Questions tagged [cofibrations]

A continuous mapping that satisfies the homotopy extension property with respect to all spaces.

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One question in Spanier's Algebraic Topology Theorem 1.4.12

In Spanier's Algebraic Topology Chapter 1 Section 4, he says that $Z_f \times I$ has the topology coinduced by the two maps as the following, which I do not understand: Why $Z_f \times I$ equipped ...
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Why are Strom-Hurewicz-cofibrations necessarily closed?

We consider the category $\mathsf{CG}$ of compactly generated spaces (since the question becomes obsolete for compactly generated weakly hausdorff spaces). A Hurewicz-fibration is a map with the ...
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$i : X^{n} \hookrightarrow X$ is a cofibration

I'd like to understand why is $X$ is $CW$ complex $i : X^{n} \hookrightarrow X$ is a cofibration when the dimension of $X$ is not finite. The proof I'd like to generalize: I'm going to use $\bigsqcup \...
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Reduced and unreduced suspension are homotopically equivalent

Definition: We say that a continuos map between topological spaces $i: A \longmapsto X$ has the Homotopy Extension Property (HEP) for $Y$ if given $h: A \times I \longmapsto Y$ and $f: X \longmapsto Y$...
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$(i_0,i_1):A\vee A\rightarrow A\wedge I_+$ is a cofibration

This is May & Ponto's More concise algebraic topology. They claimed that $(i_0,i_1):A\vee A\rightarrow A\wedge I_+$ is a cofibration. But I don't know why. Question: Why the map $(i_0,i_1):A\vee A\...
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45 views

Arrow category $\cal K^\to$ [closed]

I would like to understand here on the page $6$ in the definition $3.3$ how works the functor $F:\cal K^\to \to K$. They say ...
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42 views

Showing that homotopy and homology groups of $X=\{0\}\cup\{1/n\}$ and $Y=\{1/n\}$ are isomorphic.

Given $X = \{0\} \cup \{\frac{1}{n} |n>0, n\in Z \}$ and $Y =\{\frac{1}{n} |n>0, n\in Z \}$ with relative topology, I try to show that $H_q(X) \cong H_q(Y)$ and that $\pi_q(X,x_0)\cong\pi_q(Y,...
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100 views

generalization of fibration

A Hurewicz fibration $E\rightarrow B$ is a map so that for all maps $X\rightarrow E$ and $X\times I\rightarrow B$ making the obvious square commute, there is a lifting $X\times I\rightarrow E$ so that ...
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50 views

Homotopy on $S^{n-1}$ extends to homotopy on $D^n$

I have the following problem: I have $\beta:D^n\to Z$ such that $\beta\circ i\simeq h$, where $i:S^{n-1}\hookrightarrow D^n$ is the inclusion and $h:S^{n-1}\to Z$ some map. Then is there a $\beta':D^n\...
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Model categories: $\text{Ho}$ and $\cal C_{cf}/\sim$

I have asked this question about model categories: Why $\text{Ho} \ \cal C$ is $\cal C_{cf}/\sim$ and not $\cal C/\sim$ and I got this answer: take for cofibratiobns Iso, weak equivalences all arrows. ...
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37 views

Induced map in model categories

In the snippet below I do not understand what is $$X_i\to X_i \coprod_{L_i X}L_i Y ,$$ i.e. how is it defined and why is it a cofibration.
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37 views

Why we are sure that $h^{'} $ is a homeomorphism?

Here is the proof that every cofibration is an embedding from "Introduction to homotopy theory" by Martin Arkowitz : My question is: Why we are sure that $h^{'}$ is a homomorphism? as that ...
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Why the map of the pushout of the cone is taken from $X$ to $*$?

Here is the pushout of the cone in some notes of Colorado Univ.: I am wondering why it looks like this, especially why we took $X$ to the base point *? could anyone explain this for me, please?
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Understanding some steps in the proof that the inclusion map of a space into the cone of that space is a cofibration.

Here is the proposition and its proof (pg. 76 from the book named "Introduction to homotopy theory" by Martin Arkowitz): And here is 1.4.2(3): My questions are: 1- why the set $S$ took ...
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Comparing the definition of a cone from Wikipedia and from Marty Arkowitz.

Here is the definition of a cone (on pg.76) from "Introduction to Homotopy Theory" by Martin Arkowitz: But the definition of Wikipedia here https://en.wikipedia.org/wiki/Cone_(topology)#:~:...
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166 views

any cofibration $i:A \to B$ is a homeomorphism onto its image (question regarding the inverse map)

I was recently working on a problem that introduced the homotopy extension property as a cofibration $i:A \to B$. Let's say we are given the commutative diagram: Now, if $i:A \to B$ is the inclusion ...
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1answer
50 views

Two apparently different definitions of a path object in the model category theory

I have a question about a path object in the context of model categories. For what Hovey says see the first snippet below. But here on page 9 the definition is different: Henry says (see the second ...
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64 views

Cofibrations in diagram category

Let $\mathcal{C}$ be a model category and $\mathcal{I}$ ba a small category. Then we have the projective model category structure on the diagram category $\mathcal{C}^\mathcal{I}$ where fibrations and ...
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133 views

Closed inclusion

I have a simple question in the context of (co)fibrations in the context of Model Categories: Why on the page $52$ in the snippet below $$g^{-1}(d)$$ must be a single point not in the image of $A$ ? ...
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63 views

Express the Klein bottle as the cofibre of a map $\kappa$ between (wedges of ) copies of $S^1.$

Express the Klein bottle as the cofibre of a map $\kappa$ between (wedges of ) copies of $S^1.$ Describe the map explicitly. Could anyone help me in finding this map please?
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Is it ever useful to consider cofibrations which satisfy the HEP only with respect to a proper subclass of spaces?

