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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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Reference for Reedy weak factorization systems, not Reedy model structure?

What paper/book is appropriate as the standard reference for Reedy weak factorization systems, not Reedy model structure? Specifically, I would like a reference for the following Propositions 1 and 2: ...
gksato's user avatar
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2 votes
0 answers
66 views

Stable homotopy of vector bundles

Consider the category $\mathsf{VectBun}$ of real vector bundles over topological spaces, where the morphisms are bundle maps that are fiberwise isomorphisms. This category has a stabilization functor ...
Derived Cats's user avatar
5 votes
2 answers
305 views

Differing monoidal model structures on a fixed model category

Suppose that $\mathcal{C}$ is a model category (with a fixed model structure). Are there any known examples where $\mathcal{C}$ is a (symmetric) monoidal model category in two different ways? I.e., ...
JD1874's user avatar
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1 vote
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Is the mapping space from $\Delta[k]$ to a simplicial set $X$ weak equivalent to $X$?

Let $X$ be a simplicial set. It is well known that a model for the path object can be given by the mapping space $\mathrm{Hom}(\Delta[1], X)$. In particular this offers a fiber replacement for the ...
SetR's user avatar
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3 votes
1 answer
140 views

Homotopy coherent transformation and totalization

Let $C$ be the category of chain complexes over a field $F$ and $C^\prime$ be the subcategory of chain complexes with zero differentials. If $X:I\to C$ is a functor, there is an induced "homology&...
vap's user avatar
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6 votes
1 answer
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Fibrations in a model structure for homotopy $n$-types of simplicial sets

Consider the model structure on simplicial sets where the cofibrations are given by monomorphisms and the weak equivalences are given by $n$-equivalences (that is, maps $f \colon X \to Y$ that induce ...
Udit Mavinkurve's user avatar
5 votes
3 answers
382 views

Bar construction in commutative algebras is calculated by pushout

$\DeclareMathOperator\colim{colim}$ Also asked in MathStackexchange here This is a statement in Lurie's Higher Algebra 5.2.2.4. Proposition 3.2.4.7 in HA said that the monoidal structure on $\text{...
Xiong Jiangnan's user avatar
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0 answers
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classifications of all weak factorisation systems on a category [duplicate]

Is there an example of a category where all the weak factorisation systems have been classified ? Is this something that people tried to classify ? This can be done trivially for Sets (see the ...
user524793's user avatar
7 votes
1 answer
163 views

Reference request for equivalences between different models of lax limits

There are several models for lax limits of model categories/ $\infty$-categories in the literature. For example, within the realm of $\infty$-categories one can construct them using coCartesian ...
happymath's user avatar
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8 votes
1 answer
425 views

Model categories as a tool to resolve size issues for localizing categories

I have a rather basic question about one motivation for introducing model categories in David White's notes, as a possible way to overcome troubles with size issues appearing when localizing a ...
user267839's user avatar
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5 votes
1 answer
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Minimal cell structures in combinatorial model categories

I recently rediscovered a classical theorem from Hatcher which states that simply-connected CW complexes have a 'minimal' cell structure, where the cells correspond to spheres and disks indexed by the ...
kelly maggs's user avatar
2 votes
2 answers
272 views

Is the mapping cylinder a replacement for morphism by cofibration in model categories?

Let $M$ be a model category, consider a very good cylinder object $X \coprod X \to X \times I \overset{\operatorname{pr}}{\to} X$ (here $X \times I$ is just a notation, no object $I$ is implied), that ...
Arshak Aivazian's user avatar
5 votes
0 answers
80 views

Tensor product of modules in model vs. infinity categories

Let $C$ be a combinatorial symmetric monoidal model category and let $A$ be a associative algebra object in $C$, that is cofibrant as an object in $C$. In Higher Algebra 4.3.3.17, Lurie proves an ...
Jakob's user avatar
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6 votes
1 answer
198 views

Weakly contractible $X$, but none of the maps $*\to X$ are cofibrations

Let $\mathrm{Top}$ be the category of all topological spaces and continuous maps. The Quillen model structure on $\mathrm{Top}$ has weak equvalences $W = \{ \text{weak homotopy equivalences} \}$, ...
mathmo's user avatar
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13 votes
2 answers
990 views

Categories on which one can determine all model structures?

Famously, there are exactly nine model structures on the category of sets, which are detailed here. In this case, one can exhaustively determine all six weak factorization systems and then see which ...
Tim Campion's user avatar
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14 votes
3 answers
608 views

Strøm model structures on the category of simplicial sets

Let $X,Y$ be simplicial sets. A simplicial homotopy is a simplicial map of the form $h:X\times\Delta^1\rightarrow Y$. There are two distinguished maps $$ in_0:X\cong X\times\Delta^0\xrightarrow{1\...
Tyrone's user avatar
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3 votes
0 answers
81 views

Are fibrations of small categories fibrations?

