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Questions tagged [cofibrations]

For questions about or involving cofibrations which are maps which satisfy the homotopy extension property for all spaces.

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Is the mapping cylinder a replacement for morphism by cofibration in model categories?

Let $M$ be a model category, consider a very good cylinder object $X \coprod X \to X \times I \overset{\operatorname{pr}}{\to} X$ (here $X \times I$ is just a notation, no object $I$ is implied), that ...
Arshak Aivazian's user avatar
1 vote
1 answer
90 views

Find a functorial zig-zag of spaces

This is a rather broad question. Suppose you have an ordinary category $C$ (for example, $\Delta$), and two diagrams $X_{\bullet}, Y_{\bullet} : C \to \textrm{Top}$. Suppose also that $X_c$ is ...
Andrea Marino's user avatar
4 votes
0 answers
98 views

Simplicial spaces and reflexive coequalisers

Let $X_\bullet$ be a simplicial space. Consider the reflexive coequaliser of $X_1\rightrightarrows X_0$, which we call $X$. Then we clearly have a map $\varphi\colon |X_\bullet|\to X$, where $|X_\...
FKranhold's user avatar
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5 votes
1 answer
258 views

Universal property of the codomain fibration

Let $\mathcal{C}$ a category with pullbacks. Does $\mathsf{cod}: \mathcal{C}^{\to}\to\mathcal{C}$ have any kind of universal property in the category of (co)fibrations over $\mathcal{C}$? I'd want it ...
eta's user avatar
  • 53
1 vote
0 answers
124 views

Codimension one submanifold gives cofibrant pair

Let $M$ be a smooth manifold, and $N$ be an embedded smooth submanifold of $M$ with $\partial M=\varnothing=\partial N$. Suppose, $\dim M-\dim N=1$, and $N$ is a closed subset of $M$. Does the ...
Someone's user avatar
  • 265
3 votes
1 answer
156 views

Are cofibrations in topological monoids preserved by forming the product with the identity?

Consider the category $\mathrm{Mon}(\mathbf{Top})$ of topological monoids, together with the model structure transferred along the adjunction $F:\mathbf{Top}\rightleftarrows \mathrm{Mon}(\mathbf{Top}):...
FKranhold's user avatar
  • 1,623
8 votes
2 answers
440 views

Homotopy pushout independent of factorization and symmetric in cofibration category

$\require{AMScd}$In Algebraic homotopy, Baues defines the notion of homotopy pushout in a cofibration category in the following way: a commutative diagram \begin{CD} A @>k>> C \\ @AfAA @AAhA\...
Pedro's user avatar
  • 1,534
3 votes
1 answer
473 views

Is $\partial \Gamma\hookrightarrow \Gamma$ a Serre cofibration?

Question : Let $M$ be a (say smooth, possibly non compact) manifold with boundary. Is the inclusion $\partial M\hookrightarrow M$ a cofibration in the Serre-Quillen model structure of topological ...
Olivier Bégassat's user avatar
7 votes
0 answers
511 views

A question about cofibrations

Let $(X, A)$ be a cofibration, with $X$ compactly generated. This is equivalent to the fact that $A$ is a NDR of $X$, i.e., there exist two functions $\phi \colon X \rightarrow I$ e $H \colon X \times ...
Fabio's user avatar
  • 1,202
2 votes
1 answer
300 views

Different model structures on Top

There is at least 3 model structures on the category of topological spaces, the Quillen Model structure, the Storm model structure and the Mixed model structure. In the Mixed model structure $\mathsf{...
Ilias A.'s user avatar
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5 votes
0 answers
184 views

Actions of cofibrations and induced maps of cofibres

Working in some nice category of based topological spaces (compactly generated with CW homotopy type, say) suppose we have a homotopy commutative diagram $$ \begin{array}{ccccc} & & j & &...
Mark Grant's user avatar
1 vote
0 answers
52 views

Is the product of two categories with cofibrations still a category with cofibrations?

Given two $k$-linear categories $\mathcal C$ and $\mathcal D$ which are "categories with cofibrations" (in the Waldhausen sense), is the product category $\mathcal C\times \mathcal D$ still a category ...
Bro's user avatar
  • 11
5 votes
1 answer
749 views

Is a 'join' of two cofibrations a cofibration?

I have encountered with following problem while I was learning homotopy theory. Let $A\rightarrow X$ and $B\rightarrow Y$ are cofibrations (in a good category). Is it true that a map $A\times Y\cup_{...
Anonymous's user avatar
  • 223
6 votes
2 answers
565 views

Why is the path fibration a strong Hurewicz fibration?

In May and Sigurdsson "Parametrized homotopy theory" there is a general treatment of Hurewicz style model structures in Chapter 4, see definitions 4.2.1 and 4.2.2. I am trying to adapt these to a more ...
Andrej Bauer's user avatar
  • 48.1k
2 votes
1 answer
608 views

When is the inclusion of a relative mapping space into a mapping space a cofibration?

Let $(X,A)$ and $(Y,B)$ be pairs of spaces and subspaces, let $\operatorname{Map}(X,Y)$ the space of maps $f:X\to Y$ equipped with the compact-open topology and let $\operatorname{Map}(X,A;Y,B)$ be ...
Richard Manthorpe's user avatar
2 votes
1 answer
212 views

Cube of cofibrations II

Let $\mathcal{C}$ be a category with cofibrations in the sense of (Waldhausen, Algebraic K-Theory of Spaces) and denote by $F_n(\mathcal{C})$ the category with cofibrations consisting of sequences of $...
Martin Brandenburg's user avatar
6 votes
1 answer
373 views

When is a cube of cofibrations are "lattice"?

Let $C$ be a category with cofibrations in the sense of (Waldhausen, Algebraic K-Theory of Spaces) and denote by $F_n(C)$ the category with cofibrations consisting of sequences of $n$ cofibrations $...
Martin Brandenburg's user avatar