The model-categories tag has no wiki summary.

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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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**1**answer

260 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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243 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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**1**answer

272 views

### 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 ...

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1k views

### On the difference between a projective chain complex and a level-wise projective chain complex

Let R be an associative ring with a unit, and consider the standard projective model structure of non-negatively graded (left) R-module, $Ch_R$. A map $f:M\to N$ in $Ch_R$ is a weak equivalence if it ...

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**1**answer

109 views

### Bibliographical reference needed (characterizing the weak equivalences of a model category)

I need a bibliographical reference for this fact: let $\mathcal{M}$ be a model category such that all objects are cofibrant; then the class of weak equivalences is the class of maps f such that ...

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267 views

### Is the geometric realisation of a nerve equivalent to the classifying space of its categorisation?

The nerve functor $N:Cat\to sSet$ has a left adjoint, namely the categorisation $C$. In fact there is a natural isomorphism $\epsilon: CN\to Id$ and $N$ is a full embedding.So if I start with a ...

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416 views

### When is homotopy orbit space weakly equivalent to orbit space, other than situation of free action?

Let $M$ be a closed symmetric monoidal model category. Let $X$ be a cofibrant object (it can also be fibrant if you like) and let $\Sigma_n$ act on $X^{\otimes n}$ by permuting the factors (note that ...

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331 views

### How strong is the condition that an operad splits, i.e. O(n)=O(s)xO(n-s)?

I recently proved something for the operad $Comm$ valued in a model category $M$ and am trying to generalize it to other symmetric operads. I'm very new to operads, so please forgive me if there are ...

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209 views

### Boardman Vogt W construction for modules over an operad

The W construction of Boardman and Vogt gives a cofibrant replacement for operads. In http://arxiv.org/abs/math/9907073, Salvatore describes a cofibrant replacement for algebras over an operad. Is ...

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640 views

### Is there a cheap proof that (homotopy) endomorphisms are functorial?

This is, in some sense, the homotopy version of this question.)
If $C$ is a category with $iC$ the subcategory of isomorphisms, there is a functor $X \mapsto End(X)$ from $iC$ to the category of ...

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**1**answer

564 views

### How many model category structures are there on Top?

I recently started learning a little model category theory and in particular I found this nice exercise. I only know a little topology, but this prompted me to wonder how many model category ...

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570 views

### Limitations on model-categorical presentations

In higher category theory, it is common that a weak structure cannot be strictified in all directions simultaneously. For instance, a monoidal category is not (in general) equivalent to one that is ...

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601 views

### Connective spectra versus simplicial abelian groups - very basic question

Hello,
I have very general , "introductory" questions (It is quite hard for me to seek for specific things in the algebraic topology literature).
I guess that connective spectra have a model ...

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555 views

### Reference request - CDGA vs. sAlg in char. 0

Hello,
Are the model categories of simplicial commutative algebras over $k$ and that of commutative differential graded algebras (in negative cohmological dimension) Quillen equivalent in char. 0 (or ...

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201 views

### Technical question about cell complexes

Hello,
I have a technical question. My terminology:
I - set of standard inclusions $\partial I^n \to I^n$.
I-cell (Relative Cell Complexes) - transfinite compositions of pushouts of maps in $I$.
...

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477 views

### About a statement in Jardine and Goerss “Simplicial Homotopy Theory”

Hello,
This probably just technical, but anyway:
In "Simplicial Homotopy Theory" by Goerss and Jardine, chap. III, par. 2, after cor. 2.12, they describe a model structure on $Ch^{+}$, the category ...

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386 views

### Transporting model structures via adjunctions

Hello,
If $F$ is a left adjoint between $C$ and $D$, and $D$ has a model structure; We can define cofibrations and equivalences in $C$ to be those that are so after applying $F$. What are criterions ...

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395 views

### Counterexample in cohomology for symmetric spectra?

Symmetric spectra are a particular model for spectra, introduced by Hovey, Shipley and Smith. They have the nice property that they have a well-behaved smash product. Our interest in spectra comes ...

