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Somewhat as in the thread I am currently writing on the homotopy hypothesis and cubical homotopy theory, I would like in this thread to write up some work on what I refer to as ’categorical knot theory’.
This is a broad programme. It has a number of aspects.
One of these is an exploration of the foundations of knot theory and knot diagrammatics. A consequence of this will be the possibility of carrying out ’knot theory’ in categories other than that of topological spaces, and I have a number of plans in this direction, especially regarding knot theory in the category of categories or groupoids. Eventually, I plan to take this further, extending to ’piecewise linear topology’ in categories other than that of topological spaces.
By ’knot theory’ and ’knot diagrammatics’, I do not mean only the classical topic. There are also for instance aspects of virtual knot theory that I plan to explore. And, most significantly for the moment, I will explore ’higher dimensional knot theory’ as well: principally, at least to begin with, the theory of 2-knots and diagrams of 2-knots. In addition, I will explore braids, the Temperley-Lieb algebra, and analogues of these for 2-knots.
At the moment, the work that I wish to get to writing up is that which was announced in this n-category café comment. With two students, Therese Mardal Hagland and Marte Lovise Nilsen, I have been working on the construction of a Jones polynomial-like invariant of 2-knots. Therese gave a first construction in her master thesis in the autumn, and Marte Lovise has explored this construction from the point of view of 2-braids and a 2-Temperley-Lieb algebra, and reworked some aspects, in her master thesis, which she is currently writing up.
I am very excited about this invariant! As far as I know it will come out somewhat out of the blue to 2-knot theorists! I have had more of a ’steering hand’: Therese and Marte Lovise have carried out what I regard as the hard, creative, geometric work that is at the heart of the construction of the invariant. The geometric intuition of another of my students, Reidun Persdatter Ødegaard, has also been a crucial cog in the wheel of arriving at the invariant. Thus this is very much joint work: I am certain that I would never have been able to carry out the geometric work needed to arrive at the invariant alone!
But it will take a little time before I get to the invariant. This thread will perhaps have a little more of an ’exploratory’ feel than that of the thread on the homotopy hypothesis, and be a little less ’linear’, but, as in the latter thread, the work that is written up will be formal and in full detail. I will probably be focusing foremost on the writing up of the material in the homotopy hypothesis thread, but will write up things here every now and then concurrently.
…the possibility of carrying out ’knot theory’ in categories other than that of topological spaces,… I will explore ’higher dimensional knot theory’ as well.
In view of the various Tangle Hypotheses (generalized tangle hypothesis), will this be about internal objects, such as $k$-tuply monoidal $n$-category with duals?
Thank you for the question, David! The tangle hypothesis and its cousins are not something that have influenced this work very much. In particular, Lurie’s work on the cobordism hypothesis has had, I would say, no influence: I would regard my work in general as proceeding in exactly the opposite direction to the ’top down’, ’homotopical’ approaches to higher category theory that are very fashionable today.
Nevertheless, the work is certainly intimately related to low-dimensional higher category theory. My PhD supervisor, Raphaël Rouquier, once said to me, I think the first time we met after I had become his student, regarding his work on higher representation theory, that he saw it as guided by ’shadows’ of a rich and beautiful higher dimensional mathematics that we can’t quite see yet, but are trying to understand, to find a language to express. It struck me as a beautiful metaphor immediately then, but as time has passed, I have come to fully appreciate the truth of it, which I feel very strongly. There is is such an incredible richness in the internal structure of higher categories, even in very low dimensions, which we see ’shadows’ of everywhere. I have found that in knot theory and 2-knot theory these ’shadows’ appear particularly strongly, and one of my principal motivations is certainly to understand how these knot-theoretic shadows can lend insight into the structure of low-dimensional higher categories.
To be more specific, the ’2-Temperley-Lieb’ algebra that Marte Lovise has been working on with me involves gadgets that should be very closely related to a notion of 2-tangle in 4-dimensions (if I understand Baez and Dolan’s terminology correctly). Then, given the tangle hypothesis, our work could certainly be regarded as saying something about the internal structure of braided monoidal 2-categories with duals. The ’2-braids’ that she has been working on should I think assemble into an interesting, perhaps universal, example of a braided monoidal 2-category. And there should eventually be a close relationship to the higher representation theory of Raphaël: I hope that the ’Jones polynomial-like’ invariant of 2-knots that we have constructed should be able to constructed via the representation theory of a quantum ’2-Lie-algebra’. There should be other connections to higher representation theory as well. Just as one of the motivations of Raphaël’s work is the construction of deep 4d-TQFTs categorifying those of Reshetikin-Turaev, our work should be relevant to this story. But all this as yet remains unexplored.
