# Is there a name for this structure?

**Michael Ackerman
**
ack@nethere.com

*Tue, 26 Mar 2002 23:29:26 -0800*

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