A class of morphisms defined by a right Quillen lifting property (weak orthogonality) is always closed under pullbacks (limits); under what assumptions will it be closed under pushouts (colimits)? In a model category it makes sense to use fibrant replacement and ask when will it be closed under taking fibrant replacement of pushouts or colimits.

More particularly:

Call a morphism $f$ good iff $f \perp C \vee D \rightarrow \top $ whenever $f \perp C \rightarrow \top $ and $f \perp D \rightarrow \top $, for arbitrary objects $C$ and $D$; here $\top $ denotes the terminal object, and $\perp $ denotes the Quillen lifting property (weak orthogonality), i.e. for morphisms $f$ and $g$ $f \perp g$ says that $f$ has the right Quillen lifting property with respect to $g$, or, in another terminology, $f$ is right weakly orthogonal to $g$.

In a model category, call a morphism $f$ good iff for arbitrary objects $C$ and $D$ $f \perp C \rightarrow \top $ and $f \perp D \rightarrow \top $, implies that
$f \perp (C \vee D)^{(f)} \rightarrow \top $ where $(C \vee D)^{(f)} $ is the fibrant replacement of $C \vee D$, i.e. there is a sequence of morphisms $C \vee D\xrightarrow{(ac)} (C \vee D)^{(f)} \xrightarrow{(f)} \top $ where thet first morphism is an acyclic cofibration, and the second one is a fibration.

Is this a well-known notion? How can one describe good morphisms, either in the category of topological spaces or simplicial sets?

Motivation: in model theory, a number of classes of models with amalgamation property can be described by a lifting property of form as above (i.e. $f \perp M\longrightarrow \top$), in a certain category extending both that of topological spaces and that of simplicial sets. Thus it is interesting how to express amalgamation properties in terms of the morphism on the left. The amalgamation properties are somewhat reminiscent of being closed under pullbacks and limits, hence the question.

A class of morphisms defined by a right Quillen lifting property (weak orthogonality) is always closed under pullbacks (limits); under what assumptions will it be closed under pushouts (colimits)? In a model category it makes sense to use fibrant replacement and ask when will it be closed under taking fibrant replacement of pushouts or colimits.

More particularly:

Call a morphism $f$ good iff $f \perp C \vee D \rightarrow \top $ whenever $f \perp C \rightarrow \top $ and $f \perp D \rightarrow \top $, for arbitrary objects $C$ and $D$; here $\top $ denotes the terminal object, and $\perp $ denotes the Quillen lifting property (weak orthogonality), i.e. for morphisms $f$ and $g$ $f \perp g$ says that $f$ has the right Quillen lifting property with respect to $g$, or, in another terminology, $f$ is right weakly orthogonal to $g$.

In a model category, call a morphism $f$ good iff for arbitrary objects $C$ and $D$ $f \perp C \rightarrow \top $ and $f \perp D \rightarrow \top $, implies that
$f \perp (C \vee D)^{(f)} \rightarrow \top $ where $(C \vee D)^{(f)} $ is the fibrant replacement of $C \vee D$, i.e. there is a sequence of morphisms $C \vee D\xrightarrow{(ac)} (C \vee D)^{(f)} \xrightarrow{(f)} \top $ where thet first morphism is an acyclic cofibration, and the second one is a fibration.

Is this a well-known notion? How can one describe good morphisms, either in the category of topological spaces or simplicial sets?

Motivation: in model theory, a number of classes of models with amalgamation property can be described by a lifting property of form as above (i.e. $f \perp M\longrightarrow \top$), in a certain category extending both that of topological spaces and that of simplicial sets. Thus it is interesting how to express amalgamation properties in terms of the morphism on the left. The amalgamation properties are somewhat reminiscent of being closed under pullbacks and limits, hence the question.

A class of morphisms defined by a right Quillen lifting property (weak orthogonality) is always closed under pullbacks (limits); under what assumptions will it be closed under pushouts (colimits)  ? In a model category it makes sense to use fibrant replacement and ask when will it be closed under taking fibrant replacement of pushouts or colimits.

More particularly:

Call a morphism $f$ good iff $f \perp C \vee D \rightarrow \top $ whenever $f \perp C \rightarrow \top $ and $f \perp D \rightarrow \top $, for arbitrary objects $C$ and $D$; here $\top $ denotes the terminal object, and $\perp $ denotes the Quillen lifting property (weak orthogonality), i.e. for morphisms $f$ and $g$ $f \perp g$ says that $f$ has the right Quillen lifting property with respect to $g$, or, in another terminology, $f$ is right weakly orthogonal to $g$.

In a model category, call a morphism $f$ good iff for arbitrary objects $C$ and $D$ $f \perp C \rightarrow \top $ and $f \perp D \rightarrow \top $, implies that
$f \perp (C \vee D)^{(f)} \rightarrow \top $ where $(C \vee D)^{(f)} $ is the fibrant replacement of $C \vee D$, i.e. there is a sequence of morphisms $C \vee D\xrightarrow{(ac)} (C \vee D)^{(f)} \xrightarrow{(f)} \top $ where thet first morphism is an acyclic cofibration, and the second one is a fibration.

Is this a well-known notion  ? How can one describe good morphisms, either in the category of topological spaces or simplicial sets  ?

