Joe English wrote:
Suppose you have two morphisms f : A -> B and g : B -> A such that neither (f . g) nor (g . f) is the identity, but satisfying (f . g . f) = f. Is there a conventional name for this? Alternately, same question, but f and g are functors and A and B categories.
In some cases (g . f . g) is also equal to g; is there a name for this as well?
I believe there isn't really a standard name for this, as evidenced by the following. In Mac Lane's "Categories for the Working Mathematician", p 21 of 1st or 2nd edn, in an exercise he defines "an arrow f:a ->b in a category C is _regular_ when there exists an arrow g: b -> a such that f g f = f". But this usage is highly non-standard; in standard usage there are regular epimorphisms (and regular categories defined in terms of them) but they're rather more involved. I think I've seen it said that f is a quasi-inverse of g (or is it the other way round?), but I can't find a reference. -- Michael Ackerman