On model categories where every object is bifibrant The Next CEO of Stack OverflowWhat is the “universal problem” that motivates the definition of homotopy limits/colimits (and more generally “derived” functors)?Do we still need model categories?Constructing a “geometric” model structure on Cat by localizing the “categorical” model structureA model category of abelian categories?On combinatorial and cellular model categories and infinity categoriesFibrant-cofibrant models of Eilenberg-MacLane spectraWhy is every object cofibrant in an excellent model category?Are strict $infty$-categories localized at weak equivalences a full subcategory of weak $infty$-categories?Is there an “injective version” of the Bergner model structure?Excellent monoidal model categories admit enriched fibrant replacement functors?
On model categories where every object is bifibrant
The Next CEO of Stack OverflowWhat is the “universal problem” that motivates the definition of homotopy limits/colimits (and more generally “derived” functors)?Do we still need model categories?Constructing a “geometric” model structure on Cat by localizing the “categorical” model structureA model category of abelian categories?On combinatorial and cellular model categories and infinity categoriesFibrant-cofibrant models of Eilenberg-MacLane spectraWhy is every object cofibrant in an excellent model category?Are strict $infty$-categories localized at weak equivalences a full subcategory of weak $infty$-categories?Is there an “injective version” of the Bergner model structure?Excellent monoidal model categories admit enriched fibrant replacement functors?
$begingroup$
Most model structures we use either have that every object is fibrant or that every object is cofibrant, and we have various general constructions that allow (under some assumption) to go from one situation to the other.
But there are very few examples of model categories where every object is both fibrant and cofibrant ("bifibrant").
The only example I know is when one starts with a strict 2-category with strict 2-limits and 2-colimits there are model structures on its underlying 1-category where every object is bifibrant and whose Dwyer-Kan localization is equivalent to the $2$-category itself (where one drop non-invertible 2-cells). So this typically applies to the canonical model category on Cat or on Groupoids. I'm not even sure it can be used to model things like the $2$-category of categories with finite limits.
I don't believe there are that many other examples. But I have never seen any obstruction for this. So:
Is there any example of a model category where every object is bifibrant whose localization is not a $2$-category?
Is every presentable $infty$-category represented by a model category where every object is bifibrant? If (as I expect) this is not the case, can we give an explicit 'obstruction' or an example to show it isn't the case?
Edit : The first question has been completely answered, but I havn't accepted answer so far because I'm still hoping to get an answer (positive or negative) to the second question.
at.algebraic-topology ct.category-theory homotopy-theory model-categories
asked 2 days ago
Simon HenrySimon Henry
15.6k14991
$endgroup$
add a comment |
$begingroup$
Most model structures we use either have that every object is fibrant or that every object is cofibrant, and we have various general constructions that allow (under some assumption) to go from one situation to the other.
But there are very few examples of model categories where every object is both fibrant and cofibrant ("bifibrant").
The only example I know is when one starts with a strict 2-category with strict 2-limits and 2-colimits there are model structures on its underlying 1-category where every object is bifibrant and whose Dwyer-Kan localization is equivalent to the $2$-category itself (where one drop non-invertible 2-cells). So this typically applies to the canonical model category on Cat or on Groupoids. I'm not even sure it can be used to model things like the $2$-category of categories with finite limits.
I don't believe there are that many other examples. But I have never seen any obstruction for this. So:
Is there any example of a model category where every object is bifibrant whose localization is not a $2$-category?
Is every presentable $infty$-category represented by a model category where every object is bifibrant? If (as I expect) this is not the case, can we give an explicit 'obstruction' or an example to show it isn't the case?
Edit : The first question has been completely answered, but I havn't accepted answer so far because I'm still hoping to get an answer (positive or negative) to the second question.
at.algebraic-topology ct.category-theory homotopy-theory model-categories
asked 2 days ago
Simon HenrySimon Henry
15.6k14991
$endgroup$
$begingroup$
Hi Simon. Just wanted to say, I haven't forgotten your email, and I do plan to reply. I've just been really busy. Sorry!
$endgroup$
– David White
2 days ago
$begingroup$
Trivial examples: any complete and cocomplete category with the isomorphisms as weak equivalences, and all morphisms are both fibrations and cofibrations.
$endgroup$
– Daniel Robert-Nicoud
2 days ago
$begingroup$
@DanielRobert-Nicoud : This one is a special case of the 2-categorical example.
$endgroup$
– Simon Henry
2 days ago
$begingroup$
I think your second question is much harder than your first one, so it might be better to ask them as separate questions.