A map $j:A\rightarrow X$ is said to have the homotopy extension property with respect to a space $Z$ if whenever given a map $f:X\rightarrow Z$ and a homotopy $H:A\times I\rightarrow Z$ with $H_0=fj$, ...
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103 views

Show that a retract of a cofibration is also a cofibration.

Here is the question: Suppose that $g: A \rightarrow C $ is a retract of $f: B \rightarrow D.$ Show that if $f$ is a cofibration, then so is $g.$ Could anyone help me in answering this question, ...
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90 views

Is there any closed embedding which is not cofibration?

Is there any closed embedding which is not cofibration? I firstly think that if $X$ is Topologist's sine curve and $A$ is $(0,0)$, then embedding $i:A\rightarrow X$ might satisfy this condition. ...
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117 views

(c) Calculate the homotopy fibre of the inclusion $i_{X} : X \vee X \rightarrow X \times X. $

Here is the question: Let $F$ be the homotopy fiber of the inclusion $X \rightarrow X \times X.$ (1)Show that $\pi_{i}(F) \cong \pi_{i +1}(X).$ Here is the answer of this part: Show that $\pi_{i}(...
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Model categories, trivial fibrations and cofibrations l.l.p.

Defintion $\bf 1.1.2~$ Suppose $i:A\to B$ and $p:X\to Y$ are maps in a category $\mathcal C$. Then $i$ has the left lifting property with respect to $p$ and $p$ has the right lifting property with ...
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125 views

Show that either $X$ or $Z$ is homotopy equivalent to a point.

Prove or disprove the following statement: Suppose $X,Y,$ and $Z$ are simply connected $CW$ complexes and that $X \rightarrow Y \rightarrow Z$ is simultaneously a cofiber sequence and a fiber ...
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194 views

Showing that a sequence of a cofibration is exact.

The question is : Suppose that $A \rightarrow B$ is a cofibration with cofiber $C.$ Show that for any pointed space $X,$ the sequence $[C,X] \rightarrow [B,X] \rightarrow [A,X]$ is an exact sequence ...
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39 views

Homotopy cofinality of $\Delta^{op}$ in $\Delta^{op}\times \Delta^{op}$

There is the usual diagonal inclusion $i:\Delta^{op}\to\Delta^{op}\times \Delta^{op}$ which is easily seen to be cofinal in the $1$-categorical sense, and so one can compute colimits on $\Delta^{op}\...
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188 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 ...
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117 views

Finding the cofibration of the map $S^{1} \rightarrow *. $

I want to answer this question: What is the homotopy cofibre of the unique map $S^{1} \rightarrow * $ ? describe the homotopy cofibre of $ X \rightarrow * $ in general. My attempt: I got a hint ...
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67 views

Use of cofibration in proof of Brown representability theorem

I am looking at the proof of Brown's representability theorem (BRT) found at https://www.math.ru.nl/~gutierrez/files/Lecture13.pdf. Specifically, on pages 7-9, the author states and proves two lemmas ...
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Proving that two formulations of the Homotopy Extension Property via diagrams, are equivalent.

Suppose we are working with a collection of topological spaces for which there are a product functor $F:Set\to Set:X\to X\times I$ and an exponential functor $G:Set\to Set: X\to X^I$ such that $F\...
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344 views

On the mapping cylinder inclusion $i : X \hookrightarrow M_f$ being a cofibration.

I'm trying to prove that given a continuous map $f : X \to Y$, the inclusion $$ i : x \in X \mapsto [(x,1)] \in M_f $$ is a cofibration. Given a homotopy $H : X \times I \to W$ and $g : M_f \to W$...
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393 views

Relationship between homotopy pushout and ordinary pushout

I'm trying to understand the homotopy pushouts and currently looking at the homotopy cofiber. For two maps $f \colon C \to A$ and $g \colon C \to B$ we defined the homotopy pushout to be the regular ...
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67 views

Equivalent definition of Cofibration

Basic question from may's concise course about cofibration. In the beginning of chapter 6 (search pg 51 in this pdf: https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf), May gives two ...
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279 views

Is a closed embedding of CW-complexes a cofibration?

It is a standard fact that the inclusion of a sub-CW-complex into a CW-complex is a cofibration, it follows from the fact that the inclusions $S^k\to D^{k+1}$ are, and that they are preserved by ...
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(wrong) Proof of stability of homotopy equivalences under pullback

In studying the category Top localized by homotopies, I asked me this question: "Is homotopy equivalence stable by pullback (base change)?" I know that it is necessary to have a further condition (...
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Model category of all model categories

Is there some natural model category structure on the category of all small model categories and some specified functors among them (i.e. not necessarily all functors), say Quillen adjunctions? What ...