All Questions
Tagged with limits-colimits homotopy-theory
26
questions
2
votes
1
answer
33
views
Reference request for realizing a simplicial set as the homotopy colimit of its simplices
I know that
$$X\simeq hocolim_{Simp(X)}\Delta^n,$$
where $Simp(X)$ is the category of simplices of $X$, I know this for example because of proposition 7.5 of the nLab's page for homotopy limits. ...
- 1,495
0
votes
0
answers
70
views
$\pi_*:Ho(Top_*)\to Set^{\mathbb N}$ preserves limits
Consider $\pi_*:Ho(Top_*)\to Set^{\mathbb N}$ defined by $[X,x]\mapsto \{\pi_n(X,x)\}_n$ from the homotopy category of the pointed topological spaces. I showed that this is a conservative functor and ...
- 463
2
votes
0
answers
57
views
Notation: Interpolation between functors in extension/lifting problems with simple categories
In chapter 6 on homotopy (co-)limits of Jeffrey Strom's Modern Classical Homotopy Theory he gives the following definition on p. 156 (I'm explaining the terminology at the bottom):
[Let $\mathscr{C}$ ...
- 8,358
1
vote
0
answers
43
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The boundary of an open set as the homotopy limit of the open set minus compact subsets.
Let $M$ be a manifold and $U \subseteq M$ be a relative compact open set of $M$. I run into an equivalence
$$\partial U \cong \operatorname{holim}_{K \subseteq U} U \setminus K $$
where the inverse ...
- 1,613
2
votes
1
answer
91
views
Homotopy push-out squares and exact triangles are colimits
I read somewhere that homotopy push-out squares and exact triangles in a triangulated category can both be interpreted as special cases of higher categorical colimits. Why is this true?
Please note ...
- 1,127
0
votes
0
answers
173
views
Stable homotopy groups as a generalized (reduced) homology theory
It is known that $\pi_*^{st}$ defines a generalized homology theory, where $\pi_n^{st}(X) = \text{colim}_{k \geq 0} \pi_{n+k}(\Sigma^k X)$ is the $n$th stable homotopy group of the based space $X$. ...
- 1,503
1
vote
1
answer
227
views
Why $S^n$ is the pushout of the inclusion $S^{n-1} \rightarrow D^n$?
What will be the pushout for the following :
where $i:S^{n-1} \rightarrow D^n$ is the inclusion of the boundary $S^{n-1}$ to the n-disk $D^n$.
According to Pg 40 in Julia E. Bergner's The Homotopy ...
3
votes
0
answers
197
views
Base point of BU/BO, classifying space of U/O.
There are principal bundles $$U(n) \to V_k(\mathbb{C}^n) \to G_k(\mathbb{C}^n)$$ and $$O(n) \to V_k(\mathbb{R}^n) \to G_k(\mathbb{R}^n),$$ where $V_k(\mathbb{F}^n)$ and $G_k(\mathbb{F}^n)$ are the ...
- 51
2
votes
1
answer
159
views
Fundamental Group functor has no left Adjoint
I have a question about a remark done by Martin Brandenburg on Tyler Lawson's answer in this MO discussion: https://mathoverflow.net/questions/10364/categorical-homotopy-colimits/10399#10399
The ...
user784905
0
votes
0
answers
43
views
Construction of a corner of a diagram out of the homotopy pushout
Let \begin{array}{ccc} X & \xrightarrow{} & Y \\ \downarrow & & \downarrow\\ Z & \xrightarrow{} & W\\ \end{array} be a homotopy commutative diagram in a proper model catgeory $\...
- 881
2
votes
1
answer
549
views
Spectral sequence for homotopy (co)limits
In the accepted answer to this question, user Cary states "What made this spectral sequence tick is that homology/cohomology takes a cofiber sequence to a long exact sequence.".
However this doesn't ...
- 43.9k
4
votes
1
answer
443
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Do colimits/limits exist in category of enriched categories?
This question may be too general. I am interested in references or proofs for special cases. I follow the definition in Chapter I of Kelly's Enriched Category.
Let $V$ be a monoidal category theory. ...
- 1,027
2
votes
1
answer
345
views
Homotopy coherence
$\require{AMScd}$
I am trying to get an understanding of the meaning of homotopy coherence - in order to understand homotopy limits and colimits - in the category $\mathbf{Top}$. Often when I see this ...
- 3,326
4
votes
1
answer
454
views
coequalizer of simplicial sets
This is the statement whose first line of proof I am confused. This is on page 10, of Goerss, Jardine's Simplicial Homotopy Theory.
(i) How does one prove the "presentation" of $\partial \Delta^n$?
...
- 9,618
1
vote
1
answer
159
views
Behaviour of direct limits of topological spaces with respect to preimages
Given a continuous map $p:E\rightarrow B$ where $B$ is given by a colimit of $B_{0}\subseteq B_{1}\subseteq B_{2}\subseteq\dots$. We get the canonical induced map
$$ colim_{n\in\mathbb{N}} p^{-1}(...
- 427