#10963: Axioms and more functorial constructions
-------------------------------------+-------------------------------------
Reporter: nthiery | Owner: stumpc5
Type: enhancement | Status: needs_info
Priority: major | Milestone: sage-6.2
Component: categories | Resolution:
Keywords: days54 | Merged in:
Authors: Nicolas M. Thiéry | Reviewers: Simon King, Frédéric
Report Upstream: N/A | Chapoton
Branch: | Work issues: merge with #15801
public/ticket/10963-doc- | once things stabilize
distributive | Commit:
Dependencies: #11224, #8327, | ce2193e9d6f179d2d51812c6af002697ccfbaa8c
#10193, #12895, #14516, #14722, | Stopgaps:
#13589, #14471, #15069, #15094, |
#11688, #13394, #15150, #15506, |
#15757, #15759, #15919 |
-------------------------------------+-------------------------------------
Comment (by pbruin):
Replying to [comment:620 SimonKing]:
> Replying to [comment:619 pbruin]:
> > I strongly support your opinion. An example that annoys me a lot
(already present before this ticket) is
> > {{{
> > sage: C = Sets(); D = Groups()
> > sage: D.is_subcategory(C)
> > True # expected False
> > }}}
> > In my humble opinion, either this should return `False`, or the name
`is_subcategory()` should be changed so that it does not go against the
established meaning of the notion of subcategory.
>
> I don't get what you mean. Are you arguing that groups aren't
necessarily sets? Sure, all what we need is an object G together with a
(inversion) morphism G->G and a (multiplication) morphism GxG->G.
In some sense I am indeed saying that groups are not necessarily sets, but
not in the way you interpret it. I insist that "a group is not a set" in
the sense that a group is not ''just'' a set; it is a set ''together with
additional structure'' (namely a multiplication map satisfying certain
axioms implying existence of identity and inverse).
The fact that this extra structure is part of the definition entails that
the category '''Groups''' of groups is not a subcategory of the category
'''Sets''', in the sense that the functor sending a group to its
underlying set (and a group homomorphism to itself viewed as a map on
sets) does ''not'' give '''Groups''' the structure of a subcategory of
'''Sets'''.
In fact, part of the definition of "D is a subcategory of C" is that the
objects of D form a subclass of the objects of C. This property is not
satisfied for '''Groups''' and '''Sets''' under the forgetful functor.
For example, if ''S'' = {''a'', ''b''} is your favourite set of two
elements (say ''a'' = {} and ''b'' = {{}}), then you can make ''S'' into a
group in exactly two ways, namely by endowing it either with the
multiplication table
{{{
| a b
-------
a| a b
b| b a
}}}
or with the multiplication table
{{{
| a b
-------
a| b a
b| a b
}}}
This means that there are exactly ''two'' objects of '''Groups'''
corresponding to the object ''S'' of '''Sets''' (i.e. whose underlying set
is ''S'').
(As an aside, it is probably also true that you can realise '''Groups'''
as a subcategory of '''Sets''' as follows. Consider a group ''G'' given
by a set ''S'' and a multiplication map ''m'': ''S'' x ''S'' -> ''S''.
Then ''m'' can be identified with a set, namely the set of all ordered
triples (''a'', ''b'', ''c'') in ''S'' x ''S'' x ''S'' such that
''m''(''a'', ''b'') = ''c''. It seems to me that associating to ''G'' the
set ''m'' (not ''S'' as you might think) does realise '''Groups''' to a
subcategory of '''Sets'''; however, this is certainly not what is normally
meant by saying that "groups are sets".)
In short, "groups are sets" is only true in the loose sense that there is
a forgetful functor '''Groups''' -> '''Sets''' (''G'' -> ''S'' in the
above notation), not that this functor is a subcategory relation.
--
Ticket URL: <http://trac.sagemath.org/ticket/10963#comment:621>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/sage-trac.
For more options, visit https://groups.google.com/d/optout.