$endgroup$
– Mike Shulman
yesterday
add a comment |
$begingroup$
Most model structures we use either have that every object is fibrant or that every object is cofibrant, and we have various general constructions that allow (under some assumption) to go from one situation to the other.
But there are very few examples of model categories where every object is both fibrant and cofibrant ("bifibrant").
The only example I know is when one starts with a strict 2-category with strict 2-limits and 2-colimits there are model structures on its underlying 1-category where every object is bifibrant and whose Dwyer-Kan localization is equivalent to the $2$-category itself (where one drop non-invertible 2-cells). So this typically applies to the canonical model category on Cat or on Groupoids. I'm not even sure it can be used to model things like the $2$-category of categories with finite limits.
I don't believe there are that many other examples. But I have never seen any obstruction for this. So:
Is there any example of a model category where every object is bifibrant whose localization is not a $2$-category?
Is every presentable $infty$-category represented by a model category where every object is bifibrant? If (as I expect) this is not the case, can we give an explicit 'obstruction' or an example to show it isn't the case?
Edit : The first question has been completely answered, but I havn't accepted answer so far because I'm still hoping to get an answer (positive or negative) to the second question.
at.algebraic-topology ct.category-theory homotopy-theory model-categories
asked 2 days ago
Simon HenrySimon Henry
15.6k14991
$endgroup$
Most model structures we use either have that every object is fibrant or that every object is cofibrant, and we have various general constructions that allow (under some assumption) to go from one situation to the other.
But there are very few examples of model categories where every object is both fibrant and cofibrant ("bifibrant").
The only example I know is when one starts with a strict 2-category with strict 2-limits and 2-colimits there are model structures on its underlying 1-category where every object is bifibrant and whose Dwyer-Kan localization is equivalent to the $2$-category itself (where one drop non-invertible 2-cells). So this typically applies to the canonical model category on Cat or on Groupoids. I'm not even sure it can be used to model things like the $2$-category of categories with finite limits.
I don't believe there are that many other examples. But I have never seen any obstruction for this. So:
Is there any example of a model category where every object is bifibrant whose localization is not a $2$-category?
Is every presentable $infty$-category represented by a model category where every object is bifibrant? If (as I expect) this is not the case, can we give an explicit 'obstruction' or an example to show it isn't the case?
Edit : The first question has been completely answered, but I havn't accepted answer so far because I'm still hoping to get an answer (positive or negative) to the second question.
at.algebraic-topology ct.category-theory homotopy-theory model-categories
at.algebraic-topology ct.category-theory homotopy-theory model-categories
asked 2 days ago
Simon HenrySimon Henry
15.6k14991
asked 2 days ago
Simon HenrySimon Henry
15.6k14991
edited 2 days ago
Simon Henry
asked 2 days ago
Simon HenrySimon Henry
15.6k14991
asked 2 days ago
Simon HenrySimon Henry
15.6k14991
asked 2 days ago
Simon HenrySimon Henry
15.6k14991
15.6k14991
$begingroup$
Hi Simon. Just wanted to say, I haven't forgotten your email, and I do plan to reply. I've just been really busy. Sorry!
$endgroup$
– David White
2 days ago
$begingroup$
Trivial examples: any complete and cocomplete category with the isomorphisms as weak equivalences, and all morphisms are both fibrations and cofibrations.
$endgroup$
– Daniel Robert-Nicoud
2 days ago
$begingroup$
@DanielRobert-Nicoud : This one is a special case of the 2-categorical example.
$endgroup$
– Simon Henry
2 days ago
$begingroup$
I think your second question is much harder than your first one, so it might be better to ask them as separate questions.
$endgroup$
– Mike Shulman
yesterday
add a comment |
$begingroup$
Hi Simon. Just wanted to say, I haven't forgotten your email, and I do plan to reply. I've just been really busy. Sorry!
$endgroup$
– David White
2 days ago
$begingroup$
Trivial examples: any complete and cocomplete category with the isomorphisms as weak equivalences, and all morphisms are both fibrations and cofibrations.
$endgroup$
– Daniel Robert-Nicoud
2 days ago
$begingroup$
@DanielRobert-Nicoud : This one is a special case of the 2-categorical example.
$endgroup$
– Simon Henry
2 days ago
$begingroup$
I think your second question is much harder than your first one, so it might be better to ask them as separate questions.
$endgroup$
– Mike Shulman
yesterday
$begingroup$
Hi Simon. Just wanted to say, I haven't forgotten your email, and I do plan to reply. I've just been really busy. Sorry!
