/nCat1/n Cat, where the slicing happens in a suitably weak (n+1)(n+1)-sense.

k=1k=1, n=0n=0

A 1-tuply monoidal 0-category is a pointed 0-connected 1-category, or a 1-category with a chosen object in which all objects are isomorphic. Thus we might as well as assume there is exactly one object, in which case we just have a monoid. A functor between one-object categories, which preserves the basepoint automatically, is exactly a monoid homomorphism.

More interestingly, a natural transformation between functors f,g:XYf,g:X\to Y between one-object categories is just an object yYy\in Y (its component at the single object) such that f(x)y=yg(x)f(x) y = y g(x) for all xXx\in X. So the 2-category of 0-connected 1-categories is not equivalent to the 1-category of monoids. However, a pointed natural transformation must have its component at the basepoint being the identity; thus y=1y=1 and so the only such natural transformations are identities f=gf=g. Therefore, the 2-category of pointed 0-connected 1-categories (that is, 1-tuply monoidal 0-categories) is equivalent to the 1-category of monoids.

k=2k=2, n=0n=0

A 2-tuply monoidal 0-category is a pointed 1-connected 2-category. Interpreting things as weakly as possible, we are talking about a bicategory BB with one object ** and one 1-cell (its identity). By the usual Eckmann-Hilton argument, the set B(1 *,1 *)B(1_*,1_*) is a commutative monoid, but there is also additional structure: the associatior and unitors of the bicategory. The pentagon identity implies that the associator is the identity, and the unitor axioms imply that the two unitors are the same, but they are not necessarily the identity. Therefore, a 1-connected 2-category (if by 2-category we mean bicategory) is a commutative monoid XX equipped with a chosen invertible element d Xd_X. This was apparently first observed by Tom Leinster.

In similar vein, one can work out (see Cheng–Gurski):

  • a (weak) functor between 1-connected bicategories is a monoid homomorphism F:XYF:X\to Y equipped with a distinguished invertible element m FYm_F\in Y.
  • a (weak) natural transformation between two monoid homomorphisms is just the assertion that they are equal (and thus, every such transformation is invertible).
  • a modification between two such transformations is a distinguished not-necessarily-invertible element ΓY\Gamma\in Y.

Note that the invertible elements d Xd_X and m Fm_F play no role in the definition of the higher morphisms, so they might as well not be there, up to equivalence. However, the nonidentity modifications do screw things up, so the tricategory of 1-connected bicategories is not equivalent to the 1-category of commutative monoids. But if we add in the basepoints, then we get:

  • a pointed 1-connected bicategory is one equipped with a functor from 11, the terminal bicategory. This is just a monoid homomorphism 1X1\to X, which of course is unique, together with a distinguished invertible element in XX which we can ignore. Thus every 1-connected bicategory can be pointed in an essentially unique way.

  • a pointed functor between two pointed 1-connected bicategories is a functor F:XYF:X\to Y together with a weak natural equivalence, say t Ft_F, between 1XY1\to X\to Y and 1Y1\to Y. Since these are always equal as monoid homomorphisms, there is always a unique (invertible) transformation connecting them, so such a pointed functor is just a monoid homomorphism F:XYF:X\to Y.

  • a pointed transformation from FF to GG is a transformation aa from F:XYF:X\to Y to G:XYG:X\to Y together with an invertible modification, say c ac_a, relating at Fa t_F to t Gt_G. In other words, it is an assertion that F=GF=G together with a distinguished invertible element Γ a\Gamma_a of YY.

  • a pointed modification from aa to bb is a modification mm such that mc a=c bm c_a = c_b. In other words, it is a distinguished element ΛY\Lambda\in Y such that ΛΓ a=Γ b\Lambda \Gamma_a = \Gamma_b, or Λ=Γ a 1Γ b\Lambda = \Gamma_a^{-1} \Gamma_b since Γ a\Gamma_a is invertible. Thus, any two pointed transformations FGF\to G are related by a unique invertible modification.

We conclude that the tricategory of pointed 1-connected bicategories is equivalent to the category of commutative monoids and monoid homomorphisms, so again the delooping hypothesis is verified.

One might complain that in addition of the single weak natural equivalence t Ft_F, FF ought also to be equipped with an inverse adjoint equivalence for it. The modifications involved in this would introduce two distinguished invertible elements in YY, which (by the triangle identities) would have to be each other’s inverses. But these elements would again play no role in the higher morphisms, so they might as well be identities.

k=1k=1, n=1n=1

A 1-tuply monoidal 1-category is a pointed 0-connected 2-category, which we can identify with a bicategory with one object. It is well-known that this is precisely the data of a monoidal category. Likewise, (weak) functors between such bicategories correspond precisely (strong) monoidal functors. However, again the transformations and modifications screw things up in the merely connected case, but by using pointed objects instead we can remedy the situation.

k=1k=1, n=(,0)n=(\infty,0)

If we identify \infty-groupoids with spaces, then a 1-tuply monoidal (,0)(\infty,0)-category, or a monoidal \infty-groupoid, can be identified intuitively with an A A_\infty-space. This is a space equipped with a multiplication which is associative and unital up to all higher homotopies; see operad for one way to encode these data.

