Dual of `lift'?
Angus Rodgers
help feeling that it has a simple and familiar
answer which I've unaccountably forgotten. If
so, let me apologise in advance.)
This definition is paraphrased from:
Eric W. Weisstein. "Lift." From MathWorld--A
Wolfram Web Resource.
http://mathworld.wolfram.com/Lift.html
In certain categories --- and presumably, by
extension, in an arbitrary category, although
in the general context there may be another term
for it which I've forgotten --- given two arrows
f: A --> C and g: B --> C, a `lift' is simply an
arrow h: A --> B such that f = gh.
Dually, one may be given arrows f: A --> C and
h: A --> B, and want to know if there exists an
arrow g: B --> C such that f = gh. Is there a
dual term for this? (Don't say "co-lift"!)
--
Angus Rodgers
(angus_prune@ eats spam; reply to angusrod@)
Contains mild peril
Robin Chapman
>
> In certain categories --- and presumably, by
> extension, in an arbitrary category, although
> in the general context there may be another term
> for it which I've forgotten --- given two arrows
> f: A --> C and g: B --> C, a `lift' is simply an
> arrow h: A --> B such that f = gh.
>
> Dually, one may be given arrows f: A --> C and
> h: A --> B, and want to know if there exists an
> arrow g: B --> C such that f = gh. Is there a
> dual term for this? (Don't say "co-lift"!)
How about "extension"?
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9"
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
Angus Rodgers
<r...@ivorynospamtower.freeserve.co.uk> wrote:
>> In certain categories --- and presumably, by
>> extension, in an arbitrary category, although
>> in the general context there may be another term
>> for it which I've forgotten --- given two arrows
>> f: A --> C and g: B --> C, a `lift' is simply an
>> arrow h: A --> B such that f = gh.
>>
>> Dually, one may be given arrows f: A --> C and
>> h: A --> B, and want to know if there exists an
>> arrow g: B --> C such that f = gh. Is there a
>> dual term for this? (Don't say "co-lift"!)
>
>
>How about "extension"?
I hadn't thought of that, and it might work (it's
also quite amusing that it's an "extension" of the
existing usage), but I fear that its connotation
would be confusing in the situation I had in mind,
where the category is concrete and the function h
is surjective, but not usually injective.
(To be even more specific, I was thinking of the
definition of $\sum_{x \in S} q(x)$, where S is
a finite set, and q: S --> T is a function into
a commutative monoid written additively.
The general associative law defines an extension -
in the usual sense - of q to finite sequences of
monoid elements, and commutativity then <grunt>s
this to a function of unordered collections with
repetition ("bags" - ugh!), and a function on a
finite set determines such a collection.
This is indeed an extension of q, but it would
sound odd, to my ears, to call it an extension
of the function f induced by q on sequences.)
In such a case, to say that f "extends" to a map
of B would almost inescapably suggest that A is
somehow being embedded in B by h, which it isn't
except in trivial cases.
If a neologism is required, I suppose one might
try "drop"? It would sound OK to me to say "f
drops to a mapping of ...", but unfortunately,
a bit odd to describe g as a "dropping" of f!
Possibly "descend"? f descends to g? g is a
descent of f ...? (Semantically a bit mangled,
I fear.)
The search continues ...
Maybe "reduce"? f reduces to g; g is a reduction
of f? Sounds OK, I think. (Got it from Roget's
Thesaurus, I must confess!)
Todd Trimble
>On Mon, 10 Jan 2005 08:56:47 +0000, Robin Chapman
><r...@ivorynospamtower.freeserve.co.uk> wrote:
>
>>> In certain categories --- and presumably, by
>>> extension, in an arbitrary category, although
>>> in the general context there may be another term
>>> for it which I've forgotten --- given two arrows
>>> f: A --> C and g: B --> C, a `lift' is simply an
>>> arrow h: A --> B such that f = gh.
>>>
>>> Dually, one may be given arrows f: A --> C and
>>> h: A --> B, and want to know if there exists an
>>> arrow g: B --> C such that f = gh. Is there a
>>> dual term for this? (Don't say "co-lift"!)
>>
>>
>>How about "extension"?
>
>I hadn't thought of that, and it might work (it's
>also quite amusing that it's an "extension" of the
>existing usage), but I fear that its connotation
>would be confusing in the situation I had in mind,
>where the category is concrete and the function h
>is surjective, but not usually injective.
Actually, "extension" is the standard name for this
categorical concept. Doesn't matter what the nature
of h is.
> [...]
Todd Trimble
Angus Rodgers
trim...@optonline.net (Todd Trimble) wrote:
>On 10 Jan 2005, Angus Rodgers wrote:
>>On Mon, 10 Jan 2005 08:56:47 +0000, Robin Chapman
>><r...@ivorynospamtower.freeserve.co.uk> wrote:
[I wrote, and then deleted my own attribution!:]
>>>> [...]
