5
$\begingroup$

Is there a special name for a morphism $f : X \to Y$ in a category which doesn't factor through any proper subobject of $Y$? In other words, we have $\mathrm{im}(f)=Y$ for the image of $f$.

This property is precisely what makes an epimorphism an extremal epimorphism. But the morphism $f$ is not necessarily an epimorphism. You can prove this when the category has equalizers, but not in general.

Ideas for a name in case no name exists in the literature (this is not part of my question since it is opinion-based, but I wanted to share these thoughts nevertheless to provide some context and examples):

  • extremal morphism. Sufficient for my needs but is confusing when applied to extremal monomorphisms. We cannot change history but coextremal epimorphism would actually be a better name for what we now call extremal epimorphism, and then coextremal morphism would probably be the best answer.
  • surjective morphism. This is because in every algebraic category it is precisely a surjective homomorphism. Also the equation $\mathrm{im}(f)=Y$ looks very much like surjectivity. On the other hand, one has to be careful with non-algebraic categories, see below.
  • dense morphism. This fits better to the example of the category of complete metric spaces, since the categorical image is the closure of the set-theoretical image.

The suggestion in the title (extremal non-epimorphism) is not really serious, it is not usable.

Improve this question
$\endgroup$
8
  • 5
    $\begingroup$ One cannot change history, but I think it would be reasonable to introduce the terminology "coextremal morphism" along with the explanation you have given. "Extremal monomorphism" appears to have been introduced in Isbell's Subobjects, adequacy, completeness and categories of algebras, and the term "extremal epimorphism" in Sonner's Canonical categories. I think it is reasonable to suggest that Sonner dualised the terminology inappropriately. $\endgroup$
    – varkor
    Commented Oct 26 at 8:24
  • 3
    $\begingroup$ I prefer to reserve “surjection” for a pullback-stable property, since it is supposed to be local on the codomain and a morphism could satisfy the extremality condition “by accident” if there are not enough subobjects. $\endgroup$
    – Zhen Lin
    Commented Oct 26 at 11:18
  • $\begingroup$ @varkor I disagree that extremal epimorphisms should be called "coextremal". They are extremal in exactly the same way that extremal monomorphisms are. They don't have the dual property to being extremal. The Arctic is in the extreme North of the planet, but the Antarctic is not in the coextreme South; it's in the extreme South. $\endgroup$
    – Jeremy Rickard
    Commented Oct 28 at 18:03
  • $\begingroup$ @JeremyRickard I get your point. But this is not handled consistently. We have the notion of a cocommutative coalgebra for example. $\endgroup$
    – Martin Brandenburg
    Commented Oct 28 at 18:10
  • 1
    $\begingroup$ @JeremyRickard: it is a common convention in category theory to dualise the adjectives as well as the nouns, e.g. cofiltered limits and filtered colimits. This is typically useful, rather than being redundant, for precisely the reason mentioned in the question: when considering generalisations, one may need to drop one of the words, but still wants to maintain a distinction between the two concepts. $\endgroup$
    – varkor
    Commented Oct 28 at 19:01

1 Answer 1

Reset to default
7
$\begingroup$

In Sketches of an Elephant this kind of morphism is called a cover (section A1.3).

In case you don't like that, here are some other possibilities (although I don't know that I actually like any of them):

Improve this answer
$\endgroup$
4
  • 1
    $\begingroup$ Thanks a lot, Mike! I will stick with "cover" in my paper then (even though I prefer "coextremal morphism" since it also dualizes well) because that is already used in print, and I don't want to introduce more confusion by creating a new name for the same thing. Even though the term "cover" for this kind of morphism is not used by many others (yet). $\endgroup$
    – Martin Brandenburg
    Commented Oct 26 at 20:39
  • $\begingroup$ At some point I think the Elephant also dualizes "cover" to "cocover", possibly when talking about quasitoposes. $\endgroup$
    – Mike Shulman
    Commented Oct 27 at 23:06
  • $\begingroup$ He missed the opportunity to talk about vers. (See also: mpact spaces, ntinuous function, ...) $\endgroup$
    – Martin Brandenburg
    Commented Oct 28 at 0:36
  • $\begingroup$ @MartinBrandenburg To say nothing of nes. $\endgroup$
    – Mike Shulman
    Commented Oct 29 at 1:12

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .