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 $p : E \longrightarrow B$ is a global fibration if for every inclusion of open sets $U\subset V$ the natural map $E(V) \longrightarrow B(V) \times_{B(U)} E(U)$ is a (Kan) fibration of simplicial sets.

My problem is: why these fibrations? As far as I can see, when they make use of this definition in constructing the factorizations of the model category structure, they could have chosen the fibrations to be defined open-wise: $p : E \longrightarrow B$ is a fibration if $p(V) : E(V) \longrightarrow B(V)$ is a (Kan) fibration of simplicial sets for every open set $V$ and apply as well the small object argument they use at this point.

In other contexts I understand this kind of fibrations. For instance, for the model structure of the category of diagrams $C^I$ of a model category $C$ when $I$ is a 'very small' category (Dwyer-Spalinski, "Homotopy theories"), or a Reedy category. In this cases, this kind of fibrations ensures that you can extend your liftings by induction. But I don't see if this is their role with a category of sheaves, since no induction seems to be at hand.

A colleague of mine has said to me thas this choice of fibrations is the consequence of choosing the cofibrations to be the monomorphism, following Joyal's "Letter to Grothendieck"; that is, these are precisely the fibrations if you choose monomorphisms as cofibrations and ask fibrations to have the RLP with respect to trivial cofibrations. But I couldn't find anywhere this famous Joyal's letter, so I would also be glad if someone could tell me where I can find it.

Thanks in advance for any hints.

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 $p : E \longrightarrow B$ is a global fibration if for every inclusion of open sets $U\subset V$ the natural map $E(V) \longrightarrow B(V) \times_{B(U)} E(U)$ is a (Kan) fibration of simplicial sets.

My problem is: why these fibrations? As far as I can see, when they make use of this definition in constructing the factorizations of the model category structure, they could have chosen the fibrations to be defined open-wise: $p : E \longrightarrow B$ is a fibration if $p(V) : E(V) \longrightarrow B(V)$ is a (Kan) fibration of simplicial sets for every open set $V$ and apply as well the small object argument they use at this point.

In other contexts I understand this kind of fibrations. For instance, for the model structure of the category of diagrams $C^I$ of a model category $C$ when $I$ is a 'very small' category (Dwyer-Spalinski, "Homotopy theories"), or a Reedy category. In these cases, this kind of fibrations ensures that you can extend your liftings by induction. But I don't see if this is their role with a category of sheaves, since no induction seems to be at hand.

A colleague of mine has said to me thas this choice of fibrations is the consequence of choosing the cofibrations to be the monomorphism, following Joyal's "Letter to Grothendieck"; that is, these are precisely the fibrations if you choose monomorphisms as cofibrations and ask fibrations to have the RLP with respect to trivial cofibrations. But I couldn't find anywhere this famous Joyal's letter, so I would also be glad if someone could tell me where I can find it.

Thanks in advance for any hints.

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 $p : E \longrightarrow B$" /> is a global fibration if for every inclusion of open sets $U\subset V$" /> the natural map $E(V) \longrightarrow B(V) \times\sb {B(U)} E(U)$" /> is a (Kan) fibration of simplicial sets.

My problem is: why these fibrations? As far as I can see, when they make use of this definition in constructing the factorizations of the model category structure, they could have chosen the fibrations to be defined open-wise:$p : E \longrightarrow B$" /> is a fibration if $p(V) : E(V) \longrightarrow B(V)$" /> is a (Kan) fibration of simplicial sets for every open set $V$" /> and apply as well the small object argument they use at this point.

In other contexts I understand this kind of fibrations. For instance, for the model structure of the category of diagrams $C^I$" /> of a model category $C$" /> when $I$" /> is a 'very small' category (Dwyer-Spalinski, "Homotopy theories"), or a Reedy category. In this cases, this kind of fibrations ensures that you can extend your liftings by induction. But I don't see if this is their role with a category of sheaves, since no induction seems to be at hand.

A colleague of mine has said to me thas this choice of fibrations is the consequence of choosing the cofibrations to be the monomorphism, following Joyal's "Letter to Grothendieck"; that is, these are precisely the fibrations if you choose monomorphisms as cofibrations and ask fibrations to have the RLP with respect to trivial cofibrations. But I couldn't find anywhere this famous Joyal's letter, so I would also be glad if someone could tell me where I can find it.

Thanks in advance for any hints.

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 $p : E \longrightarrow B$ is a global fibration if for every inclusion of open sets $U\subset V$ the natural map $E(V) \longrightarrow B(V) \times_{B(U)} E(U)$ is a (Kan) fibration of simplicial sets.

My problem is: why these fibrations? As far as I can see, when they make use of this definition in constructing the factorizations of the model category structure, they could have chosen the fibrations to be defined open-wise: $p : E \longrightarrow B$ is a fibration if $p(V) : E(V) \longrightarrow B(V)$ is a (Kan) fibration of simplicial sets for every open set $V$ and apply as well the small object argument they use at this point.

In other contexts I understand this kind of fibrations. For instance, for the model structure of the category of diagrams $C^I$ of a model category $C$ when $I$ is a 'very small' category (Dwyer-Spalinski, "Homotopy theories"), or a Reedy category. In this cases, this kind of fibrations ensures that you can extend your liftings by induction. But I don't see if this is their role with a category of sheaves, since no induction seems to be at hand.

A colleague of mine has said to me thas this choice of fibrations is the consequence of choosing the cofibrations to be the monomorphism, following Joyal's "Letter to Grothendieck"; that is, these are precisely the fibrations if you choose monomorphisms as cofibrations and ask fibrations to have the RLP with respect to trivial cofibrations. But I couldn't find anywhere this famous Joyal's letter, so I would also be glad if someone could tell me where I can find it.

Thanks in advance for any hints.