#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 |
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Comment (by nthiery):
From the primer:
{{{
We have seen above that, for example, the category of groups is
considered by Sage as a subcategory of the category of sets. For
category purists, this is not quite correct: namely, a group is not a
set; instead, one can recover the underlying set by forgetting the
multiplicative structure. However, it would be impractical to have to
explicitly forget the multiplicative structure whenever one wanted to
apply on a group an operation defined on sets. In object oriented
parlance, we really want the relation "a group *is a* set", so that
groups can inherit code implemented on sets.
Therefore, in Sage, as well as in most systems with a similar category
framework, we use this slightly abusive definition of subcategory:
A category ``Ds()`` is a *subcategory* of the category ``Cs()`` if, up
to implicitly applying the appropriate forgetful functor, every object
of ``Ds()`` is an object of ``Cs()``. Reciprocally, ``Cs()`` is in
this case a *super category* of ``Ds()``.
}}}
This is the correct hierarchy relation that we want for organizing the
code.
This hierarchy relation has been called "subcategory" in Sage since
2007, following the other systems with a category mechanism (Axiom,
MuPAD, ... and actually GAP as well). If you believe that the implicit
application of the forgetful functor makes it abusive to call this
"subcategory", or if you think we *also* want to model the "pure
subcategories relation", we can debate the pros and cons. But this
belongs to a separate new ticket.
Cheers,
Nicolas
--
Ticket URL: <http://trac.sagemath.org/ticket/10963#comment:625>
Sage <http://www.sagemath.org>
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