homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
symmetric monoidal (∞,1)-category of spectra
A $k$-tuply monoidal $n$-category is an $n$-category in which objects can be multiplied in $k$ different ways, all of which interchange with each other up to equivalence. By the Eckmann–Hilton argument, this implies that these $k$ ways all end up being equivalent, but that the single resulting operation is more and more commutative as $k$ increases. The stabilization hypothesis states that by the time we reach $k = n + 2$, the multiplication has become “maximally commutative.”
While there is maybe no generally accepted definition of $k$-tuply monoidal $n$-category yet, it seems that defining it to be an $n$-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 $(n,1)$-categories (see below).
By the delooping hypothesis a $k$-tuply monoidal $n$-category can be interpreted as a special kind of $(n+k)$-category. One may take this hypothesis as a definition, but it has been verified in many low-dimensional cases; see below.
For purposes of this page, a $k$-tuply monoidal $n$-category is a pointed $(n+k)$-category such that any two parallel $j$-morphisms are equivalent for $j \lt k$. One usually relabels the $j$-morphisms as $(j-k)$-morphisms. You may interpret this definition as weakly or strictly as you like, by starting with weak or strict notions of $(n+k)$-category.
The given point serves as an equivalence between $(-1)$-morphisms? (for now, see $(n,r)$-category for these), so there is nothing to say if $k \leq 0$ except that the category is pointed. Thus we may as well assume that $k \geq 0$. Also, according to the stabilisation hypothesis, every $k$-tuply monoidal $n$-category for $k \gt n + 2$ may be reinterpreted as an $(n+2)$-tuply monoidal $n$-category. Unlike the restriction $k\ge 0$, this one is not trivial.
A $0$-tuply monoidal $n$-category is simply a pointed $n$-category, that is an $n$-category equipped with a chosen object.
A $1$-tuply monoidal $n$-category may be called simply a monoidal n-category?. See
A doubly monoidal $n$-category is a braided monoidal n-category?. See
A symmetric monoidal $n$-category is an $k$-tuply monoidal $n$-category for $k \geq n+2$. (This also called a stably monoidal $n$-category, but never a stable monoidal $n$-category.)
That this is indeed independent of $k$, as soon as $k$ is large enough, is the statement of the stabilization hypothesis, more on which below.
See
There is a periodic table of $k$-tuply monoidal $n$-categories:
$k$↓\$n$→ | $-1$ | $0$ | $1$ | $2$ | ... |
---|---|---|---|---|---|
$0$ | trivial | pointed set | pointed category? | pointed 2-category? | ... |
$1$ | trivial | monoid | monoidal category | monoidal 2-category | ... |
$2$ | " | abelian monoid | braided monoidal category | braided monoidal 2-category | ... |
$3$ | " | " | symmetric monoidal category | sylleptic monoidal 2-category | ... |
$4$ | " | " | " | symmetric monoidal 2-category | ... |
⋮ | " | " | " | " | ⋱ |
Originally the importance of pointedness was not fully appreciated, so any $n$-category was accepted as $0$-tuply monoidal, and $k$-tuply monoidal $n$-categories were identified simply with $(k-1)$-simply connected $(n+k)$-categories (those in which any two parallel $j$-morphisms are equivalent for $j \lt k$). See periodic table for this original.
As remarked above, a $0$-tuply monoidal $n$-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 $(n+1)$-category of $0$-tuply monoidal $n$-categories is the co-slice $(n+1)$-category $1/n Cat$, where the slicing happens in a suitably weak $(n+1)$-sense.
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 $f,g:X\to Y$ between one-object categories is just an object $y\in Y$ (its component at the single object) such that $f(x) y = y g(x)$ for all $x\in X$. 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 $y=1$ and so the only such natural transformations are identities $f=g$. 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.
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 $B$ with one object $*$ and one 1-cell (its identity). By the usual Eckmann-Hilton argument, the set $B(1_*,1_*)$ 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 $X$ equipped with a chosen invertible element $d_X$. This was apparently first observed by Tom Leinster.
