nLab
strong monomorphism

Contents

Definition

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

Remarks

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

  • Every regular monomorphism is strong.

  • Every strong monomorphism is extremal; the converse is true if CC 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 YY of a diffeological space XX is again a diffeological space. If smooth, the inclusion ι:YX\iota:Y \rightarrow X is always a monomorphism, but it is a strong monomorphism if and only if YY has “enough” plots, that is if φ:UY\varphi: U\rightarrow Y is a plot if and only if the composite ιφ:UX\iota\varphi: U\rightarrow X is a plot.

Properties

Proposition

pushout of strong monomorphism in quasitopos

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

Suppose that

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

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

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

Then

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

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

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

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

Revised on June 19, 2017 03:26:45 by Urs Schreiber (46.183.103.17)