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seen Oct 17 at 21:52

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
Feb
21
awarded  Good Question
Jan
29
awarded  Good Answer
Jan
25
answered classifying $\infty$-toposes for topological/localic groups?