# Questions tagged [model-categories]

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**4**

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### Cotangent complex and its distinguished triangle- a generalisation?

Associated to any ring maps $A\to B\to C$ there is the distinguished triangle
$$\mathbf{L}_{B/A}\otimes^L_BC\ \longrightarrow \ \mathbf{L}_{C/A} \ \longrightarrow \ \mathbf{L}_{C/B} \ \stackrel{+1}{\...

**4**

votes

**1**answer

167 views

### Thomason fibrant replacement and nerve of a localization

The Thomason model structure on the category $\mathrm{Cat}$ of small categories is transferred along the right adjoint of the adjunction $$\tau_1 \circ \mathrm{Sd}^2 \colon s\mathrm{Set} \...

**8**

votes

**3**answers

406 views

### On model categories where every object is bifibrant

Most model structures we use either have that every object is fibrant or that every object is cofibrant, and we have various general constructions that allow (under some assumption) to go from one ...

**6**

votes

**2**answers

247 views

### Model category structure on spectra

I have a concrete question for the algebraic category of spectra, but if there is an answer for its topological analogue I would be interested in it.
Let $S$ be a finite dimensional Noetherian scheme ...

**12**

votes

**2**answers

183 views

### Example of non accessible model categories

By curiosity, I would like to see an example of a model category with the underlying category locally presentable which is not accessible in this sense (and just in case: even by using Vopěnka's ...

**2**

votes

**0**answers

81 views

### Why is a homotopy limit of a cosimplicial space not the ordinary limit?

I've been trying to compute a homotopy limit of a cosimplicial object $X: \Delta \to \mathscr{M}$, where $\mathscr{M}$ is some simplicial model category, we may take it to be spaces. I'm wondering ...

**7**

votes

**2**answers

404 views

### What are the advantages of simplicial model categories over non-simplicial ones?

Of course, there are general results allowing one to replace a model category with a simplicial one. But suppose I want to stay in my original non-simplicial model category (say for some reason I'm a ...

**8**

votes

**0**answers

178 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

370 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

241 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

272 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

292 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

93 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

244 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

45 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 ${...