# When is a right lifting property closed under pushouts?

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.

TL;DR : I'm not sure!

(In the following, I basically elide the difference between (weak) factorization systems and classes of morphisms defined by (weak) lifting properties, which is often harmless, e.g. if one works in a locally presentable category and throws in a small-generation assumption. Basically whereever I say "(weak) factorization system", feel free to substitute "dual classes of morphisms defined by (weak) lifting properties".)

I think there are relatively few interesting full-blown model categories where the fibrations or acyclic fibrations are closed under cobase-change (although upon reflection, the projective model structure on chain complexes of $$R$$-modules does have the property that the fibrations (=levelwise surjections) are closed under cobase change, for any ring $$R$$ -- so perhaps I'm mistaken!). I think it's fruitful to consider the weaker version of the question, asking for weak factorization systems whose right class is closed under cobase-change.

A strengthening of the dual situation has a name: a modality is defined to be an orthogonal factorization system whose left class is closed under base change. Many interesting factorization systems are modalities, and in fact some of them are very naturally thought about in terms of their duals. For instance, formally etale maps of (affine, say) schemes are basically the left half of a modality; the dual statement is that formally etale maps of rings are the right half of a factorization system which is closed under cobase-change. I have the sense that many of the factorization systems appearing in algebraic geometry follow a similar pattern.

I find this phenomenon mysterious and interesting. For another example, consider the category $$Set$$, which admits a factorization system (epi,mono), but also a weak factorization system (mono, epi). This tells us that from the perspective of the (epi,mono) factorization system, the epis are unexpectedly closed under base change / the monos are unexpectedly closed under cobase-change, and vice versa from the perspective of the (mono,epi) weak factorization system. I asked a question about this "interlocking factorization system" phenomenon here but didn't arrive at anything conclusive.

In the context of that question, I did spend some time thinking about the following version of your question: given a class of morphisms $$\mathcal M$$ defined by a weak right-lifting property with respect to a set of morphisms $$I$$, what conditions on $$I$$ will ensure that $$\mathcal M$$ is closed under cobase-change / coproducts / transfinite composition, etc. I arrived at only a few fragmentary observations, such as:

• If we work in a stable $$\infty$$-category, and if the codomains of the morphisms of $$I$$ are all the zero object $$0$$, then $$\mathcal M$$ is closed under cobase-change.
• Here's another special case which may be related to "goodness": Suppose that we work in a category with disjoint coproducts, and suppose that the codomains $j$ of the morphsims $i \to j$ of $I$ are all connected (i.e. $Hom(j,-)$ preserves coproducts). Then $\mathcal M$ is closed under coproducts. Jul 22, 2021 at 15:32
• Thanks, it'll take time to read. the factorisation system (mono,epi), any "submorphism" of a mono is a mono. May a condition like this on I be hepful to show that M is based under co-base change ? Something like this is true in the examples from model theory I care about. (But not quite like this) Jul 23, 2021 at 13:37