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seen Nov 28 at 21:22

Nov
28
comment Failure of “equivariant triangulation” for finite complexes equipped with a $G$-action
Actually wait: I've misunderstood the question again, and what I said doesn't necessarily apply. (It applies when X comes from a finite object in the "genuine" equivariant category. But your finiteness condition looks like it might be weaker than that, even without allowing retracts.)
Nov
28
comment Failure of “equivariant triangulation” for finite complexes equipped with a $G$-action
Ah, misunderstood you. I don't have very much helpful to suggest, then. One comment: in the special case G = Z/p, if a space X with an action of G is to have the form you describe, then it satisfies the Sullivan conjecture. In particular, the homotopy fixed points for the action of G on the p-adic completion X_p must itself arise as the p-adic completion of a finite space. Possibly this could give you a way to test if something is a counterexample? (A finite X w/G action such that (X_p)^hG is not the p-adic completion of anything finite?)
Nov
28
comment Failure of “equivariant triangulation” for finite complexes equipped with a $G$-action
In Fun(BG,S), the orbits G/H generally won't be compact when H is nontrivial. But G itself generates the compact objects under finite colimits and retracts (this follows formally from the fact that it is a compact object which corepresents a conservative functor: in this case, the functor which forgets the G-action).
Oct
18
awarded  at.algebraic-topology
Oct
17
comment Associative Ring Spectra and Derived Completion
Let I be the fiber of the map A -> B. Then the fiber of the map A -> Tot^n is the (n+1)st smash power of I over A. For these fibers to vanish in the limit, it suffices that they are getting more connected. Since A is connective, it suffices for I to be connected, which is immediate from your hypothesis.
Oct
17
answered Associative Ring Spectra and Derived Completion
Oct
13
awarded  Good Answer
Oct
11
awarded  Enlightened
Oct
11
comment Are there noncommutative extensions of $\alpha_p$ by $\mathbb{G}_m$?
Any line bundle over the spectrum of a local ring is trivial.
Oct
11
awarded  Nice Answer
Oct
10
answered Are there noncommutative extensions of $\alpha_p$ by $\mathbb{G}_m$?
Oct
7
awarded  Enlightened
Sep
18
awarded  Pundit
Sep
18
comment Commutation of simplicial homotopy colimits and homotopy products in spaces
For a counterexample in spaces, let X_* be the simplicial set with vertices the integers and edges joining each integer n to n+1. Regard X_* as a simplicial object of spaces whose geometric realization is contractible. Then a product of infinitely many copies of |X_*| is contractible. But the geometric realization of a product of infinitely many copies of X_* is not connected: for example, there's no path joining the identity map Z -> Z with the constant map joining Z -> {0} -> Z. (Contradicting your "elementary computation".)
Sep
18
comment Commutation of simplicial homotopy colimits and homotopy products in spaces
Ah, I misunderstood you. Your original statement is false, even in the infty-category of spaces. (Also, it wouldn't follow from a general infty-topos from there, because left exact functors only preserve finite products. Also, sifted colimits don't commute with finite products in a general presentable infty-category.)
Sep
18
comment Commutation of simplicial homotopy colimits and homotopy products in spaces
The statement that colimits commute with products (in each variable) is a special case of the assumption that colimits are universal (namely, that they are preserved by pullback along the map X -> *), which is one of the axioms on your list.
Jul
18
awarded  Yearling
Jul
17
awarded  Enlightened
Jul
17
awarded  Nice Answer
Mar
21
awarded  Notable Question