The model-categories tag has no usage guidance.

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### Quillen equivalence of diagram categories

Let $C$ be a model category and $B$ a direct category. By Theorem 5.1.3 in Mark Hovey's book Model categories, there is a model category structure on the diagram category $C^B$ such that weak ...

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**1**answer

243 views

### Bousfield localization before and after taking homotopy

The ncatlab says:
Under suitable conditions it should be true that for $C$ a model category whose homotopy category $\mathrm{Ho}(C)$ is a triangulated category the homotopy category of a left ...

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**1**answer

330 views

### Unicity up to homotopy of simplicial enrichments

On the one hand, in their paper Simplicial structures on model categories and functors, Rezk, Schwede and Shipley proved that a simplicial model category structure on a given model category is unique ...

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234 views

### Cisinski-Deglise model structure on chain complexes

Let $\newcommand{\A}{\mathscr{A}}\A$ be a grothendieck abelian category. In their paper "Local and stable homological algebra in grothendieck abelian categories", Cisinski and Déglise define the ...

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### Homotopy limit-colimit diagrams in stable model categories

It is shown in Remark 7.1.12 of (a newer version of) Mark Hovey's book Model Categories that, in a stable model category, homotopy pullback squares coincide with homotopy pushout squares. The argument ...

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votes

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264 views

### Is there a class of simplicial sets whose weak homotopy type is preserved by symmetrization?

We have the categories, $S$, of simplicial sets and $SS$, of symmetric simplicial sets (whose simplices are unordered). There are functors:
$H:S\to SS$ forgetting the ordering on simplices and
$L:...

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**0**answers

162 views

### Fibrations of orthogonal G-spectra and fixed points

There are at least two fixed point functors that characterize stable equivalences of orthogonal G-spectra: the geometric fixed points and the naive fixed points of a fibrant replacement.
Is this true ...

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**1**answer

254 views

### Cellular model structures on continuous functors

The category of enriched functors from finite based CW complexes to based topological spaces
has a projective model structure. The fibrations
are the objectwise Serre fibrations and the weak ...

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**2**answers

497 views

### A category with weak equivalences which is not a model category

I'm only considering complete and cocomplete categories. A pair $(\mathfrak{X} , \mathfrak{W}) $ is, by definition, a category with weak equivalences if $ \mathfrak{X} $ is a category and $ \mathfrak{...

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**5**answers

898 views

### What is a good basic reference on model categories?

I am looking for a general reference text on model categories, that contains all the basic results and definitions. I'm perfectly happy to be pointed towards a textbook, and I'm not looking for ...

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**0**answers

181 views

### Model structure on the category of chain complexes in an abelian category gives rise to the derived category

Let $\mathcal A$ be an abelian category with enough projectives. Consider a morphism $f \colon X^\bullet \longrightarrow Y^\bullet$ in the category $C(\cal A)$ of chain complexes in $\mathcal A$.
Is ...

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**1**answer

349 views

### Homotopy left-exactness of a left derived functor

Let
$$
F: \mathcal{C} \leftrightarrows \mathcal{D} :G
$$
be a Quillen adjunction between model categories. Consider the corresponding adjunction of total derived functors
$$
\mathbb{L}F: \mathrm{Ho}(\...

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votes

**1**answer

129 views

### Equivariant versus retractive spaces: a reference request

Let $T$ be the category of compactly generated weak Hausdorff spaces with model structure given by Serre fibrations, Serre cofibrations and weak homotopy equivalences. Let $G = |G.|$ be the (geometric)...

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votes

**1**answer

193 views

### Directed colimits of maps in a combinatorial model category

I have the following situation. $M$ is a combinatorial model category, or if you like a locally presentable $(\infty,1)$-category. I have a set of maps $S$ and I let $C$ be the class of maps generated ...

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**1**answer

589 views

### Seeking errata for Berger-Moerdijk Axiomatic Homotopy Theory for Operads

The paper I'm referring to can be found here. It came out in 2003. I've been told by professors before that I should be careful relying on things from this paper, because it contained errors.
Is ...

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**2**answers

485 views

### Cofibrant replacements of a given object in a combinatorial model category

In a combinatorial model category, every $\lambda$-filtered colimit is a homotopy colimit for $\lambda$ regular big enough. So for $\lambda$ regular big enough, every $\lambda$-filtered colimit of a ...

