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Results tagged with model-categories
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user 8032
A model category is a category equipped with notions of weak equivalences, fibrations and cofibrations allowing to run arguments similar to those of classical homotopy theory.
15
votes
2
answers
1k
views
"Strøm-type" model structure on chain complexes?
Background
The Quillen model structure on spaces has weak equivalences given by the weak homotopy equivalences and the fibrations are the Serre fibrations. The cofibrations are characterized by t …
- 18.5k
4
votes
1
answer
415
views
Alternative model structure on retractive spaces
Background
For a topological space $X$, let $R(X)$ be the category of retractive spaces over $X$. An object of this category is a space $Y$ equipped with maps $s: X \to Y$ and $r: Y \to X$ such that …
- 18.5k
6
votes
Homotopy fibre of composition
$\require{AMScd}$
Here are some details. Without loss in generality, we can assume $f$ and $g$ are fibrations. I am assuming that all spaces are based. We may assume that $F_f$ now refers to the actua …
- 18.5k
2
votes
Accepted
Model categories and cellular maps
The answer to your question is yes in the case of cellular maps of topological spaces.
(I think a similar argument works in the simplicial case, but I doubt the result is true in the CW case.)
There …
- 18.5k
8
votes
Homotopic maps out of cofibration sequences
I once read in a paper of McGibbon about the following example:
Let $X = \Bbb RP^\infty$
and let $X_n$ be the $n$-skeleton. Then there is a canonical map
$\vee_n X_n \to X$. The mapping cone of thi …
- 18.5k
7
votes
Why does this construction give a (homotopy-invariant) suspension (resp. homotopy cofiber) i...
An argument showing that the two models of suspension are equivalent will probably be based on something like the following:
Assertion: Suppose we are given a commutative diagram of the form
$\requir …
- 18.5k
4
votes
1
answer
176
views
Equivariant versus retractive spaces: a reference request
Let $T$ be the category of compactly generated weak Hausdorff spaces with model structure given by Serre fibrations, Serre cofibrations and weak homotopy equivalences. Let $G = |G.|$ be the (geometric …
- 18.5k
1
vote
Existence of homotopy limits and colimits in model categories
Edit: The questioner has objected to the fact that the reference I gave to Q1 assumes functoriality.
Here is another reference which doesn't:
https://pages.uoregon.edu/ddugger/hocolim.pdf
See es …
- 18.5k
19
votes
Fibrations and Cofibrations of spectra are "the same"
The following might help answer the last part of your post:
In the late 1960s, Tudor Ganea developed technology that studies the difference between the homotopy fibers and cofibers of a map. For exam …
- 18.5k