Motivation: in model theory, a number of classes of models with amalgamation property can be described by a lifting property of form as above (i.e. $f \perp M\longrightarrow \top$), in a certain category extending both that of topological spaces and that of simplicial sets. Thus it is interesting how to express amalgamation properties in terms of the morphism on the left. The amalgamation properties are somewhat reminiscent of being closed under pullbacks and limits, hence the question.

A class of morphisms defined by a right Quillen lifting property (weak orthogonality) is always closed under pullbacks (limits); under what assumptions will it be closed under pushouts (colimits)? In a model category it makes sense to use fibrant replacement and ask when will it be closed under taking fibrant replacement of pushouts or colimits.

More particularly:

Call a morphism $f$ good iff $f \perp C \vee D \rightarrow \top $ whenever $f \perp C \rightarrow \top $ and $f \perp D \rightarrow \top $, for arbitrary objects $C$ and $D$; here $\top $ denotes the terminal object, and $\perp $ denotes the Quillen lifting property (weak orthogonality), i.e. for morphisms $f$ and $g$ $f \perp g$ says that $f$ has the right Quillen lifting property with respect to $g$, or, in another terminology, $f$ is right weakly orthogonal to $g$.

In a model category, call a morphism $f$ good iff for arbitrary objects $C$ and $D$ $f \perp C \rightarrow \top $ and $f \perp D \rightarrow \top $, implies that
$f \perp (C \vee D)^{(f)} \rightarrow \top $ where $(C \vee D)^{(f)} $ is the fibrant replacement of $C \vee D$, i.e. there is a sequence of morphisms $C \vee D\xrightarrow{(ac)} (C \vee D)^{(f)} \xrightarrow{(f)} \top $ where thet first morphism is an acyclic cofibration, and the second one is a fibration.

Is this a well-known notion? How can one describe good morphisms, either in the category of topological spaces or simplicial sets?

Motivation: in model theory, a number of classes of models with amalgamation property can be described by a lifting property of form as above (i.e. $f \perp M\longrightarrow \top$), in a certain category extending both that of topological spaces and that of simplicial sets. Thus it is interesting how to express amalgamation properties in terms of the morphism on the left. The amalgamation properties are somewhat reminiscent of being closed under pullbacks and limits, hence the question.

A class of morphisms defined by a right Quillen lifting property (weak orthogonality) is always closed under pullbacks (limits); under what assumptions will it be closed under pushouts (colimits) ?

More particularly:

Call a morphism $f$ good iff $f \perp C \vee D \rightarrow \top $ whenever $f \perp C \rightarrow \top $ and $f \perp D \rightarrow \top $, for arbitrary objects $C$ and $D$; here $\top $ denotes the terminal object, and $\perp $ denotes the Quillen lifting property (weak orthogonality), i.e. for morphisms $f$ and $g$ $f \perp g$ says that $f$ has the right Quillen lifting property with respect to $g$, or, in another terminology, $f$ is right weakly orthogonal to $g$.

Is this a well-known notion ? How can one describe good morphisms, either in the category of topological spaces or simplicial sets ?

Motivation: in model theory, a number of classes of models with amalgamation property can be described by a lifting property of form as above (i.e. $f \perp M\longrightarrow \top$), in a certain category extending both that of topological spaces and that of simplicial sets. Thus it is interesting how to express amalgamation properties in terms of the morphism on the left. The amalgamation properties are somewhat reminiscent of being closed under pullbacks and limits, hence the question.

A class of morphisms defined by a right Quillen lifting property (weak orthogonality) is always closed under pullbacks (limits); under what assumptions will it be closed under pushouts (colimits) ? In a model category it makes sense to use fibrant replacement and ask when will it be closed under taking fibrant replacement of pushouts or colimits.

More particularly:

Call a morphism $f$ good iff $f \perp C \vee D \rightarrow \top $ whenever $f \perp C \rightarrow \top $ and $f \perp D \rightarrow \top $, for arbitrary objects $C$ and $D$; here $\top $ denotes the terminal object, and $\perp $ denotes the Quillen lifting property (weak orthogonality), i.e. for morphisms $f$ and $g$ $f \perp g$ says that $f$ has the right Quillen lifting property with respect to $g$, or, in another terminology, $f$ is right weakly orthogonal to $g$.

In a model category, call a morphism $f$ good iff for arbitrary objects $C$ and $D$ $f \perp C \rightarrow \top $ and $f \perp D \rightarrow \top $, implies that
$f \perp (C \vee D)^{(f)} \rightarrow \top $ where $(C \vee D)^{(f)} $ is the fibrant replacement of $C \vee D$, i.e. there is a sequence of morphisms $C \vee D\xrightarrow{(ac)} (C \vee D)^{(f)} \xrightarrow{(f)} \top $ where thet first morphism is an acyclic cofibration, and the second one is a fibration.

Is this a well-known notion ? How can one describe good morphisms, either in the category of topological spaces or simplicial sets ?

Motivation: in model theory, a number of classes of models with amalgamation property can be described by a lifting property of form as above (i.e. $f \perp M\longrightarrow \top$), in a certain category extending both that of topological spaces and that of simplicial sets. Thus it is interesting how to express amalgamation properties in terms of the morphism on the left. The amalgamation properties are somewhat reminiscent of being closed under pullbacks and limits, hence the question.