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The Bousfield-Kan $p$-completion of a simplicial set $X_\bullet$ is the totalization (= homotopy limit) of the cosimplicial space obtained by levelwise iterating the functor $S\mapsto \mathbb F_p[S]$ (that sends a set $S$ to the free $\mathbb F_p$-vector space on that set).

I had an idea at some point (with the explicit thought that it could be something like Bousfield-Kan completion at the infinite place) of doing the same construction with the functor $S\mapsto B_1(\ell^1(S))$ in place of $\mathbb F_p[-]$. This makes sense because $B_1(\ell^1(-))$ is also a monad.

Here, $B_1(\ell^1(S))$ is the unit ball in the Banach space $\ell^1(S)$ (over the reals). A possible variant is to only use the positive part of $B_1(\ell^1(S))$.

I never pursued that idea. Tilman Bauer and I discussed it at some point, and we had the vague though that this might be related to the concept of $\ell^1$-homology. [Note that $\ell^1$-homology only sees $\pi_1$ and is completely insensitive to higher homotopy groups (and moreover, it vanishes identically when $\pi_1$ is ameanable), so it belongs more to the area of geometric group theory and less so to algebraic topology.]

The Bousfield-Kan $p$-completion of a simplicial set $X_\bullet$ is the totalization (= homotopy limit) of the cosimplicial space obtained by levelwise iterating the functor $S\mapsto \mathbb F_p[S]$ (that sends a set $S$ to the free $\mathbb F_p$-vector space on that set).

I had an idea at some point (with the explicit thought that it could be something like Bousfield-Kan completion at the infinite place) of doing the same construction with the functor $S\mapsto B_1(\ell^1(S))$ in place of $\mathbb F_p[-]$. This makes sense because $B_1(\ell^1(-))$ is also a monad.

Here, $B_1(\ell^1(S))$ is the unit ball in the Banach space $\ell^1(S)$ (over the reals). A possible variant is to only use the positive part of $B_1(\ell^1(S))$.

I never pursued that idea. Tilman Bauer and I discussed it at some point, and we had the vague though that this might be related to the concept of $\ell^1$-homology. [Note that $\ell^1$-homology only sees $\pi_1$ and is completely insensitive to higher homotopy groups (and moreover, it vanishes identically when $\pi_1$ is ameanable), so it belongs more to the area of geometric group theory and less to algebraic topology.]

The Bousfield-Kan $p$-completion of a simplicial set $X_\bullet$ is the totalization (= homotopy limit) of the cosimplicial space obtained by levelwise iterating the functor $S\mapsto \mathbb F_p[S]$ (that sends a set $S$ to the free $\mathbb F_p$-vector space on that set).

I had an idea at some point (with the explicit thought that it could be something like Bousfield-Kan completion at the infinite place) of doing the same construction with the functor $S\mapsto B_1(\ell^1(S))$ in place of $\mathbb F_p[-]$. This makes sense because $B_1(\ell^1(-))$ is also a monad.

Here, $B_1(\ell^1(S))$ is the unit ball in the Banach space $\ell^1(S)$ (over the reals). A possible variant is to only use the positive part of $B_1(\ell^1(S))$.

I never pursued that idea. Tilman Bauer and I discussed it at some point, and we had the vague though that this might be related to the concept of $\ell^1$-homology.

The Bousfield-Kan $p$-completion of a simplicial set $X_\bullet$ is the totalization (= homotopy limit) of the cosimplicial space obtained by levelwise iterating the functor $S\mapsto \mathbb F_p[S]$ (that sends a set $S$ to the free $\mathbb F_p$-vector space on that set).

I had an idea at some point (with the explicit thought that it could be something like Bousfield-Kan completion at the infinite place) of doing the same construction with the functor $S\mapsto B_1(\ell^1(S))$ in place of $\mathbb F_p[-]$. This makes sense because $B_1(\ell^1(-))$ is also a monad.

Here, $B_1(\ell^1(S))$ is the unit ball in the Banach space $\ell^1(S)$ (over the reals). A possible variant is to only use the positive part of $B_1(\ell^1(S))$.

I never pursued that idea. Tilman Bauer and I discussed it at some point, and we had the vague though that this might be related to the concept of $\ell^1$-homology. [Note that $\ell^1$-homology only sees $\pi_1$ and is completely insensitive to higher homotopy groups (and moreover, it vanishes identically when $\pi_1$ is ameanable), so it belongs more to the area of geometric group theory and less so to algebraic topology.]

The Bousfield-Kan $p$-completion of a simplicial set $X_\bullet$ is the totalization (= homotopy limit) of the cosimplicial space obtained by levelwise iterating the functor $S\mapsto \mathbb F_p[S]$ (that sends a set $S$ to the free $\mathbb F_p$-vector space on that set).

I had an idea at some point (with the explicit thought that it could be something like completion at the infinite place) of doing the same construction with the functor $B_1(\ell^1(-))$ in place of $\mathbb F_p[-]$.

Here given a set $S$, then $B_1(\ell^1(S))$ is the unit ball in the Banach space $\ell^1(S)$ (over the reals). A possible variant is to only use the positive part of $B_1(\ell^1(S))$.

I never pursued that idea. I only had the vague though that this might be related to the concept of $\ell^1$-homology.

The Bousfield-Kan $p$-completion of a simplicial set $X_\bullet$ is the totalization (= homotopy limit) of the cosimplicial space obtained by levelwise iterating the functor $S\mapsto \mathbb F_p[S]$ (that sends a set $S$ to the free $\mathbb F_p$-vector space on that set).

I had an idea at some point (with the explicit thought that it could be something like Bousfield-Kan completion at the infinite place) of doing the same construction with the functor $S\mapsto B_1(\ell^1(S))$ in place of $\mathbb F_p[-]$. This makes sense because $B_1(\ell^1(-))$ is also a monad.

Here, $B_1(\ell^1(S))$ is the unit ball in the Banach space $\ell^1(S)$ (over the reals). A possible variant is to only use the positive part of $B_1(\ell^1(S))$.

I never pursued that idea. Tilman Bauer and I discussed it at some point, and we had the vague though that this might be related to the concept of $\ell^1$-homology.