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Questions tagged [cofibrations]

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

In homotopy theory, a continuous mapping $$i: A\to X,$$ where $A$ and $X$ are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces $Y$. This definition is dual to that of a fibration, which is required to satisfy the homotopy lifting property with respect to all spaces. This duality is informally referred to as Eckmann–Hilton duality.

A more general notion of cofibration is developed in the theory of model categories.

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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 pushouts. My question is about a general closed…
Maxime Ramzi
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Is the terminal object always cofibrant?

Let $\mathcal{M}$ be a model category. Is the unique map from the initial to the terminal object $!: \mathbf{0} \rightarrow \mathbf{1}$ always a cofibration? If it helps, we can even assume that $\mathcal{M}$ is combinatorial (i.e. locally…
chickenNinja123
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Good Pairs and Pushouts

I was wondering whether the following was true: Assume we have a pushout square $$\require{AMScd}\begin{CD} A @>>> B \\ @VVV @VVV \\ X @>>> Y \end{CD}$$ with $Y$ being the pushout of $X \leftarrow A \rightarrow B$ and the left arrow is the inclusion…
Max
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Good pair vs. well-pointed space

Call a pair of topological spaces $(X,A)$ a good pair if $A\subseteq X$ is a closed subspace and there exists an open neighborhood $A\subseteq U \subseteq X$ such that $A$ is a deformation retract of $U$. I have seen two definitions of the notion of…
Peter
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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 \mathbb{S}_{\lambda}^n \hookrightarrow \bigsqcup…
jacopoburelli
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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. However, I couldn't prove there is no retraction…
afdsfasdf
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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 sequence. Show that either $X$ or $Z$ is homotopy…
Emptymind
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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 $g : M_f \to W$ so that $$ H(x,0) = gi(x) = g([(x,1)]) $$ we want…
qualcuno
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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 is the most natural choice to this question: what…
user122424
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Is this inclusion a cofibration in the cofibrantly generated model category of topological spaces?

Consider the category $\mathbf{Top}$ of topological spaces as a cofibrantly generated model category, which is generated by $I=\{S^{n-1}\to D^n; n\geq 0\}$ (boundary inclusions) and $J=\{D^n\to D^n\times[0,1]; n\geq 0\}$ (inclusions…
Frank
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NDR pairs and extensions

Assume $X$ is a nice enough metric space, if $(X,A)$ is space , $A$ closed in $X$, such that for some $U$ open in $X$, with $A\subseteq U$, $(U,A)$ is an NDR pair, does it follow that $(X,A)$ is an NDR pair? Here, $(X,A)$ NDR means that there exists…
monoidaltransform
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Cofibrations and cofibrant objects in a simplicial abelian category

For context, I am reading chapter III.2 of Goerss and Jardine's book on simplicial homotopy theory, but I am not an expert in model categories. In that chapter, they introduce the model structure on simplicial abelian groups $sAb$ with weak…
SetR
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Good pair and G-delta vs cofibration

A pair of topological spaces $(X, A)$ is a cofibred pair if $A$ is closed in $X$ and the homotopy extension property holds. Moreover, $(X, A)$ is a good pair if $A$ is closed in $X$ and there exists an open neighbourhood $U$ of $A$ such that $A$ is…
Fabio
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Pushout of equivalences along cofibrations are equivalences

I would like to show that if $i:A\to X$ is a cofibration and $f:A\to B$ is a homotopy equivalence, then the induced map $k:X\to X\cup_AB$ is again a homotopy equivalence. $\require{AMScd}$ $$ \begin{CD} A @>i>> X \\ @VfVV @VVkV \\ B @>>>…
Sardines
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Is this map a cofibration in a model category?

Consider the map $$i\colon A\sqcup A\to A$$ where $i=(Id_{A},Id_{A})$. Is this map a cofibration? Actually, I know that isomorphisms are cofibrations and pushouts of cofibrations are cofibrations. But I do not know what tools to use to prove that…
T. Wildwolf
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