>>>> Dually, one may be given arrows f: A --> C and
>>>> h: A --> B, and want to know if there exists an
>>>> arrow g: B --> C such that f = gh. Is there a
>>>> dual term for this? (Don't say "co-lift"!)
>>>How about "extension"?
>>I hadn't thought of that, and it might work [...]
>Actually, "extension" is the standard name for this
>categorical concept. Doesn't matter what the nature
>of h is.
I can't rid myself of a feeling of discomfort about
this. If I were to write, for example, that every
group homomorphism f: G --> H "extends" [uniquely],
or has a [unique] "extension", to a homomorphism
G/Ker(f) --> H, would professional readers accept
this as normal, or would they think that I was
confused? If the usage is "standard", how come
I can't find a definition of it anywhere?
I *really* hope that I'm not just being obtuse, and
I hesitated a long time before posting this, but the
feeling of discomfort wouldn't go away. This is the
first time I've experienced such uncertainty about
the meaning of a mathematical term, and I'm not even
sure how to pose the question, but, loosely speaking:
*where* is the definition?
Is it in papers read only by specialists in category
theory --- in which case, surely even professional
mathematicians in other fields might find the usage
confusing, especially as the term doesn't seem to be
defined in standard textbooks on category theory to
which they might naturally turn --- or is there some
more accessible source for it?
Finally, perhaps someone reading this might create
a Wikipedia entry for the term, because at present
there isn't one.
Marc Olschok
I am not sure, but perhaps also the term 'lifting' was
originally applied to surjective maps only. This would
not repair your discomfort, but at least might dualize it.
If pressed, you could use 'f factors through h' in the above
(this works also in the dual situation).
>
> Is it in papers read only by specialists in category
> theory --- in which case, surely even professional
> mathematicians in other fields might find the usage
> confusing, especially as the term doesn't seem to be
> defined in standard textbooks on category theory to
> which they might naturally turn --- or is there some
> more accessible source for it?
It should be in some of the textbooks, but I have nothing at
hand right now.
Marc
Todd Trimble
I confess that I'm having trouble finding a citation for
this usage. Elsewhere specialists in category theory speak of
left and right [Kan] extensions and [Kan] lifts in 2-categories
(roughly, categories in which the "homs" are not just sets
but categories, and in which composition is functorial).
Such Kan extensions specialize to "extensions" in the
present sense when the 2-category is just an ordinary
category (i.e., if the homs are just sets seen as discrete
categories). Maybe that was what was in my head when I
asserted this is standard usage -- that and related situations
in homotopy theory (where for instance one speaks of the
homotopy extension problem as dual to the homotopy lifting
problem).
Otherwise I understand your discomfort. Dually, one might be
uncomfortable using the word "lift" to describe a situation
where one has a map f: A --> C that one wants to factor
through an injective or monic map h: B --> C. With that
in mind, would you be any more comfortable with "factoring"
or "factorization"? E.g. a factoring of f: A --> C through
h: A --> B is an arrow g: B --> C such that f = gh. Not
precisely as a term for the dual of lift, but rather as a
catch-all term for both lifts and extensions.
>
>I *really* hope that I'm not just being obtuse, and
>I hesitated a long time before posting this, but the
>feeling of discomfort wouldn't go away. This is the
>first time I've experienced such uncertainty about
>the meaning of a mathematical term, and I'm not even
>sure how to pose the question, but, loosely speaking:
>*where* is the definition?
>
>Is it in papers read only by specialists in category
>theory --- in which case, surely even professional
>mathematicians in other fields might find the usage
>confusing, especially as the term doesn't seem to be
>defined in standard textbooks on category theory to
>which they might naturally turn --- or is there some
>more accessible source for it?
>
>Finally, perhaps someone reading this might create
>a Wikipedia entry for the term, because at present
>there isn't one.
>--
>Angus Rodgers
>(angus_prune@ eats spam; reply to angusrod@)
>Contains mild peril
Todd Trimble
Shmuel (Seymour J.) Metz
at 02:43 PM, trim...@optonline.net (Todd Trimble) said:
>I confess that I'm having trouble finding a citation for this usage.
Do they ever use the nomenclature "pullback?"
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
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Todd Trimble
>In <200501121416...@proapp.mathforum.org>, on 01/12/2005
> at 02:43 PM, trim...@optonline.net (Todd Trimble) said:
>
>>I confess that I'm having trouble finding a citation for this usage.
>
>Do they ever use the nomenclature "pullback?"
