User:Lethe/Range (mathematics)

from https://en.wikipedia.org/w/index.php?title=Range_(mathematics)&oldid=45394451

Category theory
In category theory, the range of a morphism can be characterized entirely in terms of morphisms. The range of a morphism f is the largest monomorphism over which f factors. More explicitly, it is an object R and a monomorphism i:A→Y such that the diagram commutes; there exists a morphism r:X→R with f=ir (note that the monic character of i implies that if f factors over i, it does so uniquely). Moreover the morphism i must be universal for this diagram; for any other R′ and monomorphism over which f factors (i.e. any other monomorphism i′ such that f=i′r′ for some r′), then there is a unique morphism u:R→R′ such that the diagram commutes. As is usual with universal properties, the range is then unique up to isomorphism. Subobjects are defined in terms of a preorder of monomorphisms: one monomorphism is greater than another if the first factors over the second, and two monomorphisms determine the same subobject if one is both greater than and less than the other. In these terms, the range is simply the largest monomorphism over which f factors, and the largest monomorphism should be though of as corresponding to the smallest subobject of the codomain; the preorder of inclusion maps for subsets is the dual of the containment preorder.

In a concrete category such as the category of sets or the category of groups, the range of a function agrees with the set-theoretic definition. In a preorder viewed as a category, the range of a morphism x ≤ y is simply x itself.