For the moment, we have just been focusing on the invariant itself, whose construction involves many interesting subtleties: both geometric and with regard to setting up the rudiments of the 2-categorical algebraic world in which the invariant lives.
My motivation goes beyond understanding the ’Jones polynomial-like’ invariant. I find the question of laying good foundations, from a categorical point of view, for knot theory, 2-knot theory, and their diagrammatics, to be an interesting one in its own right.
The idea to consider knot theory in other categories is, as far as I know, new. One specific thing that I believe to understand how to do, though I have not yet written up the details, is to give a clean, structural construction of the Wirtinger/Dehn presentations of a knot group by working entirely inside ’categorical/groupoidal’ knot theory, rather than ordinary ’topological knot theory’. I would like to understand more of knot theory and piecewise linear topology in a similar way. It also opens up the possibility of constructing knot invariants in new ways.
Most of all, I feel that ’knottedness’ is a fundamental concept, in mathematics, but also even in the wider world around us. I consider it very natural that knots and braids show up as ’universal’ examples of phenomena, as in the tangle hypothesis. Knot theory is also wonderful from a pedagogical point of view: I think that it could and indeed should be made use of in the teaching of mathematics in school, from the earliest of ages, and have concrete ideas on how this can be done (and hope to have the chance to try these ideas out one day). It is enough motivation for me just to explore knot theory for its own sake, trying to understand it more deeply.
(16th of March 2015)
To begin with, I will recall the construction of the Kauffman bracket polynomial from the point of view of braids and (certain kinds of) tangles. Although the heart of what I will describe are the original ideas of Kauffman, there will be several points of novelty in the exposition that I will give.
One important aspect will be that, rather than construct a morphism from the braid group on $n$ strands into the corresponding Temperley-Lieb algebra, I will construct ’all of these morphisms in one go’, by making use of a little category theory. I find that this approach allows us in an elegant way to focus on the essence of the construction, allowing the categorical machinery to take care of the rest.
I will describe the other points of novelty in their turn. Some of these are influenced by our work on the analogous story for 2-knots. Everything that I write until further notice is joint work with Marte Lovise Nilsen.
Our first task will be to construct a ’category of braids’ and a ’category of (certain kinds of) tangles’, the latter of which I will refer to as the ’Temperley-Lieb category’. I view this category of braids as an algebraisation of part of knot theory: we can capture the invariance of knot diagrams under R2 and R3 moves, but not under R1 moves. In a later post, I will describe a new algebraisation of knot theory in which we also capture invariance under R1 moves. For this, we need a richer structure than a category: we will use cubical 2-categories. I will then build upon the categorical construction of the bracket polynomial that I am about to give, to construct the entire Jones polynomial, using a richer, 2-categorical, version of the Temperley-Lieb category.
Let us begin, though, with the construction of a category $\mathsf{Braids}$, the aforementioned ’category of braids’.
To carry out this construction, I will make use of a notion that I have not come across in the literature before, although it is similar to the notion of a ’monoidal signature/tensor scheme’. I have had a need for such a notion quite often in my work. Let us get straight to it.
I will define a ’monoidal datum’ to consist of the following data.
1) A directed graph $\Gamma$.
2) A category $\mathcal{B}$.
3) A morphism of directed graphs as follows, where $U$ is the forgetful functor from the category of categories to the category of directed graphs.
Let $\mathcal{C}$ be a strict monoidal category. I will define a ’datum for $\mathcal{C}$’ to consist of the following data.
1) A monoidal datum $\mathbb{M} = (\Gamma,\mathcal{B},\otimes_{\mathbb{M}})$.
2) A morphism of directed graphs as follows.
3) A functor as follows.
I require that the following diagram commutes.
Here
is the canonical isomorphism of directed graphs coming from the fact that $U$ is a right adjoint.
I claim that given any monoidal datum $\mathbb{M} = (\mathcal{A},\mathcal{B},\otimes_{\mathbb{M}})$, there is a strict monoidal category $\mathcal{C}$ with the following universal property: given any strict monoidal category $\mathcal{D}$ for which $\mathbb{M}$ is a monoidal datum, there is a unique strictly monoidal functor
such that the diagrams
and
commute. I will refer to $\mathcal{C}$ as the ’free strict monoidal category on $\mathbb{M}$’.
