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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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 ...
Megan's user avatar
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4 answers
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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 ...
Akhil Mathew's user avatar
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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 ...
roger123's user avatar
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7 votes
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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-...
Dmitri Pavlov's user avatar
46 votes
8 answers
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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 ...
Peter LeFanu Lumsdaine's user avatar
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 ...
Aly's user avatar
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16 votes
2 answers
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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 ...
Simon Henry's user avatar
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14 votes
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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 ...
Dan Dreiberg's user avatar
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 ...
Tim Campion's user avatar
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9 votes
2 answers
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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 $D$...
James's user avatar
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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 ...
David White's user avatar
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22 votes
1 answer
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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 ...
Helene Sigloch's user avatar
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 $...
Akhil Mathew's user avatar
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19 votes
1 answer
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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 ...
Tim Campion's user avatar
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19 votes
4 answers
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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 ...
Chris Schommer-Pries's user avatar
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 $\infty$-...
Tim Campion's user avatar
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15 votes
2 answers
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"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 ...
John Klein's user avatar
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14 votes
3 answers
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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\...
Tyrone's user avatar
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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 ...
Joey Hirsh's user avatar
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13 votes
3 answers
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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, ...
Joey Hirsh's user avatar
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12 votes
6 answers
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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 ...
Theo Johnson-Freyd's user avatar
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 $\...
Reid Barton's user avatar
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10 votes
1 answer
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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 ...
Fernando Muro's user avatar
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) \...
Paul Slevin's user avatar
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 ...
Simon Henry's user avatar
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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 ...
fosco's user avatar
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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 $\mathcal V =\mathbf C(k)$ ...
Francesco Genovese's user avatar
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 ...
Rasmus's user avatar
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1 answer
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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 ...
David White's user avatar
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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 $Top\hookrightarrow sFilt$ ...
user117509's user avatar
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 ...
Shlomi A's user avatar
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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 ...
Shlomi A's user avatar
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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 ...
Mikhail Bondarko's user avatar
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 ...
Simon Markett's user avatar
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 ...
Tyrone's user avatar
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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 ...
J Williams's user avatar
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20 votes
2 answers
1k 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, ...
Charles Rezk's user avatar
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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 ...
Giorgio Mossa's user avatar
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 ...
Urs Schreiber's user avatar
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. ...
Tim Campion's user avatar
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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 ...
Zhen Lin's user avatar
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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 ...
Jeff Strom's user avatar
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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 ...
Reid Barton's user avatar
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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 ...
Daniel Grady's user avatar
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 : ...
Agustí Roig's user avatar
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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 ...
Tim Campion's user avatar
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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-...
Dmitri Pavlov's user avatar
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 ...
Niall Taggart's user avatar
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 ...
yasha's user avatar
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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 ...
Mike Shulman's user avatar