# 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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### A category with two notions of weak equivalence?

I have a category with two classes of weak equivalences (neither class is contained in the other, and I believe they actually both form homotopical categories). I'm wondering if this notion exists in ...

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### Fibrant replacement of an injective model category of enriched diagrams

Take a topologically enriched small category $\mathcal{P}$ and the category of enriched diagrams of spaces $[\mathcal{P},\mathrm{Top}]_0$. We work with the category of $\Delta$-generated spaces ...

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### What is the right notion of a functor from an internal topological category to a topologically enriched category?

Let $\mathcal{C}$ be a category internal to (some convenient model for) topological spaces (which I will denote by $\mathsf{Top}$). In the question Greg Arone asks:
What is the correct notion of a ...

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### Signs in dg Yoneda embedding: proof of existence of Dwyer-Kan model structure on $\mathit{dgcat}$

I'm studying a proof of the fact that the category of dg-categories admits a (Dwyer-Kan) model structure. As references, I'm using Pieter Belmans' master thesis and Goncalo Tabuada's paper Une ...

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### fibrations lifting with respect to all closed inclusions

Why can't we require in the definition of a trivial fibration that the map has the right lifting property with respect to each inclusion of a closed subset into a (hereditary) normal space, or perhaps ...

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### Is $\{M\to\Lambda\}^\text{lr}$ the class of trivial fibrations?

I am interested in the weak left-right orthogonal $\{M\to \Lambda\}^\text{lr}$ of a particular map of finite topological spaces with 5 and 3 points, depicted below, in fact it is the map of preorders ...

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### Model structure on dg-algebras over an "equivariant fundamental category"?

For purposes of $G$-equivariant rational homotopy theory one wants a Quillen adjunction which generalizes the classical one of Bousfield-Gugenheim from plain dg-algebras/simplicial-sets to (co-)...

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### Is the Reedy model structure cofibrantly generated?

I am reading about the Reedy model structure from Hovey's book and
I was wondering if the Reedy model structure on $\mathcal{M}^{\Delta}$
is cofibrantly generated by a small set of arrows and permits ...

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### Simplicial spaces and reflexive coequalisers

Let $X_\bullet$ be a simplicial space. Consider the reflexive coequaliser of $X_1\rightrightarrows X_0$, which we call $X$. Then we clearly have a map $\varphi\colon |X_\bullet|\to X$, where $|X_\...

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### The localization of the span category

Suppose one has a model category $C$ with its class of weak equivalences $W$. It is possible to form a separate homotopical category $(C_{\operatorname{span}},W_{\operatorname{span}})$ which has ...

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### example of "really" non-existent transferred model structure

I am looking for an example where a transferred model structure fails to exist, even if one is willing to work with semi-model category. But let me be more precise:
Let's say I have a combinatorial ...

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### When this coend is invariant up to homotopy?

It is a follow-up of my question Calculation of the homotopy colimit of a diagram of spaces which was badly formulated.
Consider a fixed diagram $D:I^{op}\to {\rm Top}$ where ${\rm Top}$ is
a ...

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### Calculation of the homotopy colimit of a diagram of spaces

Consider a small category $I$. There exists a small diagram $D:I^{op}\to {\rm Top}$ where ${\rm Top}$ is a convenient category for doing algebraic topology such that for all small diagrams $X:I\to {\...

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### On diagrams in model categories and rectification

For a model category $C$, I'm denoting $h_\infty(C)$ the associated $\infty$-category (for example its Dwyer-Kan localization at weak equivalences, or if $C$ is simplicial the simplicial nerve of the ...

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### On the derived functor of the tensor product in a monoidal category

Let $(\mathcal{M},\otimes)$ be a symmetric monoidal model category; I'll assume for simplicity that every object is fibrant. Suppose that the unit $I$ is NOT cofibrant. I'm interested in whether the ...

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### Fibrations of fibrant marked simplicial sets

Let $\mathrm{sSet}^+ = \mathrm{sSet}^+_{/ \Delta^0}$ be the model category of marked simplicial sets over the point. By Theorem 3.1.5.1 in Higher Topos Theory, this model category is Quillen ...

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### Does derived hom commute with homotopy limits?

Suppose that $\mathcal{V}$ is a symmetric monoidal model category, and that $\mathcal{C}$ is a $\mathcal{V}$-enriched model category. Write $\Bbb{R}\!\operatorname{Hom}(-,-)$ for the derived Hom ...

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### When is a right lifting property closed under pushouts?

A class of morphisms defined by a right Quillen lifting property (weak orthogonality)
is always closed under pullbacks (limits); under what assumptions will it be closed under pushouts (colimits)? In ...

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### A model structure on CDGAs

Assume our base ring $k=\mathbb{Z}_{(p)}$ or $\mathbb{F}_p$ and
for simplicity
let us consider non-negative-graded connected algebras $A$ with an augmentation, i.e. $A^0=k,A^{<0}=0$.
The question: ...

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### Is hammock localization a localization in the sense of Lurie?

In a series of papers ([1], [2] and [3]), Dwyer and Kan introduced the hammock localization [2] as an effective technique to compute the simplicial localization of a model category [1]. This is meant ...

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### Non-enriched Bousfield localizations

We know that whenever we have a Bousfield localization between two simplicial model categories, this gives rise to a reflective subcategory in $\infty$-categories (or coreflective, depending on the ...

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### Does the functor of Chevalley–Eilenberg cochains $ CE^\bullet:L_\infty\mathbf{Alg}^{op}\to \mathbf{dgAlg} $ map homotopy limits to homotopy colimits?

I was wondering whether the functor of Chevalley–Eilenberg cochains
$$
\operatorname{CE}^\bullet:L_\infty\mathbf{Alg}^\text{op}\to \mathbf{dgAlg}
$$
maps homotopy limits to homotopy colimits. Is still ...

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### Does the monoidal structure on semisimplicial sets preserve fibrant objects?

The category of semisimplicial sets has the structure of a monoidal category by the geometric product $\otimes$, see for example Rourke and Sanderson's paper '$\Delta$-sets I: Homotopy Theory'. This ...

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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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### 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 ...