nLab strong monomorphism

Definition

A strong monomorphism in a category $C$ is a monomorphism which is right orthogonal to any epimorphism. The dual notion is, of course, strong epimorphism.

Remarks

• If $C$ has coequalizers, then any morphism which is right orthogonal to epimorphisms must automatically be a monomorphism.

• If $C$ has kernel pairs and coequalizers of kernel pairs, then any morphism which is right orthogonal to epimorphisms must automatically be a monomorphism.

• Every regular monomorphism is strong. The converse is true if $C$ is co-regular.

• Every strong monomorphism is extremal; the converse is true if $C$ has pushouts.

Examples

• A nice example of strong monomorphisms in a category are the subspace inclusions in the category of diffeological spaces. In this setting, any subset $Y$ of a diffeological space $X$ is again a diffeological space. If smooth, the inclusion $\iota:Y \rightarrow X$ is always a monomorphism, but it is a strong monomorphism if and only if $Y$ has “enough” plots, that is if $\varphi: U\rightarrow Y$ is a plot if and only if the composite $\iota\varphi: U\rightarrow X$ is a plot.

• Let $C$ be the category whose objects are the integers $\mathbf{Z}$, and whose morphisms are generated from arrows

  $$s_i, t_i : i \to i+1$$

  subject to the relations

  $$s\circ s = s\circ t = t\circ s = t\circ t.$$

  Then the only epimorphisms and monomorphisms in $C$ are the identities, thus every map is right orthogonal to all epimorphisms but only the identities are strong monomorphisms.

Properties

Proposition

pushout of strong monomorphism in quasitopos

Suppose that $(\mathrm{T},\mathcal{C})$ is either

• (monomorphism,topos), or
• (strong monomorphism,quasitopos)

Suppose that

$$\array{ O_{0,1} & \to & O_{1,1} \\ \downarrow m &&\downarrow h \\ O_{0,0} & \to & O_{1,0} }$$

is a commutative diagram in $\mathcal{C}$ such that

• • $m$ is $\mathrm{T}$ in $\mathcal{C}$
• • the diagram is a pushout in $\mathcal{C}$

Then

• • $h$ is $\mathrm{T}$ in $\mathcal{C}$
• • the diagram is a pullback in $\mathcal{C}$

See at quasitopos this lemma. Note that the result for quasitoposes immediately implies the result for toposes, since all monomorphisms $i: A \to B$ in a topos are regular ($i$ being the equalizer of the arrows $\chi_i, t \circ !: B \to \Omega$ in

$$\array{ & & 1 \\ & \mathllap{!} \nearrow & \downarrow \mathrlap{t} \\ B & \underset{\chi_i}{\to} & \Omega }$$

where $\chi_i$ is the classifying map of $i$) and therefore strong.