$endgroup$
– David White
2 days ago
$begingroup$
Hi Simon. Just wanted to say, I haven't forgotten your email, and I do plan to reply. I've just been really busy. Sorry!
$endgroup$
– David White
2 days ago
$begingroup$
Trivial examples: any complete and cocomplete category with the isomorphisms as weak equivalences, and all morphisms are both fibrations and cofibrations.
$endgroup$
– Daniel Robert-Nicoud
2 days ago
$begingroup$
Trivial examples: any complete and cocomplete category with the isomorphisms as weak equivalences, and all morphisms are both fibrations and cofibrations.
$endgroup$
– Daniel Robert-Nicoud
2 days ago
$begingroup$
@DanielRobert-Nicoud : This one is a special case of the 2-categorical example.
$endgroup$
– Simon Henry
2 days ago
$begingroup$
@DanielRobert-Nicoud : This one is a special case of the 2-categorical example.
$endgroup$
– Simon Henry
2 days ago
$begingroup$
I think your second question is much harder than your first one, so it might be better to ask them as separate questions.
$endgroup$
– Mike Shulman
yesterday
$begingroup$
I think your second question is much harder than your first one, so it might be better to ask them as separate questions.
$endgroup$
– Mike Shulman
yesterday
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
An example of a different sort is the model structure on $R$-mod, whose homotopy category is the stable module category. A great reference is Theorem 2.2.12 in Hovey's book Model Categories. In this reference, $R$ is taken to be quasi-Frobenius. This model structure is generalized to work for any ring in the thesis of Daniel Bravo, and a resulting paper of Bravo-Gillespie-Hovey. But, you lose the property about all objects being bifibrant.
Another example is the projective (or injective) model structure on $Ch(R)$ where $R$ is a field, and where we take chain complexes to be bounded (e.g. always non-negative degree, or you could do cochain complexes in non-positive degree). A great reference is Quillen's Rational Homotopy Theory. See also Section 2.3 of Hovey's book, but this is for the situation of unbounded chain complexes. The point is that, for the projective model structure, the fibrations are surjections, and as Lemma 2.3.6 shows, bounded below complexes of projective modules are cofibrant, and if $R$ is a field (or semi-simple ring) then all modules are projective.
answered 2 days ago
David WhiteDavid White
13.1k462104
$endgroup$
add a comment |
$begingroup$
Another example is given by Strom's model structure on topological spaces where
- Fibrations: Hurewicz fibrations,
- Weak equivalences : (strong) homotopy equivalences.
answered 2 days ago
David CDavid C
7,39022140
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
An example of a different sort is the model structure on $R$-mod, whose homotopy category is the stable module category. A great reference is Theorem 2.2.12 in Hovey's book Model Categories. In this reference, $R$ is taken to be quasi-Frobenius. This model structure is generalized to work for any ring in the thesis of Daniel Bravo, and a resulting paper of Bravo-Gillespie-Hovey. But, you lose the property about all objects being bifibrant.
Another example is the projective (or injective) model structure on $Ch(R)$ where $R$ is a field, and where we take chain complexes to be bounded (e.g. always non-negative degree, or you could do cochain complexes in non-positive degree). A great reference is Quillen's Rational Homotopy Theory. See also Section 2.3 of Hovey's book, but this is for the situation of unbounded chain complexes. The point is that, for the projective model structure, the fibrations are surjections, and as Lemma 2.3.6 shows, bounded below complexes of projective modules are cofibrant, and if $R$ is a field (or semi-simple ring) then all modules are projective.
answered 2 days ago
David WhiteDavid White
13.1k462104
$endgroup$
add a comment |
$begingroup$
An example of a different sort is the model structure on $R$-mod, whose homotopy category is the stable module category. A great reference is Theorem 2.2.12 in Hovey's book Model Categories. In this reference, $R$ is taken to be quasi-Frobenius. This model structure is generalized to work for any ring in the thesis of Daniel Bravo, and a resulting paper of Bravo-Gillespie-Hovey. But, you lose the property about all objects being bifibrant.
