# 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.

686
questions

2
votes

0
answers

81
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 \...

1
vote

0
answers

38
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 ...

2
votes

0
answers

85
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 ...

4
votes

0
answers

80
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 ...

3
votes

1
answer

81
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 $\mathcal{C}$ ...

2
votes

1
answer

278
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, ...

2
votes

1
answer

167
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?
...

6
votes

2
answers

215
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 ...

1
vote

0
answers

84
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 ...

0
votes

1
answer

112
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"...

2
votes

1
answer

84
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 ...

3
votes

1
answer

132
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}_\...

1
vote

0
answers

47
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 $\mathcal{P}$ such that $\mathcal{P}(\ell,\ell')$ ...

2
votes

0
answers

85
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 ...

6
votes

0
answers

112
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 ...

3
votes

0
answers

52
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 $M$...

3
votes

0
answers

71
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 ...

4
votes

0
answers

54
views

### weak factorization systems (co)generated by an arbitrary class of morphisms

Under what assumptions can one prove existence of weak factorization systems (co)generated by a class of morphisms ?
Are there counterexamples ? I am interested both in assumptions on the class of ...

2
votes

0
answers

63
views

### "Relative cell complexes in the undercategory $B/\mathscr{M}$ are relative cell complexes in $\mathscr{M}$": why do we need $B$ to be a cell complex? [closed]

$\newcommand{\M}{\mathscr{M}}\newcommand{\I}{\mathcal{I}}\newcommand{\J}{\mathcal{J}}$The context: the following is claimed in J. May's "more concise algebraic topology".
We have some model ...

3
votes

0
answers

112
views

### What should be required from a model category that the category of algebraic objects in it has the natural model structure?

I have two reference questions
What should be required of a category with finite products so that a (multi-sorted, finitary) Lawvere theory to induce monadic adjunction in it? This should be ...

5
votes

0
answers

106
views

### Kähler differentials give a left Quillen functor

Is there a reference for the fact that the functor of Kähler differentials is a left Quillen functor on the category of $\mathrm{CDGA}_k/B$? Here $k$ is a field of characteristic $0$, and $B$ is some ...

10
votes

0
answers

146
views

### Rectifying diagrams of $\infty$-categories

If $C$ is a 1-category and $D$ is a locally presentable $(\infty,1)$-category presented by a combinatorial model (1-)category $M$, then any $(\infty,1)$-functor $C\to D$ can be represented by a strict ...

3
votes

0
answers

46
views

### Recognising absolute distributors in terms of simplicial model categories

Briefly, my question is the following:
Can we recognise when a simplicial model category $\def\cM{\mathcal M}\cM$ is an absolute distributor, using only the language of (simplicial) model categories?
...

8
votes

1
answer

200
views

### When a model category with prescribed homotopy category exists?

My question is probably insanely hard or very well-studied, but I could not find an answer, so I will ask it here.
Assume that we have a suitably complete closed module over $\operatorname{Ho}(sSet).$ ...

6
votes

1
answer

211
views

### Inexistence of a Kan–Quillen model structure on globular sets

(This is in a sense a follow-up to my earlier question on a geometric definition of globular $\infty$-groupoids)
We know by Scholie 8.4.14 of Cisinski's thesis that the globe category $\mathbb{G}$ is ...

4
votes

1
answer

141
views

### Decomposing a $\mathcal{M}$-valued presheaf into a homotopy colimit of representables

The context for this question comes from this arxiv preprint. Specifically, a remark in the final proof of the paper. To make the question more self-contained, I'll phrase this question in a slightly ...

8
votes

1
answer

224
views

### Is there a "geometric definition" of globular $\infty$-groupoids/categories?

The nLab page on $\infty$-categories splits the known definitions of $\infty$-categories into two types:
Algebraic $\infty$-categories, in which composition is expressed "externally", e.g. ...

4
votes

0
answers

62
views

### Is it possible to define the Serre mod $\mathscr{C}$ model category structure on $r$-reduced simplicial sets as Cisinski model structure?

I think it is quite straightforward to show that a $\bmod\mathscr{C}$ model structure on $r$-reduced simplicial sets is a Bousfield localization of the transferred Kan-Quillen model structure.
However,...

2
votes

0
answers

66
views

### Model structure for dga of (endormorphism) vector bundle valued differential forms

I was browsing and came across this discussion on the model structure for a dga. They mostly explain the commutative case but then say some things about the non-commutative case.
Context Consider a ...

4
votes

1
answer

221
views

### When are filtered colimits of (trivial) cofibrations still (trivial) cofibrations?

Let $\mathcal{M}$ be a
locally finitely presentable model category, cofibrantly generated by
two sets $\mathcal{I}$ and $\mathcal{J}$ of cofibrations and trivial
cofibrations with presentable domain ...

3
votes

0
answers

45
views

### subcategory of "nice" maps of topological spaces where each closed inclusion is a cofibration?

Is there a subcategory of topological spaces such that each closed inclusion is necessarily a cofibration, and which is good for homotopy theory ?
In general, what are sufficient conditions for a ...

0
votes

0
answers

35
views

### Cellular model of a locally presentable category

According to https://ncatlab.org/nlab/show/cellular+model, I call a cellular model of a locally presentable category a set of monomorphisms cofibrantly generating the monomorphisms. I am curious to ...

