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45
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6answers
3k views

What are surprising examples of Model Categories?

Background Model categories are an axiomization of the machinery underlying the study of topological spaces up to homotopy equivalence. They consist of a category $C$, together with three ...
33
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4answers
5k views

Do we still need model categories?

One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak ...
30
votes
6answers
3k views

Non-examples of model structures, that fail for subtle/surprising reasons?

An often-cited principle of good mathematical exposition is that a definition should always come with a few examples and a few non-examples to help the learner get an intuition for where the concept's ...
25
votes
6answers
2k views

How to think about model categories?

I've read about model categories from an Appendix to one of Lurie's papers. What are the examples of model categories? What should be my intuition about them? E.g. I understand the typical examples ...
19
votes
4answers
1k views

Model structure on Simplicial Sets without using topological spaces

The category of simplicial sets has a standard model structure, where the weak equivalences are those maps whose geometric realization is a weak homotopy equivalence, the cofibrations are ...
19
votes
4answers
1k views

A Peculiar Model Structure on Simplicial Sets?

I'm wondering if there is a Quillen model structure on the category of simplicial sets which generalizes the usual model structure, but where every simplicial set is fibrant? I want to use this to do ...
17
votes
5answers
1k views

Derived categories and homotopy categories

There are two constructions that look quite similar to me: the derived category of an abelian category, and the homotopy category of a model category. Is there any explicit relationship between these ...
17
votes
2answers
546 views

Limitations on model-categorical presentations

In higher category theory, it is common that a weak structure cannot be strictified in all directions simultaneously. For instance, a monoidal category is not (in general) equivalent to one that is ...
16
votes
4answers
1k views

How canonical is cofibrant replacement?

Quillen's original definition of a model category included noncanonical factorization axioms, one being that any map can be factored into a cofibration followed by an acyclic fibration. More recent ...
16
votes
1answer
1k views

Homotopy colimits/limits using model categories

A homotopy (limits and) colimit of a diagram $D$ topological spaces can be explicitly described as a geometric realization of simplicial replacement for $D$. However, a homotopy colimit can also be ...
15
votes
3answers
1k views

Is there an additive model of the stable homotopy category?

Is there a model category $C$ on an additive category such that its homotopy category $Ho(C)$ is the stable homotopy category of spectra and the additive structure on $Ho(C)$ is induced from that on ...
15
votes
2answers
759 views

Homotopy Limits over Fibered Categories

Suppose I have a small category $ \mathcal{C} $ which is fibered over some category $\mathcal{I}$ in the categorical sense. That is, there is a functor $\pi : \mathcal{C} \rightarrow \mathcal{I}$ ...
15
votes
3answers
2k views

Model category structure on Set without axiom of choice

There is a model category structure on Set in which the cofibrations are the monomorphisms, the fibrations are maps which are either epimorphisms or have empty domain, and the weak equivalences are ...
14
votes
4answers
3k views

Homotopy pullbacks and homotopy pushouts

I have a good grasp of ordinary pullbacks and pushouts; in particular, there are many categorical constructions that can be seen as special cases: e.g., equalizers/coequalizers, kernerls/cokernels, ...
14
votes
4answers
447 views

Fibrations and Cofibrations of spectra are “the same”

My question refers to a folklore statement that I have now seen a couple of times, but never really precise. One avatar is: "For spectra every cofibration is equivalent to a fibration" (e.g. in the ...
14
votes
5answers
1k views

Model structure of commutative dg-algebras inside all dg-algebras

Most of the literature considers the standard model category structure on (graded) commutative differential algebras. But this generalizes to all (not-necessarily commutative) dg-algebras. Details ...
14
votes
2answers
530 views

How many model categories have the same weak equivalences?

There are many situations which arise where one might consider different Model categories with the same underlying category. For example in (left) Bousfield localization you start with a model ...
14
votes
1answer
674 views

Is there a combinatorial way to factor a map of simplicial sets as a weak equivalence followed by a fibration?

Background on why I want this: I'd like to check that suspension in a simplicial model category is the same thing as suspension in the quasicategory obtained by composing Rezk's assignment of a ...
14
votes
2answers
1k views

Acyclic models via model categories?