In similar vein, one can work out (see Cheng–Gurski):
Note that the invertible elements $d_X$ and $m_F$ 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 $1$, the terminal bicategory. This is just a monoid homomorphism $1\to X$, which of course is unique, together with a distinguished invertible element in $X$ 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 $F:X\to Y$ together with a weak natural equivalence, say $t_F$, between $1\to X\to Y$ and $1\to Y$. 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 $F:X\to Y$.
a pointed transformation from $F$ to $G$ is a transformation $a$ from $F:X\to Y$ to $G:X\to Y$ together with an invertible modification, say $c_a$, relating $a t_F$ to $t_G$. In other words, it is an assertion that $F=G$ together with a distinguished invertible element $\Gamma_a$ of $Y$.
a pointed modification from $a$ to $b$ is a modification $m$ such that $m c_a = c_b$. In other words, it is a distinguished element $\Lambda\in Y$ such that $\Lambda \Gamma_a = \Gamma_b$, or $\Lambda = \Gamma_a^{-1} \Gamma_b$ since $\Gamma_a$ is invertible. Thus, any two pointed transformations $F\to G$ 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 $t_F$, $F$ ought also to be equipped with an inverse adjoint equivalence for it. The modifications involved in this would introduce two distinguished invertible elements in $Y$, 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.
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.
If we identify $\infty$-groupoids with spaces, then a 1-tuply monoidal $(\infty,0)$-category, or a monoidal $\infty$-groupoid, can be identified intuitively with an $A_\infty$-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, $(\infty,1)$-categories) of based connected spaces and of grouplike $A_\infty$-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 $\pi_0$ is a group, or that the multiplication has inverses up to homotopy) is because we consider only based connected $(\infty,0)$-categories, whereas we would need based connected $(\infty,1)$-categories to recover all $A_\infty$-spaces. It would be interesting to verify the hypothesis in this case using one of the known models for $(\infty,1)$-categories.
A $k$-tuply monoidal ∞-groupoid may be identified, under $k$-fold delooping, with a $(k-1)$-connected $\infty$-groupoid. This statement holds in fact even for parameterized $\infty$-groupoids, i.e. for ∞-stacks.
Let $k \gt 0$, let $\mathcal{X}$ be an (∞,1)-category of (∞,1)-sheaves and let $\mathcal{X}_*^{\geq k}$ denote the full sub-(∞,1)-category of the category $\mathcal{X}_{*}$ of pointed objects, spanned by those pointed objects thar are $k-1$-connected (i.e. their first $k$ ∞-stack homotopy groups) vanish. Then there is a canonical equivalence of (∞,1)-categories
This is EKAlg, theorem 1.3.6..
Specifically for $\mathcal{X} = Top$, this refines to the classical theorem by Peter May
(May recognition theorem)
Let $Y$ be a topological space equipped with an action of the little cubes operad $\mathcal{C}_k$ and suppose that $X$ is grouplike. Then $Y$ is homotopy equivalent to a $k$-fold loop space $\Omega^k X$ for some pointed topological space $X$.
This is EkAlg, theorem 1.3.16.
An (n,1)-category is a $n$-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
$(\infty,1)$-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 $k$-tuply monoidal $(\infty,1)$-categories in between.
(stabilization for $(n,1)$-categories)
Equipping an (n,1)-category of $k$ compatible monoidal structures for $k \geq n + 2$ (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.
One expects that
Some discussion of the peridodic table is in
Eugenia Cheng, Nick Gurski, The periodic table of n-categories for low dimensions I: degenerate categories and degenerate bicategories. arXiv:0708.1178.
John Baez and Mike Shulman, Lectures on $n$-categories and cohomology. arXiv:math.CT/0608420.
The theory of $k$-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)
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 $n$-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 $n$-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 $(\infty,1)$-category” to mean “an $(\infty,1)$-category which behaves like the $(\infty,1)$-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 $2$-category’ or maybe ‘symmetric monoidal $3$-category’ that ‘stably monoidal’ isn't subject to, although that's a bit parochial.