# 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.

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### Model categories with uniqueness

I've been learning about the construction of $(\infty,1)$-categories from simplicial sets, and more generally about the model category structure on simplicial sets, defined in terms of lifting ...

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### Cofibrancy of a right module over an operad

If I have a right module $M$ over an operad $\mathscr{O}$ in spaces, are there general methods to determine if $M$ is cofibrant with respect to the Reedy model structure? What if I know that my module ...

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### Is the category of topological operads left proper?

I just learned that there is a model structure on the category $Op_{Top}$ of topological operads, due to Berger-Moerdjik [1], obtained by right transfer of the Quillen model structure on $Top$.
Since $...

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### Enriched coends which preserve equivalences

Although this question might be formulated in higher generality, let me try to be concrete:
Let $(\mathbf{Top},\times,*)$ be the monoidal category of compactly generated weak Hausdorff spaces; and let ...

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### Left anodyne is covariant equivalence

I have a question about the proof of left anodyne between two simplicial sets over $S$, where $S$ is a simplicial set, is covariant equivalence. In the proof (HTT 2.1.4.6 or https://ncatlab.org/nlab/...

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### Deformation of a diagram preserve the homotopy limit

I have been a bit sloppy in the title, but let me be specific. I stepped again into the subtle difference between homotopy limit and limit in the homotopy category, in the following version.
Suppose ...

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### Does the homotopy category of a model category detect weak equivalences?

Let $\mathcal{C}$ be a model category. Suppose that a morphism $f:x\to y$ in $\mathcal{C}$ induces an isomorphism in the homotopy category $\textbf{Ho}(\mathcal{C})$. Is it necessarily true that $f$ ...

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### Identifying discrete points in derived hom spaces

Let M be a model category presenting an ∞-category $\mathcal{M}$, and let $f : X \to Y$ and $g : Y \to Z$ be arrows of M. Consider the following propositions:
The connected component of $f$ in $\...

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### Is the canonical model structure on strict $\infty$-Cat left proper?

Is the canonical (or Folk) model structure on the category of (strict) $\infty$-categories as constructed by Lafont, Métayer and Worytkiewicz in A folk model structure on omega-cat left proper ?
All ...

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### Checking that (hyper) sheafification is fibrant in local projective model structure on simplicial presheaves

Context and Notation
Let $X$ be a manifold and $\mathcal{C} = Op(X)$ be the category of open subsets of $X$ with inclusions. I then consider the projective model structure on the (simplicial model) ...

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### Cellularization functor and cohomological dimension

I'm little bit confused by the following problem. And I was hoping for some help.
Here is the set up: I have an associative ring $R$. Let $M$ and $N$ two $R$-modules such that
$F_{n}\rightarrow F_{n-1}...

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### Is an open subset of a cofibration a cofibration?

Suppose $A \to X$ is a cofibration in topological spaces, and $U \subseteq X$ is an open subset. Is $U \cap A \to U$ a cofibration?
Sorry if this is rather simple, but I don't have much experience ...

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### How to learn about Higher Topoi

I have been learning quasicategory theory in an attempt to understand higher topoi, and I have been trying to look at as many different sources as possible. Higher Topos Theory provides a higher ...

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### Two $\infty$-categories of chain complexes

In the literature, I've mostly seen two quasicategories coming from $\text{Ch}_R$:
By considering $\text{Ch}_R$ with weak equivalences $\mathcal W = \text{quasi-isomorphisms}$, we can consider its ...

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### Homotopy coherent colimits in chain complexes

In remark 1.2.6.2 (HTT), Lurie states that
Another possible approach to the problem of homotopy
coherence is to restrict our attention to simplicial (or topological) categories
C in which every ...

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### The derived $\infty$-category of sheaves on a site is closed symmetric monoidal

Let $X$ be a quasicompact semiseparated scheme. I am trying to recover the (closed) symmetric monoidal structure on $\mathcal{D}(\mathrm{QCoh}(X))$, the derived $\infty$-category of quasicoherent ...

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### Structure sheaf of derived intersection

Everything is over a field $k$ of characteristic $0$.
Let $X$ and $Y$ two closed dg subschemes over a dg scheme $Z$. I am trying to understand the structure sheaf of the derived intersection of $X$ ...

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### Understanding fusion categories from the perspective of anyons?

ncatlab defines a fusion category as "A fusion category is a rigid semisimple linear (Vect-enriched) monoidal category (“tensor category”), with only finitely many isomorphism classes of simple ...