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195 views

### local model structure on simplicial presheaves

Hello,
Let $\mathcal{C}$ be a (small) category equipped with a Grothendieck pretopology.
Let $sPSh(\mathcal{C})$ be the category of simplicial presheaves on $\mathcal{C}$, together with its ...

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607 views

### When is the category of pro-objects a homotopy category?

For a category $C$, there is a category Pro-$C$ whose objects are cofiltered diagrams $I \to C$ and whose morphisms are given by
$$
{\rm Hom}(\{x_s\},\{y_t\}) = \varprojlim_t\ \varinjlim_s\ {\rm ...

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207 views

### Free Monoids in Closed Symmetric Monoidal Categories

There appear to be questions perhaps tangentially related to this that have been asked already. If so a reference and a close would be heartily appreciated.
Give some category $\mathcal{C}$ with the ...

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207 views

### Fitting desired weak equivalences and cofibrations into a model category

Suppose I have a category $\mathbf{C}$ and classes of morphisms $\mathcal{W}$ and
$\mathcal{C}$, and I would like to know that $\mathcal{W}$ and $\mathcal{C}$ are
the weak equivalences and the ...

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648 views

### Alternative characterization of homotopy equivalence

Using the formalism of model categories its possible define the concept of homotopy as done here.
If we take as model category $\mathbf{Top}$ having homotopy-equivalence as weak-equivalence and ...

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521 views

### Hovey's unit axiom in monoidal model categories

Let $\mathcal{C}$ be a monoidal model category in the sense of Hovey's book. He assumes the following unit axiom not considered in other references (e.g. Schwede-Shipley): given a cofibrant ...

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308 views

### Fibration of Batanin/Leinster $\omega$-groupoids

Is there (defined somewhere) a notion of fibration between two weak $\omega$-groupoids in the sense of Batanin/Leinster?
I tried to search on Google and in Higher Operads, Higher Categories of Tom ...

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606 views

### Is the category of small categories locally presentable?

I was wondering whether the various model structures on the category of small categories are combinatorial. I think that the ones I know are at least cofibrantly generated. In order to be ...

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2k views

### Computations in $\infty$-categories

Direct to the point.
Since now I've looked a lot of presentations of $\infty$-categories, but it seems that the only way to do explicit computations on these objects is via model categories. Is that ...

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371 views

### Find weak equivalences from fibrations and cofibrations

Let's suppose I have a category $\mathcal C$ with two weak factorisation systems $(C,F_W)$ and $(C_W,F)$, where $C_W\subset C$ and $F_W\subset F$.
I would like to have a model structure on $\mathcal ...

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### Does the right adjoint of a Quillen equivalence preserve homotopy colimits?

Call a diagram $E$ in a model category a homotopy colimit diagram if the morphism $$\mathrm{hocolim}~E\to \mathrm{colim}~ E$$ is a weak equivalence. A homotopy colimit is defined as the categorical ...

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### Presheaves on a complete Segal space

Let C be an $(\infty,1)$-category, incarnated as a complete Segal space, hence in particular a bisimplicial set. Is there a model structure on the slice category of bisimplicial sets over C which ...

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### Left adjoints to several inclusions of homotopy simplicial model categories

The left adjoint to the inclusion $sGrp\hookrightarrow sPtSet$ of the category of simplicial groups into the category of pointed simplicial sets is homotopy equivalent to the loop suspension functor ...

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443 views

### Homotopic maps out of cofibration sequences

Consider a model category $\mathcal{C}$ and a sequence of cofibrations $0 \to X_0 \to X_1 \to X_2 \to \dots$ lying in $\mathcal{C}$. Let $X$ be the colimit of this sequence. Suppose furthermore that ...

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### Do simplicial objects in a Topos form a model category?

Sometimes people say "If you don't like the word 'topos', just think the category of Sets", but I'm not sure to what extent this analogy holds.
The real question here is, do simplicial object in a ...

