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0
votes
1answer
93 views
Why does the definition of homotopy cartesian involve factorisations
Setup: A diagram
\begin{matrix}
X&{\rightarrow}&Y\\
\downarrow{}&&\downarrow{f}\\
U&{\rightarrow}&V
\end{matrix}
in a (proper) model category is called homotopy cartesian if ...
5
votes
0answers
103 views
Closed model categories in the sense of Quillen [1969] vs the modern sense
The modern definition of (closed) model category differs in two ways from Quillen's 1969 definition:
Model categories are now required to be complete and cocomplete, whereas Quillen only asked for ...
3
votes
0answers
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A model structure on $\bf Cat$
Define a model structure on $\bf Cat$ by the following rules:
A weak equivalence is an equivalence of categories;
A cofibration is a functor which is injective on objects;
A fibration is a functor ...
2
votes
0answers
95 views
A fibrant-objects structure on $\bf Top$
One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological spaces $\bf Top$, called the $\pi_0$-fibrant structure:
A ...
2
votes
0answers
46 views
In a model category, is the full subcategory of fibrant objects a reflective subcategory?
I apologize in advance if my question is utterly stupid, but I can't resist asking it. So...
Is it true that in a model category ( - for example $\mbox{Set}_\Delta$ with the Joyal model structure - ) ...
2
votes
0answers
74 views
How are injective model structures cofibrantly generated?
I have a question about the injective model structure on functor categories.
As background : If $\mathcal{M}$ is a combinatorial model category and $\mathcal{C}$ is a small category, then there are ...
2
votes
0answers
76 views
left inverse to trivial fibration is trivial cofibration
It is claimed that in the model category of simplicial sets (with usual model structure), a trivial fibration $X \to Y$ has a section, which is a trivial cofibration.
Now, I see that there is a ...
1
vote
0answers
75 views
Homotopy pullback square implies weak equivalence of homotopy fibres
I am quite confused about the following situation: suppose that we have a map $f \colon X \to Y$. Its homotopy fibre is defined as the pullback of the following of diagram:
\begin{matrix}
Ff & ...