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### Are dg-modules over a cofibrant dg-category cofibrant?

Fix a commutative ring $k;$ all dg-categories will be dg-categories over $k.$ Throughout the question, I will be following the notation and conventions of Toën's "The homotopy theory of dg-...

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### Why is this condition necessary for the existence of a transferred simplicial model structure?

In chapter II of Goerss and Jardine's text on simplicial homotopy theory, they give a general theorem, Theorem 6.8, for transfer of simplicial model structures across a simplicial adjunction. This ...

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### Inducing a model structure using a cosimplicial object

In a recent paper, Hiroshi Kihara induced a model structure on the category of diffeological spaces. He generates the classes of fibrations, cofibrations, and weak equivalences by constructing a ...

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### Are cofibrations in topological monoids preserved by forming the product with the identity?

Consider the category $\mathrm{Mon}(\mathbf{Top})$ of topological monoids, together with the model structure transferred along the adjunction $F:\mathbf{Top}\rightleftarrows \mathrm{Mon}(\mathbf{Top}):...

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### Are fibered categories fibrant objects in some model structure on Cat/C?

Given a category $\mathcal{C}$, by a category over $\mathcal{C}$, I mean a category $\mathcal{D}$ along with a functor $\pi_{\mathcal{D}}:\mathcal{D}\rightarrow \mathcal{C}$.
Consider the category $...

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### On how Simpson's model structure on Tamsamani $n$-prenerves is cofibrantly generated

I was reading through "A closed model structure for $n$-categories, internal $Hom$, $n$-stacks and generalized Seifert Van-Kampen" by Simpson and was struggling to piece together the ...

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### does geometric realization factor through an endofunctor?

Does the functor of geometric realization of a simplicial set as a topological space,
factor through an endofunctor of the category of simplicial topological spaces which does something non-trivial (...

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### Boardman-Vogt construction for PROP(erads)

Let $\left\lbrace \mathsf{O}(n)\right\rbrace_{n\in \mathbb{N}} $ be an operad in a symmetric monoidal category $(\mathsf{C},\otimes, \mathbf{1})$ which in addition has the structure of a model ...

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### Trees in chain complexes

$\DeclareMathOperator{\Ch}{\mathit{Ch}}$Let $\Ch_\mathbb{Q}$ denote the model category of chain complexes over rational numbers. Let $T_\ast$ be a tree in $\Ch_{\mathbb{Q}}$ with $n$ vertices.
How to ...

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### Obstructions to presenting $\infty$-categories by Reedy categories

We know that any locally presentable $( \infty , 1 )$-category admits a presentation by a Cisinski model category. However, I'm not clear on what obstructions there are to chosing a category $A$ to ...

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### Lax monoidal fibrant replacement for marked simplicial sets

The category $\mathrm{Set}_{\Delta}^{+}$ of marked simplicial sets has a model structure (the (co)cartesian model structure) constructed by Lurie in HTT.3.1.3. I would like to know if this model ...

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### Proof of existence of Joyal model structure via Cisinski theory?

I'm looking for a proof of the existence of the Joyal model structure -- with its usual description -- which uses Cisinski theory directly. The closest thing I know of is Theorem 5.26 of Ara's Higher ...

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### About the dual of the cube lemma in homotopy theory

Consider the Reedy category $2\rightarrow 1 \leftarrow 0$. Consider a map of diagrams of topological spaces $D\to E$ over this Reedy category:
The maps which are fibrations are depicted with the ...

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### Schwänzl and Vogt, Cofibration and fibration structures in enriched categories

In [Schwänzl and Vogt, Strong cofibrations and fibrations in enriched categories], the authors refer to an earlier preprint, [Schwänzl and Vogt, Cofibration and fibration structures in enriched ...

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### Constructing a model structure without knowing the class of weak equivalences

I need to prove the existence of a model structure but I am still unable to formulate a definition of the class of weak equivalences. I have the following informations:
The underlying category is ...

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### Contractibility of the category of cosimplicial resolutions

Let $\gamma : \mathcal{C} \to \mathcal{M}$ be a functor and define a cosimplicial resoultion of $\gamma$ as a functor $\Gamma: \mathcal{C} \to \mathcal{M}^{\Delta}$ such that
$\Gamma C$ is Reedy ...