Another example is the projective (or injective) model structure on $Ch(R)$ where $R$ is a field, and where we take chain complexes to be bounded (e.g. always non-negative degree, or you could do cochain complexes in non-positive degree). A great reference is Quillen's Rational Homotopy Theory. See also Section 2.3 of Hovey's book, but this is for the situation of unbounded chain complexes. The point is that, for the projective model structure, the fibrations are surjections, and as Lemma 2.3.6 shows, bounded below complexes of projective modules are cofibrant, and if $R$ is a field (or semi-simple ring) then all modules are projective.
answered 2 days ago
David WhiteDavid White
13.1k462104
$endgroup$
add a comment |
$begingroup$
An example of a different sort is the model structure on $R$-mod, whose homotopy category is the stable module category. A great reference is Theorem 2.2.12 in Hovey's book Model Categories. In this reference, $R$ is taken to be quasi-Frobenius. This model structure is generalized to work for any ring in the thesis of Daniel Bravo, and a resulting paper of Bravo-Gillespie-Hovey. But, you lose the property about all objects being bifibrant.
Another example is the projective (or injective) model structure on $Ch(R)$ where $R$ is a field, and where we take chain complexes to be bounded (e.g. always non-negative degree, or you could do cochain complexes in non-positive degree). A great reference is Quillen's Rational Homotopy Theory. See also Section 2.3 of Hovey's book, but this is for the situation of unbounded chain complexes. The point is that, for the projective model structure, the fibrations are surjections, and as Lemma 2.3.6 shows, bounded below complexes of projective modules are cofibrant, and if $R$ is a field (or semi-simple ring) then all modules are projective.
answered 2 days ago
David WhiteDavid White
13.1k462104
$endgroup$
An example of a different sort is the model structure on $R$-mod, whose homotopy category is the stable module category. A great reference is Theorem 2.2.12 in Hovey's book Model Categories. In this reference, $R$ is taken to be quasi-Frobenius. This model structure is generalized to work for any ring in the thesis of Daniel Bravo, and a resulting paper of Bravo-Gillespie-Hovey. But, you lose the property about all objects being bifibrant.
Another example is the projective (or injective) model structure on $Ch(R)$ where $R$ is a field, and where we take chain complexes to be bounded (e.g. always non-negative degree, or you could do cochain complexes in non-positive degree). A great reference is Quillen's Rational Homotopy Theory. See also Section 2.3 of Hovey's book, but this is for the situation of unbounded chain complexes. The point is that, for the projective model structure, the fibrations are surjections, and as Lemma 2.3.6 shows, bounded below complexes of projective modules are cofibrant, and if $R$ is a field (or semi-simple ring) then all modules are projective.
answered 2 days ago
David WhiteDavid White
13.1k462104
answered 2 days ago
David WhiteDavid White
13.1k462104
answered 2 days ago
David WhiteDavid White
13.1k462104
answered 2 days ago
David WhiteDavid White
13.1k462104
13.1k462104
add a comment |
add a comment |
$begingroup$
Another example is given by Strom's model structure on topological spaces where
- Fibrations: Hurewicz fibrations,
- Weak equivalences : (strong) homotopy equivalences.
answered 2 days ago
David CDavid C
7,39022140
$endgroup$
add a comment |
$begingroup$
Another example is given by Strom's model structure on topological spaces where
- Fibrations: Hurewicz fibrations,
- Weak equivalences : (strong) homotopy equivalences.
answered 2 days ago
David CDavid C
7,39022140
$endgroup$
add a comment |
$begingroup$
Another example is given by Strom's model structure on topological spaces where
- Fibrations: Hurewicz fibrations,
- Weak equivalences : (strong) homotopy equivalences.
answered 2 days ago
David CDavid C
7,39022140
$endgroup$
Another example is given by Strom's model structure on topological spaces where
- Fibrations: Hurewicz fibrations,
- Weak equivalences : (strong) homotopy equivalences.
answered 2 days ago
David CDavid C
7,39022140
answered 2 days ago
David CDavid C
7,39022140
answered 2 days ago
David CDavid C
7,39022140
answered 2 days ago
David CDavid C
7,39022140
7,39022140
add a comment |
add a comment |
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$begingroup$
Hi Simon. Just wanted to say, I haven't forgotten your email, and I do plan to reply. I've just been really busy. Sorry!
$endgroup$
– David White
2 days ago
$begingroup$
Trivial examples: any complete and cocomplete category with the isomorphisms as weak equivalences, and all morphisms are both fibrations and cofibrations.
$endgroup$
– Daniel Robert-Nicoud
2 days ago
$begingroup$
@DanielRobert-Nicoud : This one is a special case of the 2-categorical example.
$endgroup$
– Simon Henry
2 days ago
$begingroup$
I think your second question is much harder than your first one, so it might be better to ask them as separate questions.
$endgroup$
– Mike Shulman
yesterday