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Stream: learning: reading & references

Topic: Ref Req: free resolutions in doctrines

James Deikun (Apr 11 2025 at 14:34):

I somehow have the impression that it's classical that the bar resolutions of $\mathsf{Assoc}$ and $\mathsf{Comm}$ wrt the monadic adjunction between collections and operads give good simplicial presentations of $A_\infty$ and $E_\infty$ respectively but I don't know where (or who) this is from.

James Deikun (Apr 11 2025 at 14:43):

Also can you get some kind of presentation of spectra from the bar resolution of the Lawvere theory of Abelian groups in the doctrine of Lawvere theories?

James Deikun (Apr 11 2025 at 22:42):

It seems like the [[Koszul-Tate resolution]] is commonly used across the [[Dold-Kan correspondence]] to take these resolutions of operads and I can find lots of references on that ... OTOH the simplicial [[Boardman-Vogt resolution]] is comparatively really complicated and only works for some operads (you have to use a separate technique to get started with $E_\infty$) so I'm starting to wonder if this is even true.

James Deikun (Apr 12 2025 at 03:22):

Well, it's at least certainly true when you resolve $\mathsf{Assoc}$ as a planar operad. The bar resolution of a planar operad $P$ has as $n$-simplices, planar trees with their vertices labelled by operations of $P$ and their internal edges labelled $0$ to $n$. The simplicial maps pick a $k$ from that range, contract all edges labelled strictly greater than $k$ via composition, and then relabel the remaining edges using a function that fixes $0$. The augmentation maps evaluate the whole tree, and the "extra degeneracies" increase all the edge numbers by 1. This actually seems to be the same as planar simplicial Boardman-Vogt on a discrete $P$, and certainly produces the same barycentric triangulation of the Stasheff associahedra.

As for the non-planar case, I'm not exactly sure what a good simplicial $E_\infty$ looks like for comparison.

James Deikun (Apr 12 2025 at 03:38):

(An alternate description of the above trees: A $n$-simplex is a sequence of $n+1$ formal expressions in $P$ each of which is a partial evaluation of the last, face maps drop one expression, degeneracies duplicate one, the augmentation is total evaluation, and the extra degeneracies tack on the total evaluation to the end of the sequence.)

James Deikun (Apr 12 2025 at 11:05):

I think the bar resolution for a Lawvere theory $T$ looks like this: $k$-ary points are lists of definitions $y_i = f_i(x_1,...,x_k,y_0,...,y_{i-1})$ where $f_i$ is an element of the free $T$-model on the listed variables as generators. $k$-ary $n$-simplices are sequences of $n+1$ $k$-ary points where each one comes from the previous one by substituting away some of the $y$'s and then renumbering so they're consecutive again. The face maps again throw away elements of the sequence, degeneracies again duplicate them, the augmentation substitutes away everything except the last definition, and the extra degeneracies again add the content of the augmentation to the end of the list.

I can't really think of a way a simplicial $\mathrm{Bar}(\mathsf{Ab})$-algebra can be seen as a spectrum object, so $E_\infty$ is still very much up in the air as far as I'm concerned. And I'm still looking for a reference for the known-good $A_\infty$ case.

James Deikun (Apr 12 2025 at 15:11):

It looks like the simplicial [[Barratt-Eccles operad]] comes out of a bar resolution for the monadic adjunction between symmetric operads and planar collections, which makes me interested in taking a second look at $\mathsf{Ab}$.

John Baez (Apr 12 2025 at 15:22):

I don't think you get the $E_\infty$ operad as the bar resolution for the operad $\text{Comm}$; I think you get something I may have seen called $C_\infty$.

James Deikun (Apr 12 2025 at 17:49):

From studying https://arxiv.org/abs/hep-th/9403055 it seems like $C_\infty$ is the bar resolution for $\mathsf{Comm}$ with respect to the monadic adjunction between symmetric operads and symmetric collections while $E_\infty$ appears when you take the bar resolution with respect to the monadic adjunction between symmetric operads and planar collections. In the former case the actions of the symmetric groups are "invisible" to the bar construction and don't get homotopically weakened whereas in the latter case they are visible and do get weakened.