The model-categories tag has no usage guidance.

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### 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 ...

**-1**

votes

**2**answers

681 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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votes

**1**answer

532 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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319 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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votes

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679 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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### 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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votes

**4**answers

381 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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votes

**1**answer

1k views

### 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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votes

**1**answer

396 views

### 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 ...

**5**

votes

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254 views

### 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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455 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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718 views

### 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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**0**answers

144 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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votes

**4**answers

7k views

### 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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votes

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202 views

### 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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1k views

### 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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**1**answer

443 views

### 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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**0**answers

264 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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votes

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830 views

### 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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votes

**0**answers

318 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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414 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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541 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

386 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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180 views

### 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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votes

**1**answer

876 views

### 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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votes

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321 views

### 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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145 views

### “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 ...

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378 views

### Is there a model category structure on non-negatively graded commutative chain algebras?

Let $\mathtt{DGA}$ be the category of non-negatively graded DG chain algebras, and $\mathtt{DGA}$* the category of non-negatively graded cochain algebras. Let $\mathtt{CommDGA}$ be the full ...

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310 views

### simplicial deRham complex and model category structure

To every simplicial manifold is associated its simplicial deRham complex.
Is there any literature that discusses explicitly to which extent this classical construction, regarded as a (contravariant) ...

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318 views

### Simplicial presheaves that are colimits of themselves?

Suppose $C$ is a small category and $X_{\bullet}$ is a simplicial object in $C$. In particular, by composing with Yoneda $$y:C \to Set^{C^{op}}$$ $y(X)_{\bullet}$ is a simplicial presheaf. I believe ...

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**1**answer

144 views

### Cofibrations and coequalisers in a proper model category

I have a proper model category and in it two coequalisers, $A_i \rightrightarrows B_i \to C_i$, $i=1,2$. I have a map of diagrams arising from maps $A_1 \to A_2$, $B_1 \to B_2$ where these two arrows ...

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### Model category structure on categories enriched over quasi-coherent sheaves

Gonçalo Tabuada has shown that there is a Quillen model category structure on the category of small dg-categories, i.e. the category of small categories enriched over chain complexes (for a fixed ...

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136 views

### end of a weak equivalence

I would like to get a concrete description of sufficient conditions for the end of a morphism in $\mathcal{C}^{J^{op}\times J}$ (which is a point-wise weak equivalence) to be a weak equivalence.
In ...

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270 views

### geometric realization on $\mathbf{sTop}$

Is geometric realization $|\cdot|:\mathbf{Top}^{\mathbf{\Delta}^{\textrm{op}}}\rightarrow \mathbf{Top}$ a left Quillen functor? If so, under what model structure on ...

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votes

**1**answer

293 views

### Model categories and cellular maps

A question came up on MSE and it generated, for me, the following question:
When looking at the maps of CW/cell/simplicial complexes do the cellular/simplicial maps have a model theoretic ...

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1k views

### Do homotopy groups “always” commute with filtered colimits?

It is well-known that homotopy groups, of, say, simplicial sets, commute with filtered colimits.
However, I could not find a reference for an analogous result for homotopy groups of spectra, or, ...

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519 views

### Inner hom and geometric realization.

I would like to prove the following fact, which I learned from a previous MO question.
Let $S_\cdot,T_\cdot\in\mathbf{sSET}$ be simplicial sets, and assume that $T_\cdot$ is Kan. Then there is a ...

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votes

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533 views

### When does a cosimplicial object compute homotopy colimits?

Suppose I want to compute the homotopy colimit of a diagram of spaces. There is a simple way of getting a simplicial space from this diagram, and a theorem tells me that taking the geometric ...

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2k views

### How canonical is cofibrant replacement?

Quillen's original definition of a model category included noncanonical factorization axioms, one being that any map can be factored into a cofibration followed by an acyclic fibration. More recent ...

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### “Strøm-type” model structure on chain complexes?

Background
The Quillen model structure on spaces has weak equivalences given by the weak homotopy equivalences and the fibrations are the Serre fibrations. The cofibrations are characterized by ...

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448 views

### Model categories of simplicial objects

If $\mathcal{C}$ is a category, then surely the category of simplicial
objects $s\mathcal{C}$ is not automatically a model category. What conditions
must $\mathcal{C}$ satisfy in order for ...

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604 views

### An Ex functor for the contravariant homotopy structure

I'm going to slack on the background and get to the point:
Is there a good notion of an $Sd/Ex$ adjunction for $sSet/S$ equipped with
the contravariant model structure (cofibrations are monomorphisms ...

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votes

**1**answer

234 views

### Compatibility of classifying space with inner-hom?

Let $\mathbf{sTop}$ be the functor category $\mathbf{Top}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let $\mathbf{sCat}$ be the functor category
$\mathbf{Cat}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let ...

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843 views

### Fibered/cofibered higher categories, relative model structures, slicing, and (∞,2)-category theory

Jacob Lurie defined a model structure on the category of marked simplicial sets sliced over a fixed simplicial set $S$ called the cartesian model structure. (For a definition, see here or HTT ...

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### Pointed Hurewicz model structure

In Strøm's (no relation) paper "The Homotopy Category is a Homotopy Category" he proves
that the category of unpointed topological spaces, with Hurewicz fibrations and ordinary cofibrations and ...

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### Model structure on Simplicial Sets without using topological spaces

The category of simplicial sets has a standard model structure, where the weak equivalences are those maps whose geometric realization is a weak homotopy equivalence, the cofibrations are ...

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### Does the category of topological symmetric spectra satisfy the monoid axiom ?

In the paper "Symmetric spectra"by Hovey, Smith and Shipley, they say that they don't know if the monoid axiom holds for topological symmetric spectra. This paper was written in 1998 so I am wondering ...

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### How does Berger-Moerdijk's relative Boardman-Vogt work?

In "The Boardman-Vogt resolution of operads in monoidal model categories," the authors construct factorizations of sufficiently nice operad maps $P\to Q$ into a cofibration followed by a weak ...

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### Is the injective model structure on symmetric spectra Bousfield localizable?

I am interested in injective model structures on both symmetric spectra as exposed in Hovey/Shipley/Smith and motivic symmetric spectra as in Jardine's article. Both authors take a model structure on ...