Group completion of $E_k$-algebras

Question

Let $X$ be an $E_k$-algebra. We can form the delooping $BX$, which is a $E_{k-1}$-algebra. The space $\Omega B X$ is again an $E_k$-algebra, which is grouplike (i. e. $\pi_0(\Omega B X)=\pi_1(B X)$ is a group).

The canonical map $X\to \Omega B X$ is a morphism of $E_k$-algebras and behaves like a group completion: For each grouplike $E_k$-algebra $Y$, each $E_k$-morphism $X\to Y$ extends uniquely (up to homotopy?) over $X\to\Omega B X$.

We are often in the case where $X=\coprod_{n\ge 0}X_n$, i. e. $\pi_0(X)=\mathbb{N}$ and where we have maps $X_n\to X_{n+1}$.

(The examples I have in mind are classifying spaces of braid groups $BB_n\to BB_{n+1}$, symmetric groups $B\Sigma_n\to B\Sigma_{n+1}$ and mapping class groups $B\Gamma_{g,1}\to B\Gamma_{g+1,1}$, all seen as $E_2$-algebras)

In these cases, one often finds the following construction for $\Omega B X$: Consider the limit $X_\infty:=\varinjlim_n X_n$ and take the Quillen plus-construction $X_\infty^+$. Then one identifies $$\Omega B X \simeq \mathbb{Z}\times X_\infty^+.$$

First of all: Did I understand everything correctly? Then: How is this identification done? In which sense is $\mathbb{Z}\times X_\infty^+$ an $E_k$-algebra? Is there an $E_k$-map $X\to \mathbb{Z}\times X_\infty^+$ having the universal group completion property? The only map I can think of is defined as $$X_n\to \{n\}\times X_n\to \{n\}\times X_\infty \to \{n\}\times X_\infty^+.$$ Why do we need the Quillen plus-construction? Why do we need the maps $X_n\to X_{n+1}$ while they are apparently irrelevant for the definition of $\Omega B X$? What is the main reference for this construction?

Answer 1

Group completion and the answer to your questions (for $k\geq 2$) are probaby best understood homologically. An ancient definition is that a map $X\rightarrow Y$ of homotopy commutative $H$-spaces is a group completion if $\pi_0(X) \rightarrow \pi_0(Y)$ is group completion (so the target is isomorphic to the Grothendieck group of the source abelian monoid) and if the map $H_*(X) \rightarrow H_*(Y)$ with coefficients in a commutative ring is the localization obtained by inverting the elements of $\pi_0(X)\subset H_0(X)$. For $X$ as you have it, a good construction of the $k$-fold delooping comes with a map of $E_k$-algebras $X\rightarrow Y$. On components, it gives a sequence of maps $X_n \rightarrow Y_n$ consistent with the maps $X_n \rightarrow X_{n+1}$ and $Y_n \rightarrow Y_{n+1}$ oobtained by adding $[1] \in \pi_0$ at each level. These maps for $Y$ are equivalences of components. Passing to telescopes (homotopy colimits) one obtains a homology isomorphism from $X_{\infty}$ to $Y_{\infty}\simeq Y_0$. The homological characterization of the plus construction then implies that $X_{\infty}^{+}$ is equivalent to $Y_0$, but there is no mathematically essential need to talk about the plus construction in this delooping context. Nor is there any mathematically essential need to talk about ``$BX$'', which should just mean the first delooping of $X$, where $Y$ is the $k$th delooping if you think iteratively.

Easy $H$-space arguments show that the $E_k$-space $Y$ is homotopy equivalent to $Y_0\times Z$, so that the map $X\rightarrow Y$ can be interpreted as the $E_k$-map you want. A homotopy invariance argument might show that the equivalent space $Y_0\times Z$ is an $E_k$-space, but there is little or no point in thinking about that for applications: one might as well define an $E_k$-space to be a space homotopy equivalent to one with an action by an $E_k$-operad.

An early (1974) reference is [13] on my web page. A recent (2017) reference giving the equivariant generalization is [124] there. There are altogether too many references in between. A perhaps too advanced expository account focusing on multiplicative structure in the case $k= \infty$ is given in Sections 8 and 9 of [112].