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.
96
questions
38
votes
8
answers
6k
views
Why do we need model categories?
I cannot give a good answer to this question. And
2) Why this definition of model category is the right way to give a philosophy of homotopy theory? Why didn't we use any other definition?
3) Has ...
82
votes
4
answers
21k
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 ...
18
votes
3
answers
5k
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 ...
7
votes
3
answers
2k
views
Reference for homotopy (co)limits of (co)chain complexes via totalization of double complexes
It seems to be a well-known fact that homotopy (co)limits
of (co)simplicial diagrams of nonnegatively graded
(co)chain complexes in (Grothendieck) abelian categories
can be computed by using the Dold-...
46
votes
8
answers
10k
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 ...
18
votes
3
answers
2k
views
When do colimits agree with homotopy colimits?
I'm wondering about when the colimit and the homotopy colimit agree with diagrams of simplicial sets. I know that hocolim$(F)=$colim$(F_c)$ where $F_c$ is the cofibrant replacement of $F$. However, it ...
16
votes
2
answers
893
views
Counter-example to the existence of left Bousfield localization of combinatorial model category
Is there any known example of a combinatorial model category $C$ together with a set of map $S$ such that the left Bousefield localization of $C$ at $S$ does not exists ?
It is well known to exists ...
14
votes
1
answer
3k
views
Does the right adjoint of a Quillen equivalence preserve homotopy colimits?
Call a diagram $E$ in a model category a homotopy colimit diagram if the morphism $$\mathrm{hocolim}~E\to \mathrm{colim}~ E$$ is a weak equivalence. A homotopy colimit is defined as the categorical ...
11
votes
2
answers
494
views
Are cofibrations accessible?
The category of fibrations in a combinatorial model category is accessible, accessibly embedded in the arrow category. How about the cofibrations?
More generally, let $C$ be a locally presentable ...
9
votes
2
answers
635
views
An example of two cofibrant dg categories whose tensor product is not cofibrant
I have been reading the paper by Toën "The homotopy theory of dg categories and derived Morita theory" where in chapter 4 it is stated that the tensor product of two cofibrant dg categories $C$ and $D$...
8
votes
2
answers
1k
views
On combinatorial and cellular model categories and infinity categories
I am looking for a counterexample. Let me first give the set-up. When you work with model categories, it is extremely common to assume they are cofibrantly generated. For me, this means the definition ...
22
votes
1
answer
2k
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 ...
20
votes
2
answers
2k
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 $...
19
votes
1
answer
784
views
Is there an analog of Kan's $Ex^\infty$ functor for quasicategories?
Is there a fibrant replacement functor in the Joyal model structure which can be described non-recursively, like $Ex^\infty$ for the Quillen model structure? I believe another way to put this is to ...
19
votes
4
answers
3k
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 ...
17
votes
4
answers
831
views
Do combinatorial model categories and Quillen adjunctions model presentable $\infty$-categories?
Let $Q$ be the homotopical category of combinatorial model categories and left Quillen functors, with left Quillen equivalences for weak equivalences.
Let $\mathbf Q$ be the corresponding $\infty$-...
15
votes
2
answers
1k
views
"Strøm-type" model structure on chain complexes?
Background
The Quillen model structure on spaces has weak equivalences given by the weak homotopy equivalences and the fibrations are the Serre fibrations. The cofibrations are characterized by ...
14
votes
3
answers
583
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\...
14
votes
2
answers
635
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 ...
13
votes
3
answers
967
views
Model Structure/Homotopy Pushouts in topological monoids?
Let $\mathsf C$ be the category of topological monoids, that is, the category of monoids in $(\textsf{Top}, \times)$.
Can the model category structure on $\textsf{Top}$ (Serre fibrations, ...
12
votes
6
answers
3k
views
What is a good basic reference on model categories?
I am looking for a general reference text on model categories, that contains all the basic results and definitions. I'm perfectly happy to be pointed towards a textbook, and I'm not looking for ...
11
votes
2
answers
880
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 $\...
10
votes
1
answer
685
views
Hovey's unit axiom in monoidal model categories
Let $\mathcal{C}$ be a monoidal model category in the sense of Hovey's book. He assumes the following unit axiom not considered in other references (e.g. Schwede-Shipley): given a cofibrant ...
9
votes
1
answer
532
views
Is the localisation of a product of categories the product of the localisation?
Let $\cal C, \cal D$ be model categories. Hovey says in his monograph "Model Categories" that the homotopy category $\operatorname{Ho}(\cal C \times D)$ is isomorphic to $\operatorname{Ho}(\cal C) \...
8
votes
1
answer
441
views
Left Bousfield localization without properness, what is known?