It is a well-known fact in homotopy theory that the homotopy theories (that is, (,1)(\infty,1)-categories) of based connected spaces and of grouplike A A_\infty-spaces are equivalent, via the loop space and classifying space constructions. This can be regarded as another version of the delooping hypothesis. The “grouplike” restriction (meaning that π 0\pi_0 is a group, or that the multiplication has inverses up to homotopy) is because we consider only based connected (,0)(\infty,0)-categories, whereas we would need based connected (,1)(\infty,1)-categories to recover all A A_\infty-spaces. It would be interesting to verify the hypothesis in this case using one of the known models for (,1)(\infty,1)-categories.

n=(,0)n = (\infty,0) and \infty-stacks

A kk-tuply monoidal ∞-groupoid may be identified, under kk-fold delooping, with a (k1)(k-1)-connected \infty-groupoid. This statement holds in fact even for parameterized \infty-groupoids, i.e. for ∞-stacks.

Theorem (k-tuply monoidal \infty-stacks)

Let k>0k \gt 0, let 𝒳\mathcal{X} be an (∞,1)-category of (∞,1)-sheaves and let 𝒳 * k\mathcal{X}_*^{\geq k} denote the full sub-(∞,1)-category of the category 𝒳 *\mathcal{X}_{*} of pointed objects, spanned by those pointed objects thar are k1k-1-connected (i.e. their first kk ∞-stack homotopy groups) vanish. Then there is a canonical equivalence of (∞,1)-categories

𝒳 * kMon 𝔼[k] gp(𝒳). \mathcal{X}_*^{\geq k} \simeq Mon^{gp}_{\mathbb{E}[k]}(\mathcal{X}) \,.
Proof

This is EKAlg, theorem 1.3.6..

Specifically for 𝒳=Top\mathcal{X} = Top, this refines to the classical theorem by Peter May

Theorem

(May recognition theorem)

Let YY be a topological space equipped with an action of the little cubes operad 𝒞 k\mathcal{C}_k and suppose that XX is grouplike. Then YY is homotopy equivalent to a kk-fold loop space Ω kX\Omega^k X for some pointed topological space XX.

Proof

This is EkAlg, theorem 1.3.16.

n=(n,1)n = (n,1) and the stabilization hypothesis

An (n,1)-category is a nn-truncated (∞,1)-category

In John Francis’ PhD thesis (reference EnAction below ) the theory of (∞,1)-categories equipped with an action of the little k-cubes operad is established, so that

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

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

  • (,1)(\infty,1)-categories with an E nE_n-action for 1<n<1 \lt n \lt \infty are the corresponding kk-tuply monoidal (,1)(\infty,1)-categories in between.

Proposition

(stabilization for (n,1)(n,1)-categories)

Equipping an (n,1)-category of kk compatible monoidal structures for kn+2k \geq n + 2 (which is the same as equipping it with a little k-cubes operad action) is the same as equipping it with the structure of a symmetric monoidal (n,1)-category.

An aspect of this was demonstrated in terms of Tamsamani n-categories? in

A proof of the full statement in terms of quasi-categories is sketched in section 43.5 of

Probably the first full proof in print is given in

where it appears in example 1.2.3 as a direct consequence of a more general statement, corollary 1.1.10.

Other low-dimensional cases

One expects that

  • pointed 2-connected tricategories can be identified with commutative monoids (again),
  • pointed 1-connected tricategories can be identified with braided monoidal categories,
  • pointed 0-connected tricategories can be identified with monoidal bicategories,
  • and so on.

References

Some discussion of the peridodic table is in

The theory of kk-tuply monoidal (∞,1)-categories was maybe first studied in

and later further developed in

where also the proof of the stabilization hypothesis in this context is noticed.

Related discussion can be found in the theory of iterated monoidal categories. See for example

  • C. Balteanu, Z. Fiedorowicz, R. Schwänzl, Rainer Vogt, Iterated monoidal categories (pdf)

  • Clemens Berger, Double loop spaces, braided monoidal categories and algebraic 3-type of space (pdf)

Discussion

A previous version of this entry led to the following discussion

Mike Shulman: I would like to suggest that we switch to using symmetric monoidal rather than stably monoidal, and especially avoid calling these just stable. One advantage of “symmetric monoidal” is that it has a well-established meaning in low-dimensions; if I say “symmetric monoidal nn-category” then people who are familiar with symmetric monoidal 1-categories are more likely to have an intuitive understanding of what I mean than if I say “stably monoidal nn-category.”

Use of the word “stable” here also creates confusion with its other meanings (see here and here). Algebraic topologists often use “stable” to mean “related to spectra,” and spectra are related to, but distinct from, symmetric monoidal \infty-groupoids. (Connective spectra can be identified with symmetric groupal \infty-groupoids.) Lurie is also using “stable (,1)(\infty,1)-category” to mean “an (,1)(\infty,1)-category which behaves like the (,1)(\infty,1)-category of spectra.” One might not like this, but it is not original with him; several other algebraic topologists use “stable model category” in the same sense. And since we have the perfectly good alternative term “symmetric monoidal” to use here, which has other things to recommend it as well, why create needless confusion?

Toby: Hopefully John will admit that saying ‘stable’ instead of ‘stably monoidal’ (or ‘stably groupal’) was a slip of the tongue … pen … fingers. I'm used to ‘stably monoidal’, and I don't think that it should cause confusion —if used in full. Also, I think there's some historical confusion about ‘symmetric monoidal 22-category’ or maybe ‘symmetric monoidal 33-category’ that ‘stably monoidal’ isn't subject to, although that's a bit parochial.

Revised on February 10, 2014 12:38:18 by Toby Bartels (64.89.53.123)