# Tagged Questions

2answers
77 views

### Cylinder object in the model category of chain complexes

Let $\text{Ch}⁺(R)$ be the category of non-negative chain complexes of $R$-modules where $R$ is a commutative ring. What is a cylinder object, in the sense of model categories, for a given complex ...
0answers
102 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 ...
2answers
65 views

### The empty set in homotopy theoretic terms (as a simplicial set/top. space)

I am currently confused about the empty set in terms of its path components and how this fits into the Quillen adjunction between topological spaces and simplicial sets. Probably, one of my ...
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 ...
1answer
92 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 ...
1answer
76 views

### Contractible homotopy fibre for CW complexes, categorial construction of the homotopy inverse

Let $f:X\to Y$ be a map of topological spaces. Assume further that the homotopy fibre is contractible. We get a long exact sequence on the homotopy groups and if $X$ and $Y$ are connected $f$ is a ...
2answers
148 views

### Kan fibrations and surjectivity

I have a basic question on the usual model structure on simplicial sets. What is the relation between being a Kan (trivial maybe ?) fibration and surjectivity ? Surjectivity here means either ...
1answer
134 views

### Do we implicitly consider model categories to be locally small?

Do we implicitly consider model categories to be locally small? I have the impression (but am not sure) that many references on model categories assume that all the categories are locally small, but ...
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 ...
1answer
134 views

### The Notion of “A Homotopy Theory”

Sometimes (specifically in this case I'm looking at Charles Rezk's "A Model for the Homotopy Theory of Homotopy Theory") it seems that people refer to the homotopy category of a model category as a ...
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