The isofibrations are the fibrations of the canonical model structure of the category of small categories. If I call fibration of small categories the same notion by removing the word isomorphism, i.e....
Philippe Gaucher's user avatar
2 votes
0 answers
129 views

Tensor product of objectwise weak homotopy equivalences of $\mathcal{M}$-spaces

I consider the enriched category $[\mathcal{M}^{op},\mathrm{Top}]$ of enriched functors (I call them $\mathcal{M}$-spaces) from the enriched small category $\mathcal{M}^{op}$ to the enriched category $...
Philippe Gaucher's user avatar
9 votes
1 answer
453 views

Is there a shape-independent definition of (∞,1)-categories?

For all definitions of $\infty$-categories I am aware of, an $(\infty,1)$-category is defined via reference to some shape, be it simplices in a form of a quasi-category or a cubical analogue of a ...
lemmanade's user avatar
9 votes
1 answer
211 views

Does $\infty$-categorical localization commute with taking directed fibered products?

Suppose we are given categories $\mathsf{C},\mathsf{D},\mathsf{E},$ equipped with collections of weak equivalences $\mathcal{W}_{\mathsf{C}},\mathcal{W}_{\mathsf{D}},$ and $\mathcal{W}_{\mathsf{E}},$ ...
Stahl's user avatar
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4 votes
1 answer
194 views

Model categories: "equivalence" of finite limits and finite colimits

I am needing a reference for the following statement (in case it is true): Quillen functor between stable model categories preserve finite limits iff it preserves finite colimits. For stable $\infty$-...
Alexey Do's user avatar
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9 votes
0 answers
280 views

How acyclic models led to idea of model categories

The Wikipedia article about Acyclic models notices that the way that they were used in the proof of the Eilenberg–Zilber theorem laid the foundation stone to the idea of the model category. Could ...
user267839's user avatar
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12 votes
1 answer
431 views

Is the Grothendieck construction a homotopy pullback?

The category of elements of a functor $F:\mathcal C\to\mathsf{Set}$ can be obtained as the strict pullback in with the forgetful functor of pointed sets $\mathsf{Set_*}\to\mathsf{Set}$: $$ \begin{...
Daniel Teixeira's user avatar
2 votes
1 answer
120 views

Is a left Bousfield localization of simplicial presheaves a locally cartesian closed model category?

Let $\mathcal{C}$ be a small category and let $\mathcal{M} = \operatorname{sPre}(C)$ be the model category of simplicial presheaves on $\mathcal{C}$ with the injective model structure. Let $S$ be a ...
Herman Rohrbach's user avatar
2 votes
0 answers
129 views

Geometric conditions on motivic fibrations

What are the geometric conditions on a map of varieties/schemes being a motivic fibration (i.e. a fibration in the motivic model structure on simplicial presheaves on affine schemes)? For example, are ...
Grisha Taroyan's user avatar
3 votes
1 answer
195 views

A fiber-like method to show equivalence of infinity categories

Suppose I have a functor of quasi-categories $f: \mathcal{C} \to \mathcal{D}$. I want to show a criterion like: "$f$ is an equivalence of $\infty$-categories if the homotopy fiber of $f$ ...
Andrea Marino's user avatar
3 votes
1 answer
182 views

Enriched cofibrant replacement in spectrally enriched categories

If $\mathcal{V}$ is a monoidal model category with all objects cofibrant, Theorem 13.5.2. of Categorical Homotopy Theory will guarantee that the functorial cofibrant replacement of a $\mathcal{V}$-...
Connor Malin's user avatar
  • 5,511
2 votes
1 answer
147 views

Left Proper model structure on the category of non-symmetric operads in chain complexes

It is shown in Moriya (Multiplicative formality of operads and Sinha’s spectral sequence for long knots, 2.1) that there exists a left proper model category structure on non-symmetric operads over $k$-...
Andrea Marino's user avatar
7 votes
1 answer
469 views

Different definitions of homotopy colimits

I was reading about the definition of homotopy limits and colimits, and I have seen two different approaches in "Homotopy theories and model categories" by Dwyer and Spalinski, and in "...
MikeTrooper's user avatar
3 votes
1 answer
121 views

$n$-truncation of a Simplicial Model Category

I'm working in the category of rational $CDGAs$ and trying to find a reference/construction of a natural $2$-categorical structure via truncation of the mapping spaces. In my head, the key point is ...
kelly maggs's user avatar
6 votes
2 answers
380 views

Size issues in localization $\mathcal{C}[\mathcal{W}^{-1}]$ category

When one starts with a locally small category $\mathcal{C}$ and wants to localize it at an appropriate choosen collection of morphisms $\mathcal{W}$, then in general one faces some size issues in the ...
user267839's user avatar
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0 votes
0 answers
113 views

Locally constant (homotopy) pre-factorization algebras

In my thesis, I'm using the theory of (homotopy) factorization algebras and particularly locally constant ones. While reading an article that I can't find again I read that already a locally constant ...
Alessandro Nanto's user avatar
0 votes
2 answers
277 views

Is the identity a cofibration?

