# Questions tagged [model-categories]

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**11**

votes

**1**answer

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### Which maps of simplicial sets geometrically realize to fibrations?

If $f:X\to Y$ is a Kan fibration of simplicial sets, then its geometric realization $|f| : |X|\to |Y|$ is (in some suitable convenient category of topological spaces, like compactly generated ones) a ...

**5**

votes

**1**answer

127 views

### Quillen equivalent module categories

Let $f:A \rightarrow B$ be a weak equivalence of simplicial commutative rings. There is a Quillen pair $(-\otimes_{A}B, f_{\ast})$ which is an equivalence. In this situation, $(-\otimes_{A}B, f_{\ast})...

**0**

votes

**1**answer

142 views

### A question about combinatorial model categories

I am currently reading the appendices of Higher Topos Theory, and I was puzzled by Lurie's proof of lemma A.2.6.7 (I can not make sense of the end of the proof.)
He uses this result to prove Jeff ...

**5**

votes

**2**answers

220 views

### Limit of weak equivalences in a Bousfield localization

Let $M$ be a model category and $C$ a class of maps in it, and assume the left Bousfield localization $L_CM$ exists. Suppose we are given sequences of maps $(p_{n+1}: X_{n+1}\to X_n), (q_{n+1}: Y_{n+1}...

**6**

votes

**1**answer

207 views

### Does the Dwyer-Kan model structure make dgCat a model $2$-category?

Let dgCat be the category of small dg-categories. The well-known Dwyer-Kan model structure makes dgCat a model category.
Now we consider dgCat as a 2-category, which objects small dg-categories, $1$-...

**4**

votes

**0**answers

169 views

### $\mathrm{HH}$ and $\mathrm{HC}$ as two different Taylor expansions at the same point

Let us consider the functors from dg algebras to graded spaces given by Hochshcild homology and cyclic homology. Then it is well known that in degree $0$, both functors are given by $A\longmapsto A/[A,...

**3**

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**0**answers

133 views

### filetered colimit of fibrant-cofibrant objects

Suppose that we have a $\lambda$-combinatorial model category $M$ (for some cardinal $\lambda$) such that any $\lambda$-filtered colimit of fibrant-cofibrant objects is always fibrant. My question is ...

**6**

votes

**2**answers

608 views

### Theorem 2.1.2.2 Higher Topos Theory

At the page 74 of HTT, there is the following theorem
Let $S$ be a simplicial set, $\mathcal{C}$ a simplicial category, and $\phi: \mathfrak{C}[S] \rightarrow \mathcal{C}^{op}$ a simplicial functor....

**4**

votes

**0**answers

267 views

### Quillen pair, fibrant-cofibrant objects

This question is a follow-up of the question I have asked today : left quillen functor and fibrant objects
Suppose that we have $$ L :C\leftrightarrow D: R$$
an adjoint Quillen pair. We assume that ...

**1**

vote

**1**answer

124 views

### left quillen functor and fibrant objects

Suppose that we have $$ L :C\leftrightarrow D: R$$
an adjoint Quillen pair. We assume that both model categories are combinatorial model categories.
Suppose that the functor $L$ (left adjoint) takes ...

**9**

votes

**0**answers

363 views

### Is the Thomason model structure on Cat simplicial? Is it a monoidal model category?

The Thomason model structure on the category of small categories is transferred from the Quillen model structure on simplicial sets along the right adjoint $Ex^2 \circ N$ (where $N$ is the nerve), i.e....

**4**

votes

**1**answer

107 views

### Ore localization and model structures

The question is this:
Suppose C is a category, with a given multiplicatively closed set of morphisms S ⊆ C. The role of the denominator conditions on S is rather similar to the role of a Quillen ...

**3**

votes

**1**answer

173 views

### derived functor that preserves weak equivalences

Suppose we have a functor $F:A\rightarrow B$ between model categories.
1- Assume that F takes weak equivalences to weak equivalences and cofibrations to cofibrations, can we define the derived ...

**32**

votes

**2**answers

3k views

### Why is Voevodsky's motivic homotopy theory 'the right' approach?

Morel and Voevoedsky developed what is now called motivic homotopy theory, which aims to apply techniques of algebraic topology to algebraic varieties and, more generally, to schemes. A simple way of ...

**3**

votes

**0**answers

181 views

### Can a functorial factorization be modified so that it fixes the initial object?

Consider a category $\mathcal C$ with a weak factorization system which is functorial. Let $*$ be an initial object. If $X\in \mathcal C$, denote by $\eta_X:*\to X$ the unique map. Using the given wfs,...

**2**

votes

**1**answer

160 views

### Localization of a model category with respect to a class of maps

I am little bit lost with the following (standard?) problem in model categories.
Suppose we have a Quillen adjunction between combinatorial model categories:
$$L:M\leftrightarrow N: R $$
and let $(...

