# Identifying the little disk operad with parenthesized braids

Let $D_2$ be the topological operad of little disks. This operad can be modelled "combinatorially" in terms of an operad of groupoids called $\newcommand{\PaB}{\mathbf{PaB}}\PaB$, the operad of parenthesized braids. An object of the groupoid $\PaB(n)$ is a complete parenthesization of a permutation of the symbols $\{1,...,n\}$, e.g. $$((24)(13)5) \in \mathrm{ob} \,\PaB(5),$$ and morphisms are $n$-strand braids, such that the start- and endpoints of each strand are labeled by the same element of $\{1,...,n\}$. Composition in this operad is given by "cabling". The connection to the little disk operad is that one can quite easily write down a morphism of operads $$\PaB \to \Pi_1(D_2),$$ where $\Pi_1(-)$ denotes the fundamental groupoid, such that $\PaB(n) \to \Pi_1(D_2)(n)$ is an equivalence of categories for all $n$.

If we denote by $N$ the composition of the nerve functor with the geometric realization, then $N\PaB$ is a topological operad.

Fact: $N\PaB$ is equivalent to $D_2$, i.e. there is a zig-zag of operad morphisms between $N\PaB$ and $D_2$, each of which is arity-wise a weak equivalence.

In the literature this fact is in several places proven by appealing to Fiedorowicz's recognition principle, see e.g. Dmitry Tamarkin, Formality of chain operad of little discs, Section 2. I don't understand why and I guess I am missing something: it seems to me that there is a much more direct argument. Indeed the map $\PaB \to \Pi_1(D_2)$ gives a homotopy equivalence $$N\PaB \to N\Pi_1(D_2),$$ moreover, the unit of the adjunction between $\Pi_1$ and $N$ gives a morphism $$D_2 \to N\Pi_1(D_2)$$ which is also an equivalence since $X \to N\Pi_1(X)$ is a homotopy equivalence for any $K(\pi,1)$-space, such as $D_2(n)$. What am I missing?

• Have you looked at Severa's paper on Tamarkin's proof? I think he constructs all of these equivalences explicitly. In any case, +1 for the question. – Theo Johnson-Freyd Jul 9 '13 at 21:08
• Thanks, Theo! Did you mean his paper with Willwacher, "Equivalence of formalities of the little discs operad"? On the bottom of page 9 of their paper they give exactly the argument I give, which is encouraging. – Dan Petersen Jul 9 '13 at 21:50
• In case someone wants a precise statement and proof, this is discussed at length in the book Homotopy of Operads and Grothendieck-Teichmüller Groups by Fresse, more precisely the statement is given in Proposition 6.2.2 ($\mathtt{PaB}$ is isomorphic to a suboperad of $\pi \mathtt{D}_2$). The unitary case (with nonempty arity zero) is also dealt with. – Najib Idrissi Sep 24 '15 at 13:10

It seems to me that the operad morphism from PaB to $\Pi_1(D_2)$ works in positive arity but does not respect composition with the $0$-ary operation on the object level.