Background

If $X$ is a based space then the James construction on $X$ is the space $J(X)$ given by $$ X \quad \cup \quad X^{\times 2} \quad \cup \quad X^{\times 3} \quad \cup \quad \cdots $$ in which we identify a $k$-tuple of points $(x_1,\dots ,x_k)$ whose $j$-th coordinate is the basepoint with the $(k-1)$-tuple given by dropping the $j$-th coordinate. Then $J(X)$ is the (reduced) free monoid on the points of $X$ and for $X$ a connected based CW complex, one has a natural weak equivalence $$ J(X) \quad \simeq \quad \Omega \Sigma X . $$ Let $J_k(X) \subset J(X)$ be the subspace represented by $k$-tuples of points of $X$ or less. Then $J(X) = \cup_k J_k(X)$ and $J_k(X)$ is obtained from $J_{k-1}(X)$ by means of the pushout of $$ J_{k-1}(X) \quad \leftarrow \quad X(k) \quad \overset{\subset} \to \quad X^{\times k} , $$ where $X(k) \subset X^{\times k}$ be the set of tuples such that at least one of the coordinates is the basepoint. Since is $X$ a CW complex, the inclusion $X(k) \subset X^{\times k}$ is a cofibration.

If it hadn't been a cofibration, we could have instead taken a homotopy pushout of the diagram with $X(k)$ replaced by the space $X(k)'$ which is given as a homotopy colimit of the punctured $k$-cube of spaces $X^T$, where $T \subsetneq \lbrace 1,\cdots k \rbrace$, where the maps $X^T \to X^S$ for $T \subset S$ are given by inserting the basepoint in those coordinates corresponding to those indices $i \in T-S$.

This will result in a derived version of $J(X)$. It seems to me that the derived version has the same homotopy type when $X$ is CW since in this case $X(k)' \simeq X(k)$ and the homotopy pushout has the same homotopy type as the pushout of the above displayed diagram.

More Background

I have an urge to Hilton-Eckmann dualize the above. For a based space $X$, we can define a tower of spaces $$ \cdots \to L_3(X) \to L_2(X) \to L_1(X) = X , $$ in which $L_k(X)$ is obtained from $L_{k-1}(X)$ as a homotopy pullback of a diagram of the form $$ X^{\vee k} \to X\langle k \rangle \leftarrow L_{k-1}(X) $$ where $X\langle k \rangle$ is the homotopy inverse limit of the punctured cube given by $T \mapsto X^{\vee T}$ (for $T \subsetneq \lbrace 1,\cdots k \rbrace$) where the maps $X^{\vee T} \to X^{\vee S}$ are given by projections. Here $X^{\vee T}$ means those functions $T \to X$ wish are supported on a single point in $T$ (the point is not fixed and all other points map to the basepoint of $X$), i.e., it is identified with the $|T|$-fold wedge of copies of $X$.

The map $L_{k-1}(X) \to X\langle k \rangle$ in the diagram can be given a simple description of this map here, but I will omit it for reasons of space. For example, when $k=2$, it amounts to a map $X \to X \times X$. This is just the diagonal.

In particular, $L_2(X) = \text{holim}(X \to X \times X \leftarrow X \vee X)$. It is well known that $$ L_2(X) \quad \simeq \quad \Sigma \Omega X $$ when $X$ is connected.

The Question

Define $L(X)$ to be the homotopy inverse limit $\text{holim}_k L_k(X)$. Then $L(X)$ is Hilton-Eckmann dual to the derived version of the James construction. It is a kind of "derived free comonoid on the points of $X$."

Question: What can be said about the homotopy type of $L(X)$?

That is, does $L(X)$ have a simpler description in terms of the functors we know and love?

(Disclaimer: I once asked an abbreviated version of this question on Don Davis' list about a decade ago, but received no identifiable answers.)