9
votes

0
answers

183
views

### Every locally presentable $\infty$-category can be presented by a proper model category

Is there an argument in the litterature that show that every locally presentable $\infty$-category is equivalent to the localization of proper combinatorial Quillen model category ?
Of course if one ...

1
vote

1
answer

254
views

### Applications of “Homotopical algebra” in the set up of Lie groupoids

The question is as in the title.
(What are some of the) are there any applications of Homotopical algebra (in the context of Quillen’s book “Homotopical algebra”) in better understanding (or ...

3
votes

1
answer

135
views

### A projective-cofibrant replacement in $\mathsf{sSet}^{\mathcal{C}^{\text{op}}}$ such that $\operatorname{ev}_0Q(S) \cong \operatorname{ev}_0S$

Consider the category of simplicial presheaves $\mathsf{sSet}^{\mathcal{C}^{\text{op}}}$ endowed with the projective model structure, i.e. weak equivalences and fibrations are point-wise.
I need a ...

3
votes

2
answers

268
views

### Pushout along weak equivalence gives weakly equivalent object

This question arose through reading "Interactions between homotopy theory and algebra" (the first chapter by Goerss and Schemmerhorn). In particular, I am struggling with the proof of ...

3
votes

0
answers

141
views

### (Commutative) Algebras in $\mathsf{dgCat}_k$

Suppose $k$ is a fixed commutative ring, and let $\mathsf{dgCat}_k$ denote the category of $k$-linear dg-categories. We will equip $\mathsf{dgCat}_k$ with the Morita model structure (see theorem 2.27 ...

5
votes

0
answers

257
views

### Why do homotopy orbits commute with the Eilenberg–Mac Lane spectrum functor?

Let $H: \mathsf{sAb}\to H\mathbb{Z}\text{-}\mathsf{Mod}$ denote the Eilenberg–Mac Lane functor sending a simplicial abelian group $M_\bullet$ to the infinite delooping of its geometric realization, i....

0
votes

1
answer

161
views

### Understanding the definition of left homotopy as given in Quillen’s Homotopical algebra book

Given two topological spaces $X,Y$, and two maps $f,g:X\rightarrow Y$, there is a notion of homotopy between $f$ and $g$. It is given by a continuous map $H:X\times I\rightarrow Y$ such that the ...

14
votes

2
answers

655
views

### sSet-enriched categories, quasi-categories and the model-independent theory

sSet-enriched categories are one possible model for $(\infty,1)$-categories, by the work of Bergner and others. They are probably the most important model from the point of view of getting actual ...

4
votes

2
answers

227
views

### Quillen pairs / $\infty$-adjunctions / adjunctions of homotopy categories

Some of the examples of $\infty$-categories are those arising from model categories. I would like to ask: what is the relationship between Quillen adjunctions between model categories and adjoint ...

2
votes

0
answers

57
views

### Relations between the operations of taking weak orthogonals

While reading the Hovey Model Category I learned that $I^{rlr} = I^r$ and $I^{lrl} = I^l$ for every class of morphisms $I$. What other relations are there between the operations of taking (weak) ...

0
votes

0
answers

78
views

### Décalage and the simplicial path object

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\const{const}$Let $[+1]:\Delta\to\Delta$ be the décalage endomorphism sending $n\mapsto n+1$
adding a new minimal element, i.e. $f:n\to m$ is sent to $...

3
votes

0
answers

93
views

### About homotopy weighted colimit

Let $X:I\to M$ be a functor from a small category $I$ to a simplicial accessible (not combinatorial) proper model category $M$ which is already injective cofibrant. Let $W:I^{op}\to \mathrm{Set}$ be a ...

1
vote

0
answers

56
views

### Defining null-homotopy in terms of endofunctors in an arbitrary simplicial category

I am looking for references defining contractible or null-homotopic in terms of endofunctors $\Delta\to\Delta$.
Let $[+1]:\Delta\to\Delta, n\mapsto n+1$ be the endofunctor of $\Delta$ adding a new ...

4
votes

0
answers

80
views

### Homotopy colimits in subcategories of combinatorial model categories

We know that, in a combinatorial simplicial model category $\mathbf{M}$, we can find a regular cardinal $\lambda$ large enough that all $\lambda$-filtered homotopy colimits can be computed as strict ...

3
votes

1
answer

384
views

### How to get by with only functorial cylindrical objects?

In the excellent "A Handbook of Model Categories" (2021), cofibrant and fibrant homotopies are defined exactly as it seemed natural to me: immediately through functorial cylindrical objects /...

4
votes

0
answers

52
views

### Left adjoint that preserves acyclic fibrations or weak equivalences

Given a Quillen adjunction with left adjoint $L: \mathcal{C} \longrightarrow \mathcal{D}$ and right adjoint $R:\mathcal{D} \longrightarrow \mathcal{C}$, then we know that $L$ preserves cofibrations ...

2
votes

0
answers

164
views

### Is there an interesting model structure on sSet whose fibered objects are exactly contractible Kan Complexes?

For the Kan-Quillen model structure, the fibered objects are exactly the Kan Complexes and for the Joyal model structure, the fibered objects are exactly the $\infty$-categories. This follows from the ...

6
votes

1
answer

203
views

### Is the composite of absolute derived functors a derived functor?

Let me recall the following definition. Let $F: C \to D$ be a functor between homotopical categories. Denote by $\gamma_C: C \to \mathrm{Ho} C$ the localization and similary for $D$. A total left ...