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](https://arxiv.org/abs/0712.0617) 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](https://mathoverflow.net/questions/249589/pushouts-of-injective-monoid-homomorphisms), 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 $Cat\_\omega$.



Are monomorphisms of strict $\omega$-categories stable under pushout along folk cofibrations?
No.


Let $B = \partial \mathbb G\_2 \vee \mathbb G\_2$ be obtained by gluing together the boundary of a 2-globe with a 2-globe, in such a way that the 1-morphisms are composable. Let $D = \mathbb G\_2 \vee \mathbb G\_2 = B \amalg\_{\partial \mathbb G\_2} \mathbb G\_2$ be the result of freely filling in that globe boundary. Let $x$ denote the first atomic 2-cell (the one which was glued on) and let $y$ denote the second atomic 2-cell (the one which is in $B$). Then $y \circ\_0 x$ is the unique 2-cell with its given boundary in $D$.


Let $A \subset B$ be obtained by deleting the atomic 2-cell. Then $A$ still has two nondegenerate 2-cells (in $B$ these were obtained by whiskering $y$ with the two nondegnerate 1-cells composable with it). Now $C := A \amalg\_{\partial \mathbb G\_2} \mathbb G\_2$ has two distinct 2-cells mapping to $y \circ\_0 x$ in $D$. That is, $C \to D$ is not a monomorphism, even though $A \to B$ is.


