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Praphulla Koushik
Praphulla Koushik
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Given a category $\mathcal{C}$, by a category over $\mathcal{C}$, I mean a category $\mathcal{D}$ along with a functor $\pi_{\mathcal{D}}:\mathcal{D}\rightarrow \mathcal{C}$.

Consider the category $Cat/C$, of categories over $\mathcal{C}$. I am assuming the familiarity with the notion of a fibered category over $\mathcal{C}$ (if not, please see Definition 3.5 of Angelo Vistoli's Notes on Grothendieck topologies,fibered categoriesand descent theory).

Question is the following:

  • Is there a (interesting/non-trivial) model structure on $Cat/\mathcal{C}$, in which fibered categories are the fibrant objects?

Or, are the terms "fibered category" and "fibrations" just two unrelated terms that sound same?

Given a category $\mathcal{C}$, by a category over $\mathcal{C}$, I mean a category $\mathcal{D}$ along with a functor $\pi_{\mathcal{D}}:\mathcal{D}\rightarrow \mathcal{C}$.

Consider the category $Cat/C$, of categories over $\mathcal{C}$. I am assuming the familiarity with the notion of a fibered category over $\mathcal{C}$ (if not, please see Definition 3.5 of Angelo Vistoli's Notes on Grothendieck topologies,fibered categoriesand descent theory).

Question is the following:

  • Is there a (interesting/non-trivial) model structure on $Cat/\mathcal{C}$, in which fibered categories are fibrant objects?

Or, are the terms "fibered category" and "fibrations" just two unrelated terms that sound same?

Given a category $\mathcal{C}$, by a category over $\mathcal{C}$, I mean a category $\mathcal{D}$ along with a functor $\pi_{\mathcal{D}}:\mathcal{D}\rightarrow \mathcal{C}$.

Consider the category $Cat/C$, of categories over $\mathcal{C}$. I am assuming the familiarity with the notion of a fibered category over $\mathcal{C}$ (if not, please see Definition 3.5 of Angelo Vistoli's Notes on Grothendieck topologies,fibered categoriesand descent theory).

Question is the following:

  • Is there a (interesting/non-trivial) model structure on $Cat/\mathcal{C}$, in which fibered categories are the fibrant objects?

Or, are the terms "fibered category" and "fibrations" just two unrelated terms that sound same?

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Praphulla Koushik
Praphulla Koushik
  • 6k
  • 3
  • 27
  • 68

Are fibered categories fibrationsfibrant objects in some model structure on Cat/C?

Given a category $\mathcal{C}$, by a category over $\mathcal{C}$, I mean a category $\mathcal{D}$ along with a functor $\pi_{\mathcal{D}}:\mathcal{D}\rightarrow \mathcal{C}$.

Consider the category $Cat/C$, of categories over $\mathcal{C}$. I am assuming the familiarity with the notion of a fibered category over $\mathcal{C}$ (if not, please see Definition 3.5 of Angelo Vistoli's Notes on Grothendieck topologies,fibered categoriesand descent theory).

Question is the following:

  • Is there a (interesting/non-trivial) model structure on $Cat/\mathcal{C}$, in which fibered categories are fibrationsfibrant objects?

Or, are the terms "fibered category" and "fibrations" just two unrelated terms that sound same?

Are fibered categories fibrations in some model structure on Cat/C?

Given a category $\mathcal{C}$, by a category over $\mathcal{C}$, I mean a category $\mathcal{D}$ along with a functor $\pi_{\mathcal{D}}:\mathcal{D}\rightarrow \mathcal{C}$.

Consider the category $Cat/C$, of categories over $\mathcal{C}$. I am assuming the familiarity with the notion of a fibered category over $\mathcal{C}$ (if not, please see Definition 3.5 of Angelo Vistoli's Notes on Grothendieck topologies,fibered categoriesand descent theory).

Question is the following:

  • Is there a (interesting/non-trivial) model structure on $Cat/\mathcal{C}$, in which fibered categories are fibrations?

Or, are the terms "fibered category" and "fibrations" just two unrelated terms that sound same?

Are fibered categories fibrant objects in some model structure on Cat/C?

Given a category $\mathcal{C}$, by a category over $\mathcal{C}$, I mean a category $\mathcal{D}$ along with a functor $\pi_{\mathcal{D}}:\mathcal{D}\rightarrow \mathcal{C}$.

Consider the category $Cat/C$, of categories over $\mathcal{C}$. I am assuming the familiarity with the notion of a fibered category over $\mathcal{C}$ (if not, please see Definition 3.5 of Angelo Vistoli's Notes on Grothendieck topologies,fibered categoriesand descent theory).

Question is the following:

  • Is there a (interesting/non-trivial) model structure on $Cat/\mathcal{C}$, in which fibered categories are fibrant objects?

Or, are the terms "fibered category" and "fibrations" just two unrelated terms that sound same?

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Source Link
Praphulla Koushik
Praphulla Koushik
  • 6k
  • 3
  • 27
  • 68

Are fibered categories fibrations in some model structure on Cat/C?

Given a category $\mathcal{C}$, by a category over $\mathcal{C}$, I mean a category $\mathcal{D}$ along with a functor $\pi_{\mathcal{D}}:\mathcal{D}\rightarrow \mathcal{C}$.

Consider the category $Cat/C$, of categories over $\mathcal{C}$. I am assuming the familiarity with the notion of a fibered category over $\mathcal{C}$ (if not, please see Definition 3.5 of Angelo Vistoli's Notes on Grothendieck topologies,fibered categoriesand descent theory).

Question is the following:

  • Is there a (interesting/non-trivial) model structure on $Cat/\mathcal{C}$, in which fibered categories are fibrations?

Or, are the terms "fibered category" and "fibrations" just two unrelated terms that sound same?