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 $Cat_\omega$.


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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.


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