# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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