I claim that $\mathcal{C}$ moreover has a 2-universal property upgrading the above universal property, and that there will be analogous universal property and 2-universal property when $\mathcal{D}$ is a monoidal category which is not necessarily strict. One should indeed be able to see the construction of the free strict monoidal category on $\mathbb{M}$ as defining a 2-monad on the category of monoidal data (i.e. the category in which an object is a monoidal datum, and the arrows are defined in the appropriate way). I plan to come back to the details of these claims at a later point. The only one we will need for now is the existence of the free strict monoidal category on a monoidal datum with its 1-universal property: let us take it for granted.
There are various ways in which one could vary the notion of a monoidal datum. Most significantly, one could work with a category instead of a directed graph. However, in the examples that I have in mind, this category, let us denote it by $\mathcal{A}$, would be a free category on a directed graph, or a quotient of such a free category. To define the functor
which is part of the monoidal datum, one would then need to construct a functor out of a product of free categories. In the kind of categorical foundations in which I wish to work, this is in fact a tricky matter.
At first it seems clear how to proceed: express a product of categories as a quotient of the free category on the product of the underlying directed graphs. However, to do this in general in the kind of foundations that I wish to work, one needs some kind of ’extensionality’, allowing one to define a functor out of a category by specifying how this functor behaves on individual arrows (in other words, one needs to be able to ’glue together’ one’s category out of individual objects and arrows). I do not wish to adopt this kind of ’extensionality’ as a foundational axiom.
Instead of working with the ’native’ notion of category, one could define a new, ’extensional’, notion within the categorical foundations, for instance as the algebras for the free category monad. But I do not wish to take this route either.
One could also construct, when working with very small categories as I will, the product of free categories by hand via colimits, in a way that will allow us to construct functors out of it. But this would be somewhat tedious.
Working with a directed graph instead of a category bypasses this difficulty entirely.
I’ll describe straight away the first example of the ’free strict monoidal category on a monoidal datum’ construction that we will need.
Let $\Gamma_{\mathsf{Braids}_{\leq 2}}$ be the directed graph with the following objects and arrows.
1) An object which I’ll denote by $1$.
2) An object which I’ll denote by $2$.
3) An arrow as follows. The notation $id(1)$ is exactly that: since $\Gamma_{\mathsf{Braids}_{\leq 2}}$ is not a reflexive directed graph, there are no arrows which must be designated ’identity arrows’.
4) An arrow as follows.
5) An arrow as follows.
I will depict the arrow $\mathsf{id}(1)$ as follows.
I will depict the arrow $\mathsf{OverCrossing}$ as follows.
I will depict the arrow $\mathsf{UnderCrossing}$ as follows.
Let $\Gamma_{\mathsf{Braids}_{\leq 3}}$ be the directed graph defined in the same way as $\Gamma_{\mathsf{Braids}_{\leq 2}}$, except that we in addition have the following objects and arrows, and do not include the arrow $id(1)$.
1) An object which I’ll denote by $3$.
2) An arrow
which I’ll denote by $id \otimes \mathsf{OverCrossing}$.
3) An arrow
which I’ll denote by $id \otimes \mathsf{UnderCrossing}$.
4) An arrow
which I’ll denote by $\mathsf{OverCrossing} \otimes id$.
5) An arrow
which I’ll denote by $\mathsf{UnderCrossing} \otimes id$.
I will depict the arrows from $1$ to $1$ and the arrows from $2$ to $2$ in the same way as for $\mathsf{Braids}_{\leq 2}$.
I will depict the arrow $id \otimes \mathsf{OverCrossing}$ as follows.
I will depict the arrow $id \otimes \mathsf{UnderCrossing}$ as follows.
I will depict the arrow $\mathsf{OverCrossing} \otimes id$ as follows.
I will depict the arrow $\mathsf{UnderCrossing} \otimes id$ as follows.
Let $\mathsf{Braids}_{\leq 3}$ be the free category on $\Gamma_{\mathsf{Braids}_{\leq 3}}$. I will depict the identity arrow
as follows.
I will depict the identity arrow
as follows.
I will depict the identity arrow
as follows.
I will depict composition by vertical concatenation. For example, we depict
as follows.
