From: Toby Bartels <toby+categories@ugcs.caltech.edu>
To: categories <categories@mta.ca>
Cc: David Leduc <david.leduc6@googlemail.com>
Subject: Re: Freyd Categories
Date: Sat, 25 Sep 2010 18:48:03 -0300 [thread overview]
Message-ID: <E1OzcbD-0000BQ-Ru@mlist.mta.ca> (raw)
In-Reply-To: <E1Oz9g8-0003nH-U0@mlist.mta.ca>
David Leduc wrote:
>A Freyd category is essentially a functor.
>So why is it called a category?!
As I understand it, one says a Freyd category on C,
where C is the source/domain of the functor.
So a Freyd category on C is a category K (with certain structure),
together with a functor C -> K (with certain properties).
More simply, a Freyd category on C is a category K
together with certain extra stuff.
Calling that whole business (a category together with ...) a "category"
is an abuse of language akin to terms like "partially ordered set";
a partially ordered set is really a set together with certain structure.
There may be a better reason why Power and Thielecke used this terminology,
but if so, they don't explain it in the papers that I have found
(but I have not been able to read their first paper on the subject,
Environments, Continuation Semantics and Indexed Categories,
so maybe there is an explanation there).
>More generally, is there a way to see a functor as being a category?
More generally, a functor is a pair of categories equipped with extra stuff,
which is trivial, but I don't think that there's anything deeper than that.
You could do something like the trick that encodes a function as a set
(as everything must be) in the foundation of material set theory:
a function f: X -> Y is the set {(a,b) in X x Y | f(a) = b}.
But this set is isomorphic, in the cateory of sets, to X itself,
so really we need this set together with maps from it to X and Y.
Similarly, think of a functor F: X -> Y as the category
with {(a,b) in Ob X x Ob Y | F(a) = b} as set of objects
and Hom((a,b), (c,d)) := {(f,g) in X(a,c) x Y(b,d) | F(f) = g},
with the obvious composition; but again, this is isomorphic to X,
so we really need it together with functors from it to X and Y.
So I don't think that this really helps anything.
--Toby
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
prev parent reply other threads:[~2010-09-25 21:48 UTC|newest]
Thread overview: 3+ messages / expand[flat|nested] mbox.gz Atom feed top
2010-09-23 11:01 David Leduc
2010-09-25 19:27 ` Paul Levy
2010-09-25 21:48 ` Toby Bartels [this message]
Reply instructions:
You may reply publicly to this message via plain-text email
using any one of the following methods:
* Save the following mbox file, import it into your mail client,
and reply-to-all from there: mbox
Avoid top-posting and favor interleaved quoting:
https://en.wikipedia.org/wiki/Posting_style#Interleaved_style
* Reply using the --to, --cc, and --in-reply-to
switches of git-send-email(1):
git send-email \
--in-reply-to=E1OzcbD-0000BQ-Ru@mlist.mta.ca \
--to=toby+categories@ugcs.caltech.edu \
--cc=categories@mta.ca \
--cc=david.leduc6@googlemail.com \
/path/to/YOUR_REPLY
https://kernel.org/pub/software/scm/git/docs/git-send-email.html
* If your mail client supports setting the In-Reply-To header
via mailto: links, try the mailto: link
Be sure your reply has a Subject: header at the top and a blank line
before the message body.
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox;
as well as URLs for NNTP newsgroup(s).