# Are monomorphisms of strict $\omega$-categories stable under pushout along folk cofibrations?

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 $$Cat_\omega$$.

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.