Let $f : A \to B$ be a monomorphism of strict $\omega$-categories, and let $d : \partial \mathbb G_n \to A$ be an attaching map. There is an induced map $g : A \cup_{\partial \mathbb G_n} \mathbb G_n \to B \cup_{\partial \mathbb G_n} \mathbb G_n$. Here the pushout is taken in strict $\omega$-categories.

**Question:** Is $g$ a monomorphism?

Since folk cofibrations are monomorphisms, this is the case when $f$ (and hence $g$) is a folk cofibration, at least. I suspect the answer is *yes* in general though.

Here a *folk cofibration* is a map which is a retract of transfinite composites of cobase-changes of maps of the form $\partial \mathbb G_n \to \mathbb G_n$, i.e. a cofibration in the folk model structure on strict $\omega$-categories. $\mathbb G_n$ denotes the $n$-globe, and $\partial \mathbb G_n$ denotes its boundary. The title question is equivalent to the question as I've stated it above because folk cofibrations are generated by maps of the form $\partial \mathbb G_n \to \mathbb G_n$ under transfinite composition and retracts, and monomorphisms are stable under transfinite composition and retracts.

**EDIT:** Monomorphisms of strict $\omega$-categories (or even of strict 1-categories) are not stable under pushout along an arbitrary map. For example, push out the monomorphism $B\mathbb N \to B\mathbb Z$ along the map $B\mathbb N \to B(\mathbb N / (2=1))$, and you get the map $B \mathbb Z \to \ast$ (in other words, an invertible idempotent is automatically an identity), which is not a monomorphism.

According to this MO question, the pushout of one injective monoid homomorphism along another injective monoid homomorphism need not be injective. Since monoids are 1-object categories (and the inclusion preserves pushouts), this means that the pushout of a monomorphism of 1-categories along a monomorphism of 1-categories need not be a monomorphism. The inclusion of 1-categories into $\omega$-categories preserves pushouts, so this is also the case in $\omega$-categories. Thus the retreat I've made in my question to pushing out a monomorphism along a folk cofibration seems warranted. But it would be nice to have an *explicit* example illustrating that the pushout of a monomorphism along a monomorphism need not be a monomorphism in $Cat_\omega$.