Re: Is there a name for this structure?

[dominic . j . steinitz]

The name escapes me but any category theory book should supply the answer.

Dominic.




Tom Pledger <[EMAIL PROTECTED]>@haskell.org on 27/03/2002 04:22:29

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Subject:  Is there a name for this structure?


Joe English writes:
 :
 | 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?

Is it equivalent to saying that (f . g) is the identity on the range
of f?  That's shorter, though still not a snappy single word term.
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Re: Is there a name for this structure?

[Michael Ackerman]

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
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Is there a name for this structure?

[Tom Pledger]

Joe English writes:
 :
 | 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?

Is it equivalent to saying that (f . g) is the identity on the range
of f?  That's shorter, though still not a snappy single word term.
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Is there a name for this structure?

[Joe English]

Not really a Haskell question, but someone here might know the answer...

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 find myself running into pairs of functions with this property
over and over again, and am looking for a short way to describe
the property...

Thanks,


--Joe English

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