In our pictorial notation, we see that two arrows are the same if the one can be obtained from the other by deleting pairs of parallel strands. For instance, the arrow pictured as
is the same as the arrow pictured as follows.
Here, as an aside, I would like to make a remark about my point of view on pictorial notation such as that I have described (we will see other examples in later posts). It is perfectly reasonable to describe a proof that two arrows (or other gadgets in a more complicated algebraic structure than a category) are equal by arguing pictorially. Bu t I will always understand such an argument as a recipe for carrying out a formalised proof without pictures.
This is not to say that a formal system could not be pictorial in nature: ultimately, a pictorial notation is made up of marks on a piece of paper, or bits of light on a screen, in just the same way as letters or other symbols which we conventionally use to communicate a mathematical proof, one human to another. But I have in mind a foundations expressed in letters and other symbols in the usual way, and a pictorial argument can only then be exactly that: something that we understand how to translate into a formal proof, but which is not a formal proof in itself.
In particular, I completely disagree with the suggestion, made for instance in the third sentence of the remark before Lemma 2.1 in the paper Traced monoidal categories of Joyal, Street, and Verity, that formalising pictorial notation in some topological way gives proofs in the pictorial notation a rigorous meaning in themselves. In fact, translating the pictorial argument into a rigorous argument within the topological formalisation is at least as difficult (I would say more difficult) than translating it directly into an argument in the category one is working in (i.e. where the notation comes from). And one then has to further appeal to significantly non-trivial results to pass from the topological formalisation to an argument in the category one is working in. I much prefer to simply pass backwards and forwards between a category and the pictorial notation we have for it.
This is not at all to say that topological formalisations of pictorial notations are not interesting. Quite the contrary, such formalisations can increase our understanding of the notation significantly, and the mathematics involved can be highly non-trivial. But one should not deceive oneself: results in such formalisations are ’meta-results’ about the notation, and cannot be any more than this. These meta-results do not and cannot magically ensure that pictorial arguments are in themselves formal arguments.
My point of view is the same concerning commutative diagrams, and, more generally, pasting diagrams. One can formalise these notions, and prove interesting things which improve our understanding of them. But this does not mean that a proof involving manipulating commutative diagrams, or more general pasting diagrams, such as the proofs in the homotopy hypothesis thread that I am writing, is in itself a formal proof. They must always be understood as recipes for constructing formal proofs. I do claim, of course, that this recipe is clear, and that the corresponding formal proofs are correct!
End of aside! Let us now get back to the construction of our category $\mathsf{Braids}$.
Let
be the morphism of directed graphs defined by the following.
1) We send $(1,1)$ to $2$.
2) We send $(1,2)$ to $3$.
3) We send $(2,1)$ to $3$.
4) We send $\big( id(1), id(1) \big)$ to $id(2)$.
5) We send $\big( id(1), \mathsf{OverCrossing} \big)$ to $id \otimes \mathsf{OverCrossing}$.
6) We send $\big( id(1), \mathsf{UnderCrossing} \big)$ to $id \otimes \mathsf{UnderCrossing}$.
7) We send $\big( \mathsf{OverCrossing}, id(1) \big)$ to $\mathsf{OverCrossing} \otimes id$.
8) We send $\big( \mathsf{UnderCrossing}, id(1) \big)$ to $\mathsf{UnderCrossing} \otimes id$.
Let $\mathsf{Braids}$ denote the free strict monoidal category on the monoidal datum $(\Gamma_{\mathsf{Braids}_{\leq 2}}, \mathsf{Braids}_{\leq 3}, \otimes)$. We shall think of the unit object of $\mathsf{Braids}$ as the ’braid on zero strings’, an ’empty braid’. This category $\mathsf{Braids}$ is the ’category of braids’ which we have been aiming to define. It is exactly the usual ’free braided monoidal category on an object’ obtained by taking the coproduct of all the braid groups $B_{n}$ for $n \geq 0$. We shall not, however, make use of this point of view in our construction of the bracket polynomial, and the construction we have given of $\mathsf{Braids}$ is more suitable for our purposes than the ’coproduct construction’ just described.
Here are a couple of other examples of free strict monoidal categories on monoidal data.
1) There is a unique monoidal datum $\mathbb{M} = (\Gamma, \mathcal{B}, \otimes)$ in which $\Gamma$ is the initial object of the category of directed graphs (i.e. the empty graph). The free strict monoidal category on this monoidal datum is the free strict monoidal category on $\mathcal{B}$ in the usual sense.
