# Questions tagged [model-categories]

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.

508
questions

**3**

votes

**1**answer

111 views

### Proving a Kan-like condition for functors to model categories?

I've been trying to prove this version of the Kan condition for a project that I'm thinking about, and I'm pretty stuck. My experience asking questions about this stuff on MO in the past has been ...

**8**

votes

**2**answers

433 views

### Non-small objects in categories

An object $c$ in a category is called small, if there exists some regular cardinal $\kappa$ such that $Hom(c,-)$ preserves $\kappa$-filtered colimits.
Is there an example of a (locally small) ...

**2**

votes

**1**answer

186 views

### Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers

Let $M$ be a model topos and $S$ a set of morphisms, there exists a set of morphism $\bar{S}$ which is generated by the $S$-local equivalences which is closed under homotopy pullbacks in $M$. Suppose ...

**5**

votes

**0**answers

99 views

### When do zigzags of weak equivalences detect isomorphisms in the localization?

The usual way to prove that two model categories are equivalent is to construct a zigzag of Quillen equivalences between them, but is it always possible? We can ask a more general question.
...

**3**

votes

**1**answer

117 views

### Strøm model structure on nonnegatively graded chain complexes

Let $\newcommand{\Ch}{\mathsf{Ch}}\Ch_{\ge 0}(R)$ be the category of $\mathbb{N}$-graded chain complexes over some ring $R$, and $\Ch(R)$ the category of $\mathbb{Z}$-graded chain complexes.
The ...

**4**

votes

**1**answer

107 views

### Bousfield localization of a left proper accessible model category

What is known about the Bousfield localization of a left proper accessible model category by a set of maps ? (I mean not combinatorial which is already known)

**3**

votes

**0**answers

164 views

### The k-ification of the compact-open topology for weak Hausdorff compactly generated spaces

Let CGWH be the category of weak Hausdorff compactly generated spaces; see e.g.
N.P. Strickland. THE CATEGORY OF CGWH SPACES: Preprint available from
https://neil-strickland.staff.shef.ac.uk/courses/...

**6**

votes

**1**answer

243 views

### Homotopy pullbacks and pushouts in stable model categories

There are lots of similar questions that have been answered on this topic (particularly Homotopy limit-colimit diagrams in stable model categories), but I have a specific question that I do not ...

**1**

vote

**0**answers

90 views

### Existence of tensor product of infinity operads

I am trying to show, or find a reference, for the following fact:
"Given O,P two infinity operads [in the sense of Lurie, HA, Definition 2.1.1.10], there always exist a tensor product".
In other ...

**3**

votes

**0**answers

58 views

### On cofibrations of simplicially enriched categories

Let $\mathbb{C}$ be an strict 2-category and denote by $C$ is underlying 1-category viewed as as a 2-category only having identity 2-cells.
We have a canonical inclusion functor ,
$$i: C \...

**5**

votes

**1**answer

194 views

### Compact objects in the $\infty$-category presented by a simplicial model category

Let $\mathsf{M}$ be a simplicial model category presenting an $\infty$-category $\mathcal{M}$. I'm interested in a general statement relating compact objects in $\mathcal{M}$ (in the $\infty$-...

**2**

votes

**0**answers

87 views

### Homotopy colimits of simplicial objects

Given a simplicial combinatorial model category $\mathcal{M}$ and a simplicial diagram $F\colon \Delta^{\mathrm{op}} \rightarrow \mathcal{M}$, is there a nice (i.e. explicitely computable) way of ...

**4**

votes

**0**answers

83 views

### Right adjoint preserving (trivial) cofibrations between cofibrant objects

Consider a right Quillen adjoint which is not a categorical left adjoint which takes (trivial resp.) cofibrations between cofibrant objects to (trivial resp.) cofibrations between cofibrant objects.
...

**1**

vote

**0**answers

84 views

### Which set of compact objects generates the subcategory of a compactly generated stable model category?

I couldn't find any info on what set of compact objects generates the following subcategory:
Let $k$ be a field of positive characteristic and let $G$ be either a finite group or a finite group ...

**2**

votes

**1**answer

84 views

### Is the Hurewicz model category left proper?

A model structure is left proper if the pushout of a weak equivalence along a cofibration is a weak equivalence. In the Hurewicz (or Strom) model structure on the category of topological spaces, weak ...