In a closed model category, is the identity $\textrm{id}: A \to A$ a cofibration? Does it only hold on some special cases? Or is it never true?
groupoid's user avatar
  • 215
2 votes
0 answers
94 views

Projective model categories on homotopy equivalent index categories

Consider a fixed proper simplicial combinatorial model category $\mathcal{M}$. Consider a functor $F:I\to J$ between small categories. It induces a right Quillen functor $F^*:\mathcal{M}^J \to \...
Philippe Gaucher's user avatar
0 votes
0 answers
50 views

Finitely continuous fibrant replacement functor for localization of simplicial presheaves with projective model structure

Let $C$ be a model category given by generators and relations in the sense of Dugger (that is, $C$ is a left Bousfield localization of a global projective model model structure on simplicial ...
Arshak Aivazian's user avatar
3 votes
0 answers
161 views

Direct images commute with homotopy colimits

In Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (II), Ayoub defined the notion of a stable homotopical algebraic derivators; roughly, for a ...
Alexey Do's user avatar
  • 813
4 votes
0 answers
101 views

Dugger's theorem for enriched model categories

We know that a combinatorial model category has a small presentation. Is an enriched version of this theorem known? The closest I could find is: Guillou, May - Enriched model categories and presheaf ...
Arshak Aivazian's user avatar
3 votes
1 answer
106 views

Injective model structure for simplicial presheaves

I am reading the paper by Jardine and Goerss, Localization theories for simplicial presheaves and having troubles with understand an argument. In this paper, the two authors considered $\mathcal{C}$ ...
Alexey Do's user avatar
  • 813
2 votes
1 answer
398 views

Why do we need enriched model categories?

As far as I understand, model categories mainly provide tools for studying the "homotopy theories" (that is, $\infty$-categories) that are ubiquitous in mathematics. From this point of view, ...
Arshak Aivazian's user avatar
1 vote
0 answers
251 views

Homotopy coherent nerve for algebraic model categories

Is a homotopy coherent nerve defined for algebraic model category that returns algebraic quasi-categories as Urs Schreiber wrote about? Or do we not know how to determine it / does it seem impossible? ...
Arshak Aivazian's user avatar
6 votes
1 answer
235 views

Left Bousfield localisation of trivial model structures

Let $\mathcal{M}$ be a category, let $\mathcal{M}^\circ$ be a reflective (full) subcategory of $\mathcal{M}$, and let $L : \mathcal{M} \to \mathcal{M}^\circ$ be the reflector. Question. Does there ...
Zhen Lin's user avatar
  • 15k
1 vote
0 answers
93 views

Chain complexes indexed over measurable subsets of $\mathbb{R}$: Towards a measurable notion of Euler Characteristic

I have for a while tried to generalize the notion of a chain complex in a way to obtain a "continuous" or at least "measurable" notion of Euler Characteristic. I have come up with ...
The Thin Whistler's user avatar
0 votes
1 answer
135 views

Examples of cartesian-closed model categories

One of the main settings of my research are Cartesian-closed model categories. I would like to know as many interesting and/or important examples of such categories as possible. "Interesting"...
Arshak Aivazian's user avatar
2 votes
1 answer
101 views

DK equivalences are Reedy equivalences for complete Segal spaces

$\require{AMScd}$ Dear all, I have a question concerning Charles Rezk's paper "A model for the homotopy theory of homotopy theory ", precisely Proposition 7.6 in this paper. It is proven ...
Igor Sikora's user avatar
  • 1,759
3 votes
1 answer
228 views

Monoidal structure on simplical model category of chain complexes

For $k$ a field (the case I am interested in, but the question makes sense over any dga), $\mathrm{Ch}_\bullet(k)$ its projective model category of unbounded chain complexes (here), $\mathrm{sCh}_\...
Urs Schreiber's user avatar
3 votes
0 answers
99 views

Homotopy theory for small strict semimonoidal topologically enriched categories

I work with the category of $\Delta$-generated spaces. I call reparametrization category a small strict semimonoidal topologically enriched category $(\mathcal{P},\otimes)$ such that $\mathcal{P}(\ell,...
Philippe Gaucher's user avatar
2 votes
0 answers
104 views

homotopy theory for simplicial profinite topological spaces, and the décalage shift endofunctor

Is there a homotopy theory for simplicial profinite sets where the notion of contractibility can be defined in terms of the decalage endomorphism ? Specifically, I need a homotopy theory where notion ...
user494312's user avatar
7 votes
0 answers
149 views

Presenting geometric morphisms by geometric morphisms

It's known that any $\infty$-topos $\mathcal{E}$ can be presented by a Quillen model category $\mathbf{E}$ that is itself a 1-topos. For instance, if $\mathcal{E}$ is a left exact localization of a ...
Mike Shulman's user avatar
  • 65.6k
3 votes
0 answers
57 views

Right transferred model structure on the category of algebras in the Grothendieck topos

Let M be a model category that is a Grothendieck topos. Let $T$ be a multisorted finitary algebraic theory. Does there exist the right transferred model structure on the category of $T$-algebras in $M$...
Arshak Aivazian's user avatar
4 votes
1 answer
104 views

Simplicial enrichment on unbounded algebras over an operad

In his paper "Homological Algebra of Homotopy Algebras" V.Hinich introduced a simplicial structure on algebras in unbounded chain complexes over arbitrary $\Sigma$-split operads. Not to get ...
Grisha Taroyan's user avatar

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