Recall the acyclic models theorem: given two functors $F, G$ from a "category $\mathcal{C}$ with models $M$" to the category of chain complexes of modules over a ring $R$, a natural transformation ...
14
votes
1answer
688 views

Homology in the $A_\infty$ World

This question is turning out to be a little long so let me start off with the headline. Given a differential graded algebra $A$, we can recover $A$ from its homology $HA$ if we know "the" ...
13
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3answers
1k views

What determines a model structure?

It is easy to prove that a model structure is determined by the following classes of maps (determined = two model structures with the mentioned classes in common are equal). cofibrations and weak ...
13
votes
1answer
738 views

Joyal's letter to Grothendieck

Mostly out of curiosity: Where do I find Joyal's letter to Grothendieck in which he defines a model structure on simplicial sheaves? The question was already asked in this MO post, but that ...
12
votes
3answers
611 views

Model Structure/Homotopy Pushouts in topological monoids?

Let C be the category of topological monoids, that is, the category of monoids in (Top, $\times$). Can the model category structure on Top (Serre fibrations, cofibrations, weak homotopy ...
12
votes
1answer
367 views

How to localize a model category with respect to a class of maps created by a left Quillen functor

Let $M$ and $N$ be "nice" model categories. I'm happy to have "nice" mean combinatorial model category. Consider a Quillen pair $$ L: M\rightleftarrows N: R.$$ I want the following result: There ...
12
votes
2answers
665 views

Model category structure on categories enriched over quasi-coherent sheaves

Gonçalo Tabuada has shown that there is a Quillen model category structure on the category of small dg-categories, i.e. the category of small categories enriched over chain complexes (for a fixed ...
12
votes
1answer
1k views

Do homotopy groups “always” commute with filtered colimits?

It is well-known that homotopy groups, of, say, simplicial sets, commute with filtered colimits. However, I could not find a reference for an analogous result for homotopy groups of spectra, or, ...
12
votes
2answers
506 views

Pointed Hurewicz model structure

In Strøm's (no relation) paper "The Homotopy Category is a Homotopy Category" he proves that the category of unpointed topological spaces, with Hurewicz fibrations and ordinary cofibrations and ...
12
votes
1answer
560 views

When is the category of pro-objects a homotopy category?

For a category $C$, there is a category Pro-$C$ whose objects are cofiltered diagrams $I \to C$ and whose morphisms are given by $$ {\rm Hom}(\{x_s\},\{y_t\}) = \varprojlim_t\ \varinjlim_s\ {\rm ...
12
votes
1answer
799 views

A Model Category of Segal Spaces?

So in Julie Bergner's work on (infty, 1)-categories arXiv:0610239, she considers several model categories which model (infty, 1)-categories, which are known to be equivalent. I'm guessing that there ...
11
votes
2answers
636 views

Are generalized cohomology theories a homotopy category of some category of invariants?

I was taught to think of generalized cohomology theories as the homotopy category of (symmetric) spectra. But is there also a category of 'invariants', that is, some category of contravariant functors ...
11
votes
1answer
404 views

The weak equivalences in the covariant model structure

Let $S$ be a simplicial set. Recall that there is a model structure, called the covariant model structure (see HTT ch. 2 and this question), on $\mathbf{SSet}/S$ such that: The cofibrations are the ...
11
votes
1answer
369 views

Model structure on the category of small $A_\infty$ categories, hocolims.

I strangely could not find a reference for this. What are some (if any) model structures on the category of small $A_{\infty}$ categories, with weak equivalences quasi-equivalences. Same question in ...
11
votes
2answers
321 views

The definition of Reedy category

The common definition of Reedy category seems to be this one that a Reedy category is a small category $R$ with two wide subcategories $R_+$ and $R_-$ and an ordinal-valued degree function on its ...
11
votes
1answer
313 views