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**0**answers

232 views

### Is there a local projective model structure on simplicial sheaves? What are its fibrant objects?

Consider a site S (I am mostly interested in hypercomplete sites, e.g., the site of smooth manifolds).
The category of simplicial presheaves SPSh(S) on S can be equipped with the local projective ...

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395 views

### Can one make the category of pairs of topological spaces a model category?

Can you make the category whose objects are pairs of spaces $(X,A)$, and morphisms the obvious diagrams, into a model category? Of course I want this to be done in a meaningful way, that is, agreeing ...

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vote

**1**answer

312 views

### Coequalizer in category of dg-algebras

It is known that there is a model structure on category of dg algebras (non-commutative over arbitrary commutative ring). In particular it is complete and co-complete category. My question is how to ...

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**1**answer

569 views

### The category theory of $(\infty, 1)$-categories

There are many proposed models for the theory of $(\infty, 1)$-categories and it has now been shown that many of these theories have Quillen-equivalent model categories, i.e. that they are equivalent ...

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### Homotopy excision and homotopy pushout

I have three related questions.
I understand homotopy pushouts via the standard model structure on the diagrams - and taking the derived functor of the pushout.
I'm not sure, but I believe that in ...

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**1**answer

459 views

### Model structure on the category of small $A_\infty$ categories, hocolims.

I strangely could not find a reference for this. What are some (if any) model structures on the category of small $A_{\infty}$ categories, with weak equivalences quasi-equivalences. Same question in ...

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vote

**0**answers

161 views

### Homotopy colimits over a certain subset category.

Hi!
Let $I$ denote all the finite subsets of some set (infinite or finite) S. For each n, let $I_n$ be the subset consisting of objects of cardinality n, so that there are no morphisms between the ...

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784 views

### Homology in the $A_\infty$ World

This question is turning out to be a little long so let me start off with the headline. Given a differential graded algebra $A$, we can recover $A$ from its homology $HA$ if we know "the" $A_\infty$-...

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259 views

### About the Cole-Ström model category structure with a locally presentable category

In "Many homotopy categories are homotopy categories" (M. Cole / Topology and its Applications 153 (2006) 1084–1099), Cole generalizes the construction of the model category of Ström to any bicomplete ...

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### When is the projective model structure cartesian? When is the internal hom invariant?

If M is a sufficiently nice model category and D is a small category then there are two natural model structures we can impose on the functor category $Fun(D,M)$ where the weak equivalences are the ...

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401 views

### When do reflexive coequalizers preserve weak equivalences?

In my work I've run into the following situation. In a model category, I have two reflexive coequalizers $A_i \stackrel{\to}{\to} B_i \to C_i$ and a map of diagrams which is levelwise a weak ...

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624 views

### Monoidal model category structure on a functor category.

Let $A$ be a small simplicial category. The category $Fun(A,s\mathrm{Set})$ of simplicial functors from $A$ to simplicial sets can be given the projective model structure in which fibration and weak ...

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161 views

### Reedy model structures on oplax limits

Suppose $R$ is a category and $F:R\to Cat$ is a functor (or pseudofunctor). The oplax limit of $F$ is the category whose objects consist of an object $x_r \in F(r)$ for all $r$ together with a ...

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### Model category structures on dga's in a ringed topos

In the introduction to his paper "Towards a non-abelian $p$-adic Hodge theory", Olsson says that for any ringed topos $(\mathcal{T},\mathcal{O})$ with $\mathcal{O}$ a sheaf of $\mathbb{Q}$-algebras, ...

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### Model for the (infinity,1)-category of functors preserving certain homotopy limits

This question is a follow up to: Model for the (infinity,1)-category of (homotopy-)limit preserving functors.
Warm-up Question: Given a simplicial model category $M$, what model category models ...

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478 views

### Model for the (infinity,1)-category of (homotopy-)limit preserving functors

I've got a simplicial model category $M.$ I'd like to get my hands on the (infinity,1) category of homotopy limit preserving functors from M to Spaces in order to compare it to another simplicial ...

