# Questions tagged [model-categories]

The model-categories tag has no usage guidance.

**7**

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**0**answers

151 views

### $\Gamma$-sets vs $\Gamma$-spaces

I know that every $\Gamma$-space is stably equivalent to a discrete $\Gamma$-space, i.e. a $\Gamma$-set.
For example, Pirashvili proves, as theorem 1.2 of Dold-Kan Type Theorem for $\Gamma$-Groups, ...

**12**

votes

**1**answer

317 views

### Counter-example to the existence of left Bousfield localization of combinatorial model category

Is there any known example of a combinatorial model category $C$ together with a set of map $S$ such that the left Bousefield localization of $C$ at $S$ does not exists ?
It is well known to exists ...

**6**

votes

**1**answer

216 views

### Is the Thomason model structure the optimal realization of Grothendieck's vision?

In Pursuing Stacks, Grothendieck uses the category $Cat$ of small categories to model spaces. A recurring theme is the question of whether there is a Quillen model structure supporting this homotopy ...

**3**

votes

**1**answer

253 views

### Definition A.3.1.5 of Higher Topos Theory

I am now reading the book Higher Topos Theory. In A.3.1.5, it gives the definition
of a $\mathbf{S}$-enriched model category, where $\mathbf{S}$ is a monoidal model category. But in the book model ...

**7**

votes

**2**answers

274 views

### For which categories of spectra is there an explicit description of the fibrant objects via lifting properties?

How explicit are the model structures for various categories of spectra?
Naive, symmetric and orthogonal spectra are obtained via left Bousfield localization of model structures with explicit ...

**3**

votes

**1**answer

91 views

### Monoidalness of a model category can be checked on generators

If $C$ is a cofibrantly generated model category which is also monoidal biclosed, then to check that $C$ is a monoidal model category, it suffices to check that the Leibniz product of generating ...

**8**

votes

**2**answers

240 views

### Localization, model categories, right transfer

Suppose that we have a locally presentable category $M$ and $N$ is a locally presentable full subcategory of $M$. Both categories are complete and cocomplete. Lets suppose that we have an adjunction $...

**2**

votes

**0**answers

44 views

### Simplicial models for mapping spaces of filtered maps

Let $S$ and $K$ be simplicial sets with $K$ Kan. Given simplicial sets $S$ and $K$, we let $SIMP(S,K)$ be the internal hom in the category of simplicial sets. Hence $SIMP(S,K)$ is Kan.
Suppose that ${...

**3**

votes

**1**answer

59 views

### Simplicial models for fibrations between mapping spaces

Let $S,T$ and $K$ be simplicial sets with $K$ Kan. Given simplicial sets $S$ and $K$, we let $SIMP(S,K)$ be the internal hom in the category of simplicial sets. We let $|S|$ denote the geometric ...

**4**

votes

**1**answer

183 views

### simplicial objects in a model category

Suppose that we have a (combinatorial if necessary) model category $M$, and let $F:\Delta^{op}\rightarrow M$ a simplicial object in $M$, such that for any natural number $n$, $F([n])$ is a fibrant ...

**6**

votes

**1**answer

125 views

### Simplicial localization of the cofibrant-fibrant objects

Let $M$ be a model category. I don't assume that $M$ has functorial factorizations or that $M$ is simplicial. Write $M^{c}$ (respectively, $M^{cf}$) for the full subcategory of $M$ on the cofibrant ...

**3**

votes

**1**answer

155 views

### Cofibrations of functors

Let $\cal M$ and $\cal N$ be model categories, $S,T:\cal M\to N$ functors, and $\alpha:S\to T$ a natural transformation. Say that $\alpha$ is a <blank> cofibration if for any cofibration $i:A\to B$...

**3**

votes

**0**answers

114 views

### Why is the Straightening functor the analogue of the Grothendieck construction?

In classical category theory, there is the notion of functor (co)fibered in groupoids. Furthermore, via Grothendieck construction we have an equivalence between peseudo functors into the category of ...

**6**

votes

**1**answer

272 views

### Model structure on the category of topological groups

Consider the category $TopGr$ of topological groups. I want to know that this is a model category (can one understand its model structure by understanding a model structure on the category of enriched ...

**11**

votes

**1**answer

222 views

### Which maps of simplicial sets geometrically realize to fibrations?

If $f:X\to Y$ is a Kan fibration of simplicial sets, then its geometric realization $|f| : |X|\to |Y|$ is (in some suitable convenient category of topological spaces, like compactly generated ones) a ...