All Questions
Tagged with limits-colimits algebraic-topology
41
questions
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34
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The universal bundle $\gamma_k\rightarrow BO_k$ is a real vector bundle
For all $n$, let $\gamma_k^n$ bet the tautological bundle over $Gr_k(\mathbb R^n)$, i.e.
$$\gamma_k^n=\{(V,v):V\in Gr_k(\mathbb R^n), v\in V\}$$
This is also naturally identified with the associated ...
- 3,431
1
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1
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68
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On the topology of $BO_k$
Let $BO_k$ be the classifying space given by:
$$BO_k=\varinjlim_{\mathbb N\ni n}Gr_k(\mathbb R^n)$$
I am trying to determine aspects about the topology of this space, but cannot find any sources that ...
- 3,431
0
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1
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49
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Colimits of full subdiagrams vs topological subspaces
Let $F:J \rightarrow \mathrm{Top}$ be a diagram in the category of topological spaces and $X:=\mathrm{colim} F$ be its colimit. For each $a \in \mathrm{ob}(J)$, denote by $\imath_a:F(a)\rightarrow X$ ...
- 151
9
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191
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Fundamental groupoid of a filtered union
Let $X$ be a topological space and let $(X_i)_{i\in I}$ be a filtered family of subspaces. Let $X =\bigcup_{i \in I} X^°_i$ be the union of the interiors of the $X_i$. I want to prove the following ...
0
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32
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Help studying the derived functor of the limit functor
I am currently working through the second half of Hatcher's algebraic topology and through Weibel's Homological algbera. In the latter the next chapter I am going to read is the derived functor of the ...
- 1,495
2
votes
1
answer
90
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Inclusion functor is final?
For $n\geq 1$ let $K_n$ be the partially ordered set of compact subsets of $\mathbb{R}^n$ ordered by inclusion. Let $B_n$ be the totally ordered set of closed balls in $\mathbb{R}^n$ centered at the ...
- 1,769
1
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1
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195
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Computing the homotopy limit of a constant diagram.
Let $X$ be a nice space, and view it as an $\infty$-groupoid via its singular simplicial set. Consider the constant functor $\mathbb{S}$ valued functor to Spectra, mapping all simplices to the sphere ...
- 6,900
2
votes
1
answer
134
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Direct limit of nested fundamental groups
Let $M$ be a compact submanifold (with boundary) of $S^n$ realised as the intersection of some other compact manifolds $M_i\subset S^n$ so $M=\cap M_i$ and $M_i\subset Int(M_{i-1})$. Then we have:
$\...
- 349
0
votes
0
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61
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Density of a subspace in certain topological spaces.
Let $(X_i)_{i \in \mathbb{N}},(f_{i,j})_{i\leq j \in \mathbb{N}})$ be an inverse system in $Top$ with inclusion $\iota_i:U \hookrightarrow X_i$ for all $i$, such that $f_{i-1,i}\circ\iota_i=\iota_{i-1}...
- 308
2
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57
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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
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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
1
vote
1
answer
240
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Question about Hatcher's definition of CW-complex
In page 519 of Hatcher's book, we can find the following definition of CW complex:
[...] Let us first recall from Chapter 0 that a CW complex is a space $X$ constructed in the following way:
Start ...
- 861
4
votes
1
answer
147
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Complete compactness and retract
Defn: Let $\mathsf{C}$ be a locally small category which admits all small colimits. We say that $X \in \mathsf{C}$ is completely compact if the functor $\mathrm{Hom}_{\mathsf{C}}(X, \cdot): \mathsf{C}...
- 2,177
2
votes
1
answer
162
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Quotient by the action of a group commuting with sequential colimits
Let $G$ be a group and suppose $X_1 \to X_2 \to X_3 \to \cdots$ is a sequence of topological $G$-spaces and continuous $G$-maps. The colimit of this sequence inherits then a $G$-action.
Under what ...
- 1,562
0
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173
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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