Model for the (infinity,1)-category of (homotopy-)limit preserving functors

I've got a simplicial model category $M.$ I'd like to get my hands on the (infinity,1) category of homotopy limit preserving functors from M to Spaces in order to compare it to another simplicial ...
11
votes
2answers
521 views

global fibrations of simplicial sheaves

I'm reading the classical Brown-Gersten's paper "Algebraic K-theory as generalized sheaf cohomology" and I'm stuck with their choose of global fibrations. Namely, a morphism of simplicial sheaves $p : ...
11
votes
1answer
377 views

Model category structures on dga's in a ringed topos

In the introduction to his paper "Towards a non-abelian $p$-adic Hodge theory", Olsson says that for any ringed topos $(\mathcal{T},\mathcal{O})$ with $\mathcal{O}$ a sheaf of $\mathbb{Q}$-algebras, ...
10
votes
5answers
2k views

Computations in $\infty$-categories

Direct to the point. Since now I've looked a lot of presentations of $\infty$-categories, but it seems that the only way to do explicit computations on these objects is via model categories. Is that ...
10
votes
3answers
1k views

What is the “universal problem” that motivates the definition of homotopy limits/colimits (and more generally “derived” functors)?

The ordinary notions of limit and colimit are universal solutions to a problem, specifically, finding terminal/initial objects in slice/coslice categories. In the context of homotopy right Kan ...
10
votes
1answer
812 views

Is the simplicial completion of a localizer always a bousfield localization of the injective model structure?

Background Recall (from Cisinski's Astérisque volume 308) that given a small category $A$, we define an $A$-localizer to be a class $W$ of morphisms of $\mathrm{Psh}(A)$ satisfying the following ...
10
votes
2answers
456 views

Fubini theorem for hocolim.

I wanted to ask the following question, Suppose $\mathbf{M}$ a cof generated model category and $I,~J$ two small categories. Suppose that $F:J\rightarrow \mathbf{M}^{\mathrm{I}}$ is a functor. Is it ...
10
votes
1answer
326 views

Is the model category of Complete Segal Spaces right proper?

Well, the title is self-explaining, I guess - I am referring to the complete Segal space model structure of Theorem 7.2 in Rezk's article "A model for the homotopy theory of homotopy theories". Has ...
10
votes
3answers
767 views

What are the fibrant objects in the injective model structure?

If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structure on simplicial ...
9
votes
2answers
779 views

Are non-empty finite sets a Grothendieck test category?

A "test category" is a certain kind of small category $A$ which turns out to have the following property: the category $\widehat{A}$ of presheaves of sets on $A$ admits a model category structure, ...
9
votes
2answers
636 views

When do the Reedy and injective model category structures agree?

Let $R$ be a Reedy category and consider the category $\mathcal{P}(R) = \mathbf{sSet}^{R^{\mathrm{op}}}$ of simplicial presheaves on $R$. When are the Reedy and injective model structures on ...
9
votes
2answers
776 views

Model category structures on categories of complexes in abelian categories

Section 2.3 of Hovey's Model Categories book defines a model category structure on Ch(R-Mod), the category of chain complexes of R-modules, where R is a ring. Lemma 2.3.6 then essentially states (I ...
9
votes
2answers
418 views

Model structure for cooperads and for coalgebras

I am studying the homotopy theory of (algebraic) operads and I came up with several questions I am unable to answer to. I would like to stress that I don't have applications in mind, I just would like ...
9
votes
1answer
414 views

Is there a notion of a “model category which admits left Bousfield localization?”

At a conference not too long ago I gave a talk on (left) Bousfield localization and was asked an interesting question afterwards. The question was whether I knew any examples of model categories which ...
9
votes
1answer
508 views

How many model category structures are there on Top?

I recently started learning a little model category theory and in particular I found this nice exercise. I only know a little topology, but this prompted me to wonder how many model category ...
9
votes
1answer
287 views

Which statements and arguments of Hovey's “Model categories” fail without functorial factorizations of morphisms?

I would like to study the homotopy theory of the category of pro-objects over a proper model category $M$. $Pro-M$ is endowed with the strict model structure; it seems that functorial functorizations ...
9
votes
1answer
422 views

Schemes as a model category

I'm just learning some basics of model categories, so please forgive me if my question turns out to be trivial. I hope it does at least make sense. A natural temptation is to relate this machinery to ...