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### monoidal (∞,1)-categories from weakly monoidal model categories

In Higher Algebra section 4.1.7, Jacob Lurie proves that the underlying $(\infty,1)$-category of a monoidal model category is a monoidal $(\infty,1)$-category.
Dominic Verity and Yuki Maehara have (...

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### Homotopy descent and cohomology

I've been reading "Structured Brown representability via concordance" by D.Pavlov (https://dmitripavlov.org/concordance.pdf) and I'm strugeling with a point and was wondering if someone ...

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### Whitehead Theorem in $\mathbb{A}^1$-homotopy theory

I'm reading "UNSTABLE MOTIVIC HOMOTOPY THEORY" by Kirsten Wickelgren and Ben Williams (https://arxiv.org/pdf/1902.08857.pdf). There they have a version of Whitehead's Theorem, namely Prop ...

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### When is an increasing union a colimit?

Let's consider a diagram $\Phi: \lambda \to \mathcal{T}_*$
$$
X_0 \to X_1 \to \cdots \to X_\xi \to X_{\xi+1} \to \cdots
$$
of pointed spaces,
indexed by some ordinal $\lambda$, in which each $X_\xi$ ...

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### If $A$ is a cofibrant commutative dg-algebra over a commutative ring of characteristic $0$, then its underlying chain complex is cofibrant

Let $R$ be a commutative ring with characteristic $0$, namely it contains the field of rational numbers. Higher Algebra Proposition 7.1.4.10 tells that the category of commutative $R$-dg-algebras $\...

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### Homotopy Kan extensions, formally coherent functors and derived Schlessinger criterion

Let $k$ be a finite field. Denote by $discArt_k$ the category of Artinian rings with residue field $k$ and $Art_k$ the category of Artinian simplicial rings. Consider a functor $\mathcal{F}:disArt_k\...

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### Proper model category for “categories with finite limits”

I'm looking for a Quillen model category which model the $2$-category of 'small category with finite limits (and functors between them preserving finite limits)':
Left proper,
right proper,
Enriched ...

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### Preserves naively $\mathbb{A}^{1}$-homotopic maps

I've been studying $\mathbb{A}^{1}$-homotopy recently and would like some guidance with the question below. Thank you so much.
Setup
Fix $k$ a field of characteristic zero. Let $Sm_{k}$ denote the ...

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### Almost combinatorial accessible model categories

Theorem: Assume VP. Let $\mathcal{M}$ be an accessible model category
such that there exists a set of generating cofibrations $I$ and such
that all objects are fibrant. Then it is combinatorial.
...

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### Modelling rational spaces of finite $\mathbb{Q}$-type with spaces with finite simplices in every simplicial dimension

EDIT 2
Original question below. I will award the outstanding bounty for an answer to the following question (question (2) in the OP).
Let $X$ be a Kan complex which is connected, nilpotent, and of ...

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### Do limits in Waldhausen categories commute with ordinary limits?

Disclaimer : I asked this question on MSE, I have no answer and I think it's better to ask it here.
Let $(A,\mathcal{W}, \mathcal{C})$ be a Waldhausen category with $A$ an additive category.
On one ...

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### Why does this construction give a (homotopy-invariant) suspension (resp. homotopy cofiber) in an arbitrary pointed model category?

In their text Foundations of Stable Homotopy Theory, Barnes and Roitzheim define the suspension of a cofibrant object X of a pointed model category to be the pushout of the diagram $*\leftarrow X\...

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### How do you prove that the category of weak equivalences of sSet is accessible?

I am trying to prove that the category of simplicial sets is a combinatorial model category by using Proposition A.2.6.15 of Lurie's book, and this requires proving that the weak equivalences are ...

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### Explaining the “free left fibration” functor for infinity categories

This is a cross-post from here
I am reading A. Mazel-Gee's paper "All about the Grothendieck construction". In that paper he explains that the left adjoint ${\mathrm{Cat}_{\infty}}_{/\mathcal{C}}\to \...

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### Applications of model categories

I was wondering if someone could explain some of the concrete applications of model categories. My possibly naive understanding of the motivation is that one wants to mimic the category of ...

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### Why does this construction give a weak factorization system in the category of span diagrams?

In Dwyer and Spalinski's classic paper Homotopy Theories and Model Categories, they describe homotopy pushouts by defining a model structure on the category of span diagrams in a given model category $...