I'm interested in the existence of several example of left Bousfield localization of model categories that are not left proper (nor simplicial). I'm relatively convince that I can construct all those ...
7
votes
3
answers
884
views
A fibrant-objects structure on Top
(Sorry for the crossposting, but I'm really interested in this question).
One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological ...
6
votes
2
answers
1k
views
A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categories, with $\mathcal V$ monoidal model category
I employ the vast majority of the post to develop the notion of quasi-functor between dg-categories: I think it is important to get the idea.
Let $k$ be a field, and let $\mathcal V =\mathbf C(k)$ ...
5
votes
2
answers
2k
views
Homotopy limit-colimit diagrams in stable model categories
It is shown in Remark 7.1.12 of (a newer version of) Mark Hovey's book Model Categories that, in a stable model category, homotopy pullback squares coincide with homotopy pushout squares. The argument ...
5
votes
1
answer
818
views
When is homotopy orbit space weakly equivalent to orbit space, other than situation of free action?
Let $M$ be a closed symmetric monoidal model category. Let $X$ be a cofibrant object (it can also be fibrant if you like) and let $\Sigma_n$ act on $X^{\otimes n}$ by permuting the factors (note that ...
4
votes
2
answers
413
views
Small model categories?
All the standard examples for model categories are large categories. Is it possible to have a small model category? Are there any interesting examples?
EDIT:
Since a complete small category is a ...
4
votes
0
answers
233
views
category of simplicial filters
I am looking for references discussing the category $Filt$ of filters (in the sense of set theory, details below),
its simplicial category $sFilt$ and its full subcategory $Top\hookrightarrow sFilt$ ...
4
votes
1
answer
350
views
Model structure on category of endofunctors
Let $\mathcal C$ be model category, perhaps even cofibrantly generated. I don't assume that $\mathcal C$ is small. Recall that $End(\mathcal C)$ is the category of endofunctors on $\mathcal C$, with ...
3
votes
0
answers
152
views
Connecting Quillen functors between motivic homotopy categories (of different "types"): references?
For a perfect base field $k$ there exists the following collection of "motivic homotopy" categories related to it:
(a) the homotopy category of simplicial presheaves (from smooth $k$-varieties); here ...
31
votes
4
answers
3k
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 ...
29
votes
4
answers
4k
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 ...
29
votes
3
answers
3k
views
The homotopy category is not complete nor cocomplete
I understand that the homotopy category of (pointed) topological spaces and continuous maps is not complete. Nor is it cocomplete. In particular it neither has all pullbacks nor all pushouts.
What ...
18
votes
5
answers
2k
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 ...
18
votes
5
answers
3k
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 ...
16
votes
1
answer
489
views
Is there an "injective version" of the Bergner model structure?
The Bergner model structure on $sCat$ (simplicially enriched categories) has a "projective" flavor: fibrations are "levelwise" while cofibrations satisfy stringent relative "freeness" conditions.
...
15
votes
3
answers
3k
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 ...
15
votes
1
answer
1k
views
Examples of differential cohomology in cohesive $\infty$ topos
I might direct this question to Urs Schreiber directly, but just in case someone else has some interesting examples, I'll make the question public.
The formulation of differential cohomology in ...
15
votes
2
answers
957
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 ...
15
votes
1
answer
875
views
The state of the art in the rectification of homotopy-coherent structures
My question concerns rectification theorems for homotopy-coherent structures. As the meaning of this may be unclear, let me list a few examples of what I am thinking of:
Cordier and Porter proved a ...
12
votes
2
answers
388
views
Property-like structure in a model category
In a model category, I have tools to show that mapping spaces are contractible. But if I want to show a mapping space is empty or contractible, is there anything I can do on general grounds?
The idea ...
12
votes
2
answers
799
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 : ...
12
votes
2
answers
839
views
When did the Joyal model structure on simplicial sets originate?
Some of the earliest writings on the Joyal model structure on simplicial sets include Jacob Lurie's account in Higher Topos Theory from 2006,
as well as Joyal's own account in The Theory of Quasi-...
12
votes
1
answer
355
views
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 ...
11
votes
1
answer
480
views
Which maps of simplicial sets geometrically realize to fibrations?
If $f:X\to Y$ is a Kan fibration of simplicial sets, then its geometric realization $|f| : |X|\to |Y|$ is (in some suitable convenient category of topological spaces, like compactly generated ones) a ...
11
votes
1
answer
730
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
3
answers
1k
views
A category with weak equivalences that is not a model category
I'm only considering complete and cocomplete categories. A pair $(\mathfrak{X} , \mathfrak{W}) $ is, by definition, a category with weak equivalences if $ \mathfrak{X} $ is a category and $ \mathfrak{...