nLab
strong monomorphism

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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.

  • If CC 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 CC is co-regular.

  • 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.

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

    s i,t i:ii+1 s_i, t_i : i \to i+1

    subject to the relations

    ss=st=ts=tt. s\circ s = s\circ t = t\circ s = t\circ t.

    Then the only epimorphisms and monomorphisms in CC 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 (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.

Last revised on August 26, 2019 at 21:24:59. See the history of this page for a list of all contributions to it.