2) The category of ’cubes with connections’, and other ’decorated categories of cubes’ in which there are some generating arrows in dimension 2 or greater, can be described as the free strict monoidal category on a monoidal datum with unit. The notion of a ’monoidal datum with unit’ is a variation on the notion of a monoidal datum which gives us the possibility of specifying the unit object in the ’free strict monoidal category’ upon such a gadget (again I claim that such a free construction again exists, in the same various senses as the construction of the free strict monoidal category on a monoidal datum discussed earlier). I omit the details of this example here, as we will not need them.
We will see a further example of a free strict monoidal category on a monoidal datum in the next post in this thread, in which I will define a ’Temperley-Lieb’ category.
As a final remark, finding ways to understand canonical constructions of monoidal structures crops up in other places in my work. One important and very interesting one is the question of ’truncating’ a monoidal structure: the case that I particularly have in mind is the ’truncation’ of the monoidal structure of strict cubical $\infty$-groupoids or categories, coming from the monoidal structure on cubical sets, to a monoidal structure (the ’Gray tensor product’) on strict cubical $n$-groupoids or categories, for any $n \geq 1$. I plan to eventually address this construction in the thread on the homotopy hypothesis.
[End of this post]
(17th of March 2015)
Continuing from #9, I will now construct a category $\mathsf{Temperley}-\mathsf{Lieb}$ of (certain kinds of) tangles.
Let $\Gamma_{\mathsf{Temperley}-\mathsf{Lieb}_{\leq 2}}$ be the directed graph with the following objects and arrows.
1) An object which I’ll denote by $1$.
2) An object which I’ll denote by $2$.
3) An arrow as follows. The notation $id(1)$ is exactly that: as this is not a reflexive directed graph, there is no requirement that certain arrows be desigated ’identities’.
4) An arrow as follows.
I will denote the arrow $id(1)$ as follows.
I will denote the arrow $\mathsf{CupAndCap}$ as follows.
Let $\Gamma_{\mathsf{Temperley}-\mathsf{Lieb}_{\leq 3}}$ be the directed graph defined in the same way as $\Gamma_{\mathsf{Temperley}-\mathsf{Lieb}_{\leq 2}}$, except that we have the following additional objects and arrows, and do not include the arrow $id(1)$.
1) An object which I will denote by $3$.
2) An arrow as follows.
3) An arrow as follows.
I will depict the arrow $id \otimes \mathsf{CupAndCap}$ as follows.
I will depict the arrow $\mathsf{CupAndCap} \otimes id$ as follows.
Let $\mathsf{Temperley}-\mathsf{Lieb}_{\leq 3}$ denote the free category on $\Gamma_{\mathsf{Temperley}-\mathsf{Lieb}_{\leq 3}}$. I will depict the identity arrows on $1$, $2$, and $3$ in the same way as the identity arrows on the objects of the same denotation in $\mathsf{Braids}$. I will depict composition by vertical concatenation.
Let
be the morphism of directed graphs defined by the following.
1) We send $(1,1)$ to $2$.
2) We send $(1,2)$ to $3$.
3) We send $(2,1)$ to $3$.
4) We send $\big( id(1), id(1) \big)$ to $id(2)$.
5) We send $\big( id(1), \mathsf{CupAndCap} \big)$ to $id \otimes \mathsf{CupAndCap}$.
6) We send $\big( \mathsf{CupAndCap}, id(1) \big)$ to $\mathsf{CupAndCap} \otimes id$.
Let $\mathsf{Temperley}-\mathsf{Lieb}$ denote the free strict monoidal category on the monoidal datum $(\Gamma_{\mathsf{Temperley}-\mathsf{Lieb}_{\leq 2}}, \mathsf{Temperley}-\mathsf{Lieb}_{\leq 3}, \otimes)$. This category is the ’Temperley-Lieb category’ that we were looking to define.
Just a quick note to say that a preliminary manuscript (of a little under 400 pages!) taking up the thread of the work begun here is now available on request. The Kauffman bracket invariant is constructed in a category theoretic framework, and categorifying this framework to the setting of cubical 2-categories, a Kauffman bracket-like invariant of 2-braids is constructed.
It will be some time before this work is made publically available, because there are various things which we wish to tidy up and investigate first. But anyone interested is welcome to get in touch, and I will send you the preliminary version.
I’d be interested.
Now sent to your Adelaide email address!
Thanks!
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