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143 views

### Can one recover an A-algebra from its cotangent complex?

Given an A-algebra B, one can define the cotangent complex $L_{B/A}$ as $\Omega^1_{P/A}\otimes_PB$, where $P$ can be taken as the canonical resolution of $B$ associated to the pair of adjoint functor ...

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### Do we still need model categories?

One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak ...

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### A model structure on the category of “dualizing maps”

Let $C$ be the category with objects being maps $h:M\to A$ where $A$ is a commutative graded $\mathbb{Q}$-algebra (cdga), $M$ is a differential graded (dg) $A$-module and $h$ is an $A$-dg-module map, ...

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### Acyclic models via model categories?

Recall the acyclic models theorem: given two functors $F, G$ from a "category $\mathcal{C}$ with models $M$" to the category of chain complexes of modules over a ring $R$, a natural transformation ...

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### The weak equivalences in the covariant model structure

Let $S$ be a simplicial set. Recall that there is a model structure, called the covariant model structure (see HTT ch. 2 and this question), on $\mathbf{SSet}/S$ such that:
The cofibrations are the ...

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257 views

### When is the cofibrant replacement of a product the product of the cofibrant replacements?

I'm in a situation where I'd like to prove $Q(E\otimes E) \simeq QE \otimes QE$ for a monoid $E$ in a symmetric monoidal model category. I know it's not true in general that $Q(E\otimes F)\simeq QE ...

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### univalent axiom as a property of a model category?

I am interested to understand the univalence axiom of Voevodsky; however, I know
very little type theory. Thanks to response below, I now understand what is being univalent
means for a morphism. A ...

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306 views

### Equivariant sheaves and simplicial varieties

I would like to proof the following theorem:
Let $\pi:X\rightarrow X/G$ be a principal $G$-bundle (say of varieties, Zariski locally trivial), then $\pi^*$ induces an equivalence between modules on ...

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396 views

### Fiber sequences in proper model categories

I am confused about the notion of a fiber sequence (or dually a cofiber sequence) in a general pointed and proper model category $\mathcal{C}$.
Following Hovey, we can define, like in topology, a map ...

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523 views

### When are “diagrams of cofibrations” projectively cofibrant?

Let $P$ be a small category, and let $F:P\to M$ be a diagram in a left-proper combinatorial model category $M$. We say that $F$ is a diagram of cofibrations if for every object $p\in P$, $F(p)$ is ...

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### Closed Model Category Structure on Chain Complexes Related to A Left-exact Functor

Let $F:A \to B$ be an additive left-exact functor of abelian categories (Do not assume that they have enough injectives / projectives.) Suppose we are given a class of objects $R$ adapted to $F$ (see ...

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**1**answer

360 views

### Resolutions by Adapted Class of Objects and Model Categories

My question is about the construction of derived functor in the language of model categories. (As it is done for example the paper by Dwyer and Spalinski "Homotopy Theories and Model Categories".) ...

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### Mapping into a geometric realization.

Suppose $S$ is a simplicial set, $X$ is a space, and we are given a map
\[
f: \text{Sing}\,X\to S.
\]
When is is possible to produce a map $X\to |S|$?
We can take ...

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### Is the simplicial completion of a localizer always a bousfield localization of the injective model structure?

Background
Recall (from Cisinski's Astérisque volume 308) that given a small category $A$, we define an $A$-localizer to be a class $W$ of morphisms of $\mathrm{Psh}(A)$ satisfying the following ...

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### Does adding degeneracies to a semi-simplicial diagram change the homotopy colimit?

Let $\Delta_{+}$ be the sub-category of the simplex category $\Delta$ containing only injective functions, and take $M$ to be a nice model category. I'll write $i \colon \Delta_{+} \hookrightarrow ...

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### “non-Bousfield” localisations of model categories

When can I localise a model category by a set(or class) of morphisms,
and how do I describe the localised model category ?
By 'localise' I mean to find a localisation functor $q_S : M \rightarrow ...