>
Eh? Lots of people use the word "pullback", often in reference
to a certain sort of finite categorical limit, and sometimes in
reference to precomposing with an arrow (for example, one can
pull back a subset U of Y along a function f: X --> Y to get a
subset of X, or equivalently pull back a characteristic function
u: Y --> {0, 1} to a function uf: X --> {0, 1} ). But "pullback"
as a term for "dual to lift"??
>--
>Shmuel (Seymour J.) Metz, SysProg and JOAT <<a href="http://patriot.net/~shmuel">http://patriot.net/~shmuel</a>>
>
>Unsolicited bulk E-mail subject to legal action. I reserve the
>right to publicly post or ridicule any abusive E-mail. Reply to
>domain Patriot dot net user shmuel+news to contact me. Do not
>reply to spam...@library.lspace.org
Todd Trimble
Angus Rodgers
trim...@optonline.net (Todd Trimble) wrote:
>[...] Angus Rodgers wrote:
>>>>>> Dually, one may be given arrows f: A --> C and
>>>>>> h: A --> B, and want to know if there exists an
>>>>>> arrow g: B --> C such that f = gh. Is there a
>>>>>> dual term for this? (Don't say "co-lift"!)
>>>Actually, "extension" is the standard name for this
>>>categorical concept. Doesn't matter what the nature
>>>of h is.
>>[...]
>I confess that I'm having trouble finding a citation for
>this usage. Elsewhere specialists in category theory speak of
>left and right [Kan] extensions and [Kan] lifts in 2-categories
>(roughly, categories in which the "homs" are not just sets
>but categories, and in which composition is functorial).
>Such Kan extensions specialize to "extensions" in the
>present sense when the 2-category is just an ordinary
>category (i.e., if the homs are just sets seen as discrete
>categories). Maybe that was what was in my head when I
>asserted this is standard usage -- that and related situations
>in homotopy theory (where for instance one speaks of the
>homotopy extension problem as dual to the homotopy lifting
>problem).
>
>Otherwise I understand your discomfort. Dually, one might be
>uncomfortable using the word "lift" to describe a situation
>where one has a map f: A --> C that one wants to factor
>through an injective or monic map h: B --> C.
Any discomfort I might feel would be attributable to my lack
of familiarity with the relevant mathematics! I'm familiar
with the idea of defining the winding number of a path in C
by `lifting' the argument (considered as function into the
unit circle) to a function into R; and that's about all.
But from what I can see, it looks as if:
Kan extensions in ordinary categories are extensions in the
usual narrow sense; lifting of homotopies, paths, and other
continuous maps is defined relative to a surjective mapping
of a covering space; lifting of homomorphisms of projective
modules is defined relative to a surjective homomorphism of
modules.
With the possible exception of 2-category theory, it doesn't
look as if `lifts' are ever considered except in relation to
epimorphisms, or `extensions' except in relation to mono-
morphisms; so my discomfort might be widely shared.
(One curious exception: Dummit & Foote, "Abstract Algebra",
2nd ed., p.367, uses `extends' as a synonym for `lifts'!
Shome mishtake, shurely?)
In most concrete situations, consideration of the problem
of factorisation through a given mapping p: X --> Y would
probably first factorise p itself into an epi q: X --> X'
and a mono r: X' --> Y, thus dividing the problem into two
halves, one of which has a simple criterion for solution
in terms of either kernels (in the case of "extensions")
or images (in the case of "lifts"), leaving the usual case
(extension/mono, lift/epi) as the tough part.
Perhaps in some more abstract situation such factorisations
are unavailable (or not even well defined), and the problem
has to be handled all in one gulp?
>With that
>in mind, would you be any more comfortable with "factoring"
>or "factorization"? E.g. a factoring of f: A --> C through
>h: A --> B is an arrow g: B --> C such that f = gh. Not
>precisely as a term for the dual of lift, but rather as a
>catch-all term for both lifts and extensions.
That's probably the only way not to risk being misunderstood.
But I still think that it's a pity not to have a nice use of
language dual to the way that the word `lift' can be used.
Angus Rodgers
>But I still think that it's a pity not to have a nice use of
>language dual to the way that the word `lift' can be used.
Damn, I'm following up one of my own posts again - sorry!
But this sentiment was so inappropriate, in context, that
I'd like to retract it immediately, before getting rightly
flamed.
I'd just gone to some trouble to observe that `lift' seems
normally to be used in relation to a surjective function,
and 'extension' normally relates to an injective function:
so the uses of these words already *are* dual - in *every*
respect, including their limitations.
The MathWorld definition of `lift' now seems to involve a
generalisation of normal usage dual to Robin's suggestion
for `extension'.
Although I'm uncomfortable with Robin's suggestion of this,
er, extension of the use of a familiar term, I can't claim
that there is any existing usage of `lift' dual to what I
wanted. My initial question was based on a false premise.
I think I'm becoming obsessed! But don't blame me, blame
MathWorld. :)