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356 views

### The Quillen model structure on simplicial sets as a Bousfield localization

Starting with the trivial model structure on the category of simplicial sets (that is the weak equivalences are exactly the isomorphisms and the cofibrations and fibrations are arbitrary maps), is it ...

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### An explicit description of injective fibrations

If $M$ is a combinatorial model category and $C$ a small category, then the category $M^C$ admits an injective model structure in which the cofibrations and weak equivalences are levelwise. I would ...

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586 views

### A fibrant-objects structure on Top

(Sorry for the crossposting, but I'm really interested in this question).
One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological ...

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### Homotopical Galois theory of coverings

In the hope this won't turn into a trivial problem (I couldn't find a similar discussion here), here's my question.
I'm studying a little homotopical algebra in this article by Brown. You can easily ...

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289 views

### Monoidal Model Categories with Suspension Functor

This is basically just me trying to find out what such categories are called, and where they are written about. If I think of some model category of spectra being a "stabilization" of some model ...

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578 views

### Is there a notion of a “model category which admits left Bousfield localization?”

At a conference not too long ago I gave a talk on (left) Bousfield localization and was asked an interesting question afterwards. The question was whether I knew any examples of model categories which ...

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526 views

### Fubini theorem for hocolim.

I wanted to ask the following question,
Suppose $\mathbf{M}$ a cof generated model category and $I,~J$ two small categories. Suppose that $F:J\rightarrow \mathbf{M}^{\mathrm{I}}$ is a functor. Is it ...

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329 views

### On the natural (bigraded) homotopy groups of a simplicial object in a model category

$\def\mc{\mathcal} \def\sm{\wedge}$
This question stems from the Goerss-Hopkins paper Moduli Problems for Structured Ring Spectra. Let me begin by attempting to summarize the relevant framework -- ...

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254 views

### Induced model structure?

Let $H$ be a set with a binary operation $\cdot _H$ on it. To show that it is a group, one has to show that $\cdot _H$ is associative, find an identity element in $H$, and so forth; it might take ...

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votes

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293 views

### Small model categories?

All the standard examples for model categories are large categories. Is it possible to have a small model category? Are there any interesting examples?
EDIT:
Since a complete small category is a ...

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### Left Properness of Simplicial Commutative Algebras

A bit of light googling turns up several sources asserting that the model structure on simplicial commutative algebras over a ring is left proper (for example, 2.9 in Charles Rezk's paper Every ...

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369 views

### [Reference Request] The Definition of Adjoint Functors between dg-categories

Let $A$ and $B$ be two dg-categories, $F: A \rightarrow B$ and $G: B \rightarrow A$ are two functors. Then what is the definition that $F$ and $G$ form an adjoint pair?
In my mind $F\dashv G$ ...

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358 views

### Does the Monoid Axiom hold for k-spaces?

In “Algebras and Modules in Monoidal Model Categories” Schwede and Shipley introduced the monoid axiom. If a cofibrantly generated monoidal model category $M$ satisfies this axiom and some smallness ...

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477 views

### Topologically enriched homotopy colimits commuting with homotopy pullbacks

Hi,
I am looking for an enriched analogon of Proposition 4.4 in https://www.google.de/url?q=http://hopf.math.purdue.edu/Rezk-Schwede-Shipley/simplicial.pdf
Concretely, I would like to prove the ...

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### relationships between properties of model categories

I've recently found myself running up against all sorts of adjectives that can describe a model category: cofibrantly generated, combinatorial, tractable, stable, locally (finitely) presentable, (left ...

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275 views

### Model structure on category of endofunctors

Let $\mathcal C$ be model category, perhaps even cofibrantly generated. I don't assume that $\mathcal C$ is small. Recall that $End(\mathcal C)$ is the category of endofunctors on $\mathcal C$, with ...

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257 views

### Need M combinatorial for existence of injective model structure on $M^G$?

I'm doing some work with model categories and operads, and to check a certain hypothesis I've had to learn a bit of equivariant homotopy theory. Let $M$ be a model category and $G$ be a finite group. ...

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### State of knowledge on the Commutative W-spaces which appear in “Model Categories of Diagram Spectra”

This is a follow-up question to another question I asked last month. In MMSS's "Model Categories of Diagram Spectra," the authors consider many different models for spectra and prove monoidal Quillen ...