**3**

votes

**1**answer

131 views

### Left lifting property and pushout

Let's say we are working in category $\mathcal{C}$, and that the three morphisms $ f: X \rightarrow X'$, $ g: Y \rightarrow Y'$ and $ h: Z \rightarrow Z'$ have the left lifting property with respect ...

**5**

votes

**1**answer

115 views

### Inductive folk model structure on strict ω-categories

There is a paper of Lafont, Metayer, and Worytkiewicz [1] that constructs a model structure on the category of strict $\omega$-categories that they call the folk model structure. This model structure ...

**7**

votes

**0**answers

108 views

### DG functors along which contractions can be lifted

For an object $X$ in a DG category, its contraction is $r \in Hom^{-1}(X,X)$ such that $d(r)=1_X$. Let us say that contractions lift along a DG functor $F: \mathcal{C} \to \mathcal{D}$, if for a ...

**1**

vote

**0**answers

40 views

### Maps having the right lifting property against cofibrations of compact spaces

I would like to know the properties of the maps that have the right lifting property against cofibrations of compact spaces. By definition, they are acyclic Serre fibrations, but I would hope to be ...

**2**

votes

**0**answers

41 views

### sub relative cell complex

This is a question concerning the proposition 11.4.8 of P.Hirschhorn's book Model categories and their localizations.
The proposition 11.4.8 is an analogous, in my opinion, to the well known ...

**7**

votes

**0**answers

144 views

### Is $\text{DGA}^{-}$ a monoidal model category?

Let $\text{DGA}^{-}$ denote the category of non-positively graded differential graded algebras with differentials of degree $+1$. It is well-known that $\text{DGA}^{-}$ has a model structure with
...

**2**

votes

**1**answer

66 views

### A monoidal model structure on pointed spaces

Do the classes of pointed Hurewicz cofibrations, pointed Hurewicz fibrations and pointed homotopy equivalences give a model structure on pointed (compactly generated weak Hausdorff) topological spaces ...

**6**

votes

**1**answer

172 views

### About a canonical model structure on topologically enriched categories

Consider the discrete combinatorial model structure on the category of $\Delta$-generated spaces: all maps are cofibrations and fibrations and the weak equivalences are the homeomorphisms. Applying ...

**6**

votes

**0**answers

154 views

### About a zig-zag of Quillen adjunctions

I have the following situation:
Three combinatorial model categories $\mathcal{K}_1, \mathcal{K}_2, \mathcal{K}_3$ such that all objects are fibrant (so no need to bother with the fibrant ...

**7**

votes

**0**answers

104 views

### Explicit data for $E_n$-monoidal model and simplicial categories

The definition of a monoidal model category requires that the tensor product is biclosed (such is needed to ensure that the tensor product is derivable in both variables). Obviously, in the situation ...

**4**

votes

**1**answer

197 views

### Simplicial model categories and simplicial equivalence

Let $M$ and $N$ two very nice simplicial model categories and let $F:N\rightarrow M$ be a (nice) simplicial functor which induces an equivalence of homotopy categories, i.e. $Ho(F): Ho(N)\rightarrow ...

**4**

votes

**1**answer

366 views

### detecting weak equivalences in a simplicial model category II

The question is related to the question: detecting weak equivalences in a simplicial model category
Suppose that we have a simplicial model category $M$ and denote by $M^{f}$ the full simplicial ...

**0**

votes

**1**answer

121 views

### detecting weak equivalences in a simplicial model category

Suppose that we have a simplicial model category $M$. The simplicial enrichment will be denoted by $map_{M}$. Let $f:A\rightarrow B$ be a morphism in the category $M$ such that $A$ is cofibrant. ...

**5**

votes

**1**answer

169 views

### Localization of a model category

Let $M$ be a very nice model category (cofibrantly generated, combinatorial or cellular and left proper simplicial model category). Let $f: X\rightarrow Y$ and $g: X\rightarrow Z $ be two morphisms ...

**3**

votes

**2**answers

114 views

### Excellent monoidal model categories admit enriched fibrant replacement functors?

Let $\mathbf{S}$ be an excellent model category in which all objects are cofibrant, viewed as an $\mathbf{S}$-enriched category by its canonical self-enrichment. Then we know that there is an obvious ...

**2**

votes

**0**answers

110 views

### Enriched homotopy colimit and space of paths

I work in the category of $\Delta$-generated spaces. Let $G$ be the group of strictly increasing continuous bijections from $[0,1]$ to itself (they necessarily take $0$ to $0$ and $1$ to $1$). I call ...

**7**

votes

**1**answer

220 views

### Proposition in HTT on cofibrations of categories

Proposition A.3.3.9. in Higher Topos Theory is as follows:
Let $S$ be an excellent model category and let $f:C\rightarrow C'$ be a cofibration of small $S$-enriched categories. Then (1) for every ...

**2**

votes

**1**answer

101 views

### Two model structures of some category

Let $\mathscr{C}$ be a category and $(W,C,F)=$(weak equivalence,cofibration,fibration) is some model structure of $\mathscr{C}$. Another model structure of $\mathscr{C}$, $(W_{S},C,F_{S})$ are called ...

