nLab jointly monic morphisms

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Definition

Given a category CC and objects a,bOb(C)a, b \in Ob(C), a pair of morphisms f,gC(a,b)f, g \in C(a, b) is jointly monic if for every object c:Ob(C)c: Ob(C) and pair of morphisms h,kC(c,a)h, k \in C(c, a), fh=fkf \circ h = f \circ k and gh=gkg \circ h = g \circ k imply that h=kh = k.

In a well-pointed category CC, given objects a,bOb(C)a, b \in Ob(C), a pair of morphisms f,gC(a,b)f, g \in C(a, b) is jointly injective if for every global element h,kC(1,a)h, k \in C(1, a), fh=fkf \circ h = f \circ k and gh=gkg \circ h = g \circ k imply that h=kh = k.

More generally, we can consider a jointly monic family of morphisms, where we indexed over a set II. When II is a singleton set, this reduces to a monomorphism, and when II is a two-element set, this reduces to a jointly monic pair. (And similarly for joint injectivity.)

In tablular allegories

In every tabular allegory, a relation RR could be factored into jointly monic maps ff and gg such that f g=Rf^\dagger \circ g = R.

Examples

  • Taking C=SetC = Set, and F i:XY iI{ F_i : X \to Y }_{i \in I} to be a family of functors, joint monicity of (F i) x,x:X(x,x)Y(F ix,F ix)(F_i)_{x, x'} : X(x, x') \to Y(F_i x, F_i x') is the notion of joint faithfulness (which specialises to the notion of faithful functor when II is a singleton).

See also

References

Last revised on October 18, 2023 at 11:53:20. See the history of this page for a list of all contributions to it.