« first day (1934 days earlier)      last day (1469 days later) » 

Saal Hardali
Saal Hardali
5:02 PM
I have a feeling this was asked here before once. The loop space functor on pointed connected space induces the $\Omega, B$ adjunction between pointed connected spaces and $\mathbb{E}_1$-algebras in pointed connected spaces (w.r.t. the product monoidal structure). It turns out that pointed connected spaces are in fact the full subcategory on the grouplike $\mathbb{E}_1$-spaces. The functor $B$ is the two sided bar construction of a loop space acting trivially on a point from left and right.

Now repeat the entire story with the reduced suspension functor, replace algebras with coalgebras, b…
Now $\Sigma X$ is a coalgebra in pointed spaces w.r.t. the wedge symmetric monoidal structure. Form the cobar construction $Cobar(\ast, \Sigma X, \ast)$
There's a map $X \to Cobar(\ast, \Sigma X, \ast)$ is it an equivalence?
(just to clarify the coalgebra structure comes from the "pinch" map $\Sigma X \to \Sigma X \vee \Sigma X$)
What I mean by "wedge" symmetric monoidal structure is the monoidal structure coming from the categorical coproduct which is the wedge sum.
(one last correction: at the start "$\mathbb{E}_1$ algebras in pointed spaces" not necessarily connected)
 
Dylan Wilson
Dylan Wilson
5:20 PM
I cited two different papers by the same author in the same year and Bibtex used the same abbreviation in the document for both like "[XXX17, XXX17]" instead of like "[XXX17a] and [XXX17b]" or something. Has this happened to anyone else? Anyone know how to fix it?
jk fixed it
 
 
3 hours later…
Marc Hoyois
Marc Hoyois
8:27 PM
@SaalHardali Bousfield showed that the map X → Cobar(*,ΣX,*) is an equivalence if X is nilpotent. This is not true for every X though, for instance because there are nontrivial acyclic spaces (i.e. such that ΣX is contractible).
There was some discussion of this question here: nforum.ncatlab.org/discussion/1967/…
 
 
3 hours later…
Saal Hardali
Saal Hardali
11:42 PM
@MarcHoyois To what extent do we need "nilpotent" here? Is it just true that this map is a homology equivalence? (i.e. HZ-local equivalence, then since nilpotent spaces are HZ local its an equivalence for them)
 

« first day (1934 days earlier)      last day (1469 days later) »