14
votes
Accepted
Deformation of a diagram preserve the homotopy limit
This is false.
Consider the two $C_2$-spaces $S^{2\sigma}$ and $S^2$, where $\sigma$ is the sign representation and $S^V$ denotes the one-point compactification. Then the two underlying spaces are the ...
- 13.5k
13
votes
Accepted
Homotopy coherent colimits in chain complexes
The result is not only true for simplicial model categories, but for plain combinatorial model categories too - this is Higher Algebra 1.3.4.25..
In fact, for this you can reduce to the case of ...
- 15.8k
11
votes
Accepted
Homotopy fibers of infinity functors
In general the homotopy pullback of the diagram given by $i:\{y\} \to \mathcal{D}$ and $f:\mathcal{C} \to \mathcal{D}$ is given by first replacing $i$ and $f$ by fibrations between fibrant objects (so ...
- 9,481
9
votes
Accepted
Can filtered colimits be computed in the homotopy category?
No, for a diagram $X: I \to \mathcal{S} \to h\mathcal{S}$ the colimit in $h\mathcal{S}$ would satisfy $[\mathrm{colim} X(i),Y] \cong \lim [X(i),Y]$ where brackets denote morphisms in $h\mathcal{S}$. ...
- 106
7
votes
Homotopy limit over a diagram of nullhomotopic maps
For the general question, the answer is no. Let $X$ be a pointed topological space. Consider the diagram used to construct $* \times_X *$, it is a poset and all the transition maps are null-...
- 3,938
7
votes
Accepted
Reference for homotopy (co)limits of (co)chain complexes via totalization of double complexes
I wrote a note for referential purposes. I hope that this will be helpful.
Arakawa, K. (2023). Homotopy Limits and Homotopy Colimits of Chain Complexes. arxiv.2310.00201
- 2,292
6
votes
Can filtered colimits be computed in the homotopy category?
As has already been said, the homotopy category does not admit filtered colimits in general, but it’s much worse than that. Even colimits in an $\infty$-category which don’t give rise to colimits in ...
- 3,411
5
votes
Can filtered colimits be computed in the homotopy category?
Ironically, I was wondering something similar earlier this week (the irony is that I was sitting next to the OP while doing my wondering). Here's another reason why this can't be the case. In general, ...
- 63.9k
5
votes
Deformation of a diagram preserve the homotopy limit
Let me try to complement Dylan Wilson's great answer with an example which is easier to analyze at the point-set level.
Take $I = B\mathbb N^2$, so that a functor $I\to\mathcal S$ is a space $X$ ...
- 4,709
4
votes
Accepted
Homotopy totalization and chains - reference
The first part of the question was previously asked and answered here: Reference for homotopy colimit = total complex.
The second part can be easily reduced to the first part by rectifying homotopy ...
- 37.8k
3
votes
Find a functorial zig-zag of spaces
If the diagrams $X_\bullet$ and $Y_\bullet$ are constructed canonically, but not necessarily naturally, from spaces $X,Y$ where $X \simeq Y$, one technique to construct such a zigzag is pick $f: X \...
- 5,829
3
votes
Homotopy colimit commutes with homotopy groups
Giving details along the hint given, note that given a (discrete) commutative ring $A$ and an element $x\in A$, the colimit of the sequence $$
A \xrightarrow{x} A \xrightarrow{x} A \xrightarrow{x} A \...
- 213
1
vote
Why does this construction give a weak factorization system in the category of span diagrams?
they seem to implicitly use the fact that (in their notation) if a map f is such that fa, fb, and fc are acyclic cofibrations, then ia(f) and ic(f) are again acyclic cofibrations.
No, that's not what ...
- 37.8k
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