A bimorphism is a morphism which is simultenously epic and mono. Such a morphism in general does not need to be iso. If every bimorphism in a category is an isomorphism we say that the category is balanced.
Mike: I don’t like the word “bimorphism.” If I’ve been thinking about bicategories then it sounds like something that is “a morphism up to isomorphism,” while if I’ve been thinking about biproducts it sounds like something that is “both a morphism and a comorphism.” Are monic epics really an important enough concept to deserve its own word? Can’t we just say “monic and epic” or even just “monic epic”?
Toby: It makes sense to me. Using ‘bi’ in bicategories is bad anyway. But it’s much like ‘bi’ in ‘biproduct’, although I suppose technically one ought to say ‘bimonomorphism’. You're right that they may not be that important; I see them mostly as a way to say give counterexamples to the naÃ¯ve student’s guess that all categories are balanced. But then, are balanced categories an important concept?
Mike: Of course, “bi” in bicategories is bad, but because of people who are used to “bi-” meaning “weak” I think it is better to avoid “bi-” altogether whenever possible. I wouldn’t object too strongly to “bimonomorphism.” But “monic epic” is the same number of syllables as “bimorphism,” fewer syllables than “bimonomorphism,” and its meaning is (to me) more obvious than either.
I don’t think that balanced categories are a very important object of study in their own right, but it is sometimes useful to know that a particular category, or a particular type of category (such as a pretopos), is balanced.
Toby: As it doesn't matter too much, we can avoid ‘bimorphism’ here. But surely you realise that it's cheating to compare ‘monic epic’ and ‘bimonomorphism’; you can compare the adjectives ‘monic epic’ and ‘bimonic’ or compare the nouns ‘monic epimorphism’ and ‘bimonomorphism’ (or compare their abbreviations ‘monic epi’ and ‘bimono’ if you like those, although I tend not to use those in writing).
Mike: Yes, of course you’re right for the syllable count. I still maintain that the meaning of “monic epic” is more obvious than either “bimorphism” or “bimonic,” which is the more important thing, especially for a little-used concept.
Mike: I just remembered another reason I dislike “bimorphism.” One context where people use it is in the study of reflective subcategories, where one talks about “mono-reflective” and “epi-reflective” subcategories to mean that the unit of the reflection is monic or epic, and some people say “bireflective” to mean that the unit is a bimorphism. But to me, bireflective “obviously” means “reflective and coreflective;” this confused me for quite a whole once.
Toby: O yeah, I agree with you there! The best that I can say for ‘bimorphism’ is that it's got some sense to it; but as the right way to follow that sense would be ‘bimonomorphism’, there's still something wrong with it.
Zoran I am used to say “reflective and coreflective”, never bireflective. Looks esoteric term. I never likes bimorphism either.
Todd: Late to this discussion, but I agree with Mike and Zoran that ‘bimorphism’ is terrible. There is no indication of “monicity” or “epicity”.
Qiaochu: Super late to this discussion, but the term I use is “fake isomorphism.” What do you guys think?