**6**

votes

**0**answers

124 views

### Product-preserving fibrant replacement functor for the Joyal model structure

There are a few fibrant replacement functors for the Quillen model structure on simplicial sets that preserve finite cartesian products, namely $\operatorname{Ex}^\infty$ and $\operatorname{Sing}(|\...

**5**

votes

**0**answers

67 views

### When is a bisimplicial set diagonal fibrant

Let $sSet^2$ be the category of bisimplicial sets.
In the diagonal model structure on $sSet^2$ weak equivalences are diagonal weak equivalence (i.e.$ X \rightarrow Y$ is a weak equivalence if $dX \...

**3**

votes

**2**answers

291 views

### Quillen equivalence, fibrant objects

Suppose that $G:M\leftrightarrow N: U$ is a Quillen equivalence between two model categories. Suppose that $a\in M$ is a fibrant object, is it true that there exists always a fibrant object $b\in N$ ...

**4**

votes

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82 views

### Classification of combinatorial model categories presentable by simplicial presheaves on a Reedy category

Dan Dugger proved that every combinatorial model category can be obtained up to Quillen equivalence as the localization of a model structure on simplicial presheaves on a small category $C$.
Is there ...

**5**

votes

**1**answer

136 views

### Model category of diagrams with the colimit detecting the weak equivalences

Let $I$ be a small category and $\mathcal{K}$ be a combinatorial model category.
Is it known a model category structure on the functor category
$\mathcal{K}^I$ such that a map of diagrams $D\to ...

**3**

votes

**2**answers

183 views

### Best notation for fibrant/cofibrant replacement

In Quillen's original text on model categories (homotopical algebra) he uses $Q$ and $R$ to denote cofibrant and fibrant replacement respectively.
This notation has been used by several other ...

**15**

votes

**1**answer

440 views

### Homotopy theories of operads

I know of three homotopy theories of colored operads.
The (derived) localization category of Berger-Moerdijk's model structure on the category of strict simplicial (or topological) operads, with weak ...

**7**

votes

**1**answer

121 views

### Homotopy pushout independent of factorization and symmetric in cofibration category

$\require{AMScd}$In Algebraic homotopy, Baues defines the notion of homotopy pushout in a cofibration category in the following way: a commutative diagram
\begin{CD}
A @>k>> C \\
@AfAA @AAhA\...

**6**

votes

**0**answers

197 views

### Flat Model Structure on $\mathbf{Ch}(\mathbf{Mod}(\mathcal{O}_X))$ computes pullback / pushforward

Given a ringed space $(X,\mathcal{O})$ of can construct the flat model structure on chain complexes of $\mathcal{O}$-modules:
Weak equivalences are quasi-isomorphisms
The fibrations are epimorphisms ...

**7**

votes

**2**answers

279 views

### Are cofibrations accessible?

The category of fibrations in a combinatorial model category is accessible, accessibly embedded in the arrow category. How about the cofibrations?
More generally, let $C$ be a locally presentable ...

**3**

votes

**0**answers

145 views

### Comparing two simplicial resolution in a model category

Suppose that is a nice model category $\mathbf{M}$ (cofibrantly generated,...).
Suppose we have a two diagrams
$$F,G: \Delta^{op}\rightarrow \mathbf{M} $$
and a natural transformation $\nu: F\...

**10**

votes

**0**answers

176 views

### Is there an “injective version” of the Bergner model structure?

The Bergner model structure on $sCat$ (simplicially enriched categories) has a "projective" flavor: fibrations are "levelwise" while cofibrations satisfy stringent relative "freeness" conditions.
...

**6**

votes

**1**answer

208 views

### Homotopy limit of model categories in the category of categories

Say $$\mathcal{C'}\to \mathcal{C}\leftarrow \mathcal{D}$$ is a diagram of model categories and (e.g. Left) Quillen functors. I want to write down a (hopefully simple) model category $\mathcal{D}'$, or ...

**5**

votes

**0**answers

142 views

### Extending a functor up to homotopy

Consider a functor between small categories $i:\mathcal{A}\to \mathcal{B}$ which is faithful, not full, and bijective on objects. Consider a functor $F:\mathcal{A}\to \mathcal{M}$ where $\mathcal{M}$ ...

**3**

votes

**0**answers

54 views

### Calculating the intersection of the saturations of a decreasing sequence of morphisms

I have a sequence $W_0\supset W_1\supset \ldots$ of classes of morphisms in a presentable $(\infty,1)$-category $\mathcal{M}$. I can present this $(\infty,1)$-category as a model category, but I'm ...

**6**

votes

**1**answer

208 views

### How are simplicial sets with Quillen model structure a simplicial model category?

I got very lost in checking that simplicial sets with Quillen model structure are indeed a simplicial model category.
Recall that a model category $\mathcal{M}$ is simplicial if it is enriched in $\...