# 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:
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### 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
...

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### 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., ...

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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 ...

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### 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&...

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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 ...

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### 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{...

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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 ...

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### 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 ...

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### 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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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 $W = \{ \text{weak homotopy equivalences} \}$, ...

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### 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 ...

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### 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\...

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### 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....

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### 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 $...

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### 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 ...

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### 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 $\mathcal{W}_{\mathsf{E}},$ ...

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### 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 $\infty$-...

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### 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 ...

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### 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{...

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### 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 ...

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### 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 ...

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### 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 $f$ ...

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### 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 $\mathcal{V}$-...

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### 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 $k$-...

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### 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 "...

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### $n$-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 ...

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### 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 ...

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### 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 ...

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### 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?

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### 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 \...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\mathcal{C}$ ...

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### 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, ...

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### 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?
...

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### 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 ...

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### 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 ...

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### 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"...

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### 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 ...

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### 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}_\...

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### 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,...

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### 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 ...

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### 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 ...

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### 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 $M$...

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### 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 ...