So for the past **six or seven months** I've been trying to get a copy of the paper ``On quasitriangular Quasi-Hopf algebras and a group closely connected with ${\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})}$'' by V.G. Drinfel'd. My online search uncovered only a Russian copy (which, unfortunately, I don't read). My library searches were equally unfruitful: the British library's copy is ``unavailable'' and I was unable to find a copy at another UK institution. My supervisor was sure he had a copy, but when he discovered that it mysteriously disappeared he encouraged me to request an inter-library loan with my university. I did, and my university said they found a copy in Leeds, but the last ten pages are missing. So - if any readers have a copy of the aforementioned paper: I'd greatly appreciate your lending me it!

In the meanwhile, I've been reading ``On Associators and the Grothendieck-Teichmuller Group'' by Dror Bar-Natan. Among other things, that paper described the construction of said group via a ``Parenthesized Braid Category''. I'll discuss that construction in this post.

## 1. Braids and Categories

I really don't want to give a precise definition of a braid (I've not yet seen one that doesn't make my head hurt), but hopefully I won't be too wrong if I propose the following:

Def1 1Abraid on ${n}$ strandsis a set of ${n}$ oriented curves with distinct starting and ending points where the starting points lie on the same line, each endpoint lies on a separate line, and no curve contains a loop.

For example, the following is a braid:

Clearly we can enumerate the starting and ending points, and treat a braid as a permutation on them. In this way, braids form a category where the braid itself is the morphism and the objects are simply sets to be permuted.

We're going to change our class of objects slightly - we're going to consider *parenthesizations* of sets. Id est, pretend that the sets have some underlying multiplicative structure. We want to order the sets and denote the order of multiplication. Exempli gratia, the ordering ${(1 \, 2 \, 3)}$ has two parenthesizations: ${((1 \, 2) \, 3)}$ and ${(1 \, (2 \, 3))}$. Our braid morphisms are allowed to act on any parenthesizations.

We aren't done complicated things yet: we're going to modify the morphisms themselves so that they become formal sums:

$\displaystyle \sum_{i=1}^{k} \beta_i B_i$

Where each ${B_i}$ is a braid on ${n}$ strands, all ${B_i}$'s have the same effective permutation, and ${\beta_i}$ belongs to some ${\mathbb{Q}}$-algebra, usually ${\mathbb{Q}}$ or ${\mathbb{C}}$. We're forming the ``algebriod'' over ${\mathbb{Q}}$ or ${\mathbb{C}}$ over the set of braids on ${n}$ strands, with composition defined to be a bilinear map in the same way you would do for a group algebra.

This category is our **Parenthesized Braid Category** ${\textbf{PaB}}$.

## 2. Fibred Linear Categories

Our category is *fibred*. To see this, consider the category ${\textbf{PaP}}$ of parenthesized objects as before, but with simple permutations as the morphisms. Then consider the functor ${\textbf{S} : \textbf{PaB} \rightarrow \textbf{PaP}}$ that maps each object to itself, but takes each formal sum of braids to that one effective permutation. Clearly the set of morphisms in ${\text{PaB}}$ from ${O_1}$ to ${O_2}$ is:

$\displaystyle \text{Hom}_\textbf{PaB}(O_1, O_2) = \textbf{S}^{-1} \left( \text{Hom}_\textbf{PaP}(O_1, O_2) \right)$

Where P is the effective permutation.

One might also note that for a fixed permutation ${P}$, the fibre of morphisms ${\textbf{S}^{-1}(P)}$ is a vector space (actually, its an algebra, because we can compose morphisms as our product). This allows us to define some very ``linear'' notions, for instance:

We can define *subcategories*: a category is a subcategory of ${\textbf{PaB}}$ if it shares the same collection of objects and each ${\text{Hom}(O_1, O_2)}$ is a subspace of the corresponding ${\text{Hom}_\textbf{PaB}(O_1, O_2)}$.

A subcategory is an *ideal* if whenever one of two composable ${B_1}$ or ${B_2}$ belongs to it then the composition belongs to it, too. Exempli gratia, let ${\textbf{I}}$ be the ideal where all morphisms ${\sum \beta_i B_i}$ have ${\sum \beta_i = 0}$. Then we can define the *quotient* ${\textbf{PaB}/\textbf{I}}$ in an obvious way: each set of morphisms ${\text{Hom}(O_1, O_2)}$ is the corresponding quotient in the bigger set of morphisms.

Additionally, we can define idealic powers: ${\textbf{I}^n}$ has morphisms that can be written as compositions of ${n}$ morphisms in ${\textbf{I}}$. We can continue this and take an inverse systems with an inverse limits, in fact, we can even do the ${\textbf{I}}$-${adic}$ completion:

$\displaystyle \widehat{\textbf{PaB}} = \varprojlim_n \textbf{PaB}/\textbf{I}^n$

Finally, we can do a tensor product of ${\textbf{PaB}}$ with itself (in the same way we would do tensor powers of an algebra). For ${\textbf{PaB}^{2\otimes}}$ we define the morphism as the disjoint union:

$\displaystyle \text{Hom}_{\textbf{PaB}^{2\otimes}}(O_1,O_2) = \coprod_{P \in \text{Hom}_{\textbf{PaB}}(O_1, O_2)} \textbf{S}^{-1}(P) \otimes \textbf{S}^{-1}(P)$

Now we can even do a coproduct ${\Delta: \textbf{PaB} \rightarrow \textbf{PaB}^{2\otimes}}$ by ${B \mapsto B \otimes B}$. This coproduct is a functor. (I suspect that there is a bialgebra/hopf algebra structure hidden somewhere in here but I've not thought about it enough yet. Not for this post.)

## 3. Some Functors

In addition to ${\textbf{S} : \textbf{PaB} \rightarrow \textbf{PaP}}$, we also have a few functors ${\textbf{PaB} \rightarrow \textbf{PaB}}$. We can describe these by their action on a single braid and one can expand them as a linear action on the formal sums we use. First, we have the extension functors ${d_0}$ and ${d_{n+1}}$ that add a single straight strand to the left or right of the braid, respectively.

Second, we have cabling functors ${d_i}$ for ${1 \leq i \leq n}$ on a braid of n strands. This functor simply doubles the ${i}$-th strand.

Finally, we have a strand-removal functor ${s_i}$, which, as you might guess, removes the ${i}$-th strand.

We can now define the Grothendieck-Teichmuller group ${\widehat{\text{GT}}}$. This is the group of all invertible functors ${a: \textbf{PaB} \rightarrow \textbf{PaB}}$ such that ${\textbf{S} \circ a = S}$, ${d_i \circ a = a \circ d_i}$, ${s_i \circ a = a \circ s_i}$, ${\Delta \circ a = a \otimes a \circ \Delta}$. We also require that $a$ leaves the braid constant:

Right! There we are. Sooner or later I actually do some maths, but for now I'm stuck understanding trying to understand definitions, so hence this post.

## 4. Source

This is all just a reiteration of the material in the aforementioned paper by Bar-Natan.

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