From: Rob Goldblatt <Rob.Goldblatt@mcs.vuw.ac.nz>
To: categories@mta.ca
Subject: Re: Laws
Date: Wed, 9 Aug 2006 14:24:01 +1200 [thread overview]
Message-ID: <E1GAud2-00021R-PN@mailserv.mta.ca> (raw)
One aspect of the categorisation of "equation" is the approach of
Banaschewski and Herrlich. This replaces an equation as a pair (s,t)
of terms that belong to some free algebra F by the least quotient
(coequaliser)
e: F -->> F/E
identifying s and t.
An algebra A satisfies equation (s,t) precisely when every
homomorphism F --> A lifts across e to a homomorphism F/E --> A.
Thus the notion of an equation becomes that of a regular epi e with
free domain, and an object satisfies such an e when it is injective
for e.
To obtain a notion intrinsic to a given category, the free domains
were replaced by domains that are regular-projective, i.e. projective
for all regular epis, this being a property enjoyed by free algebras
in categories of universal algebras.
The approach has been dualised in the coalgebra literature, with a
"coequation" being defined as a regular mono with regular-injective
codomain, and a "covariety" as the class of coalgebras that are
projective for some given class of coequations.
Some (not all) relevant references are below.
cheers,
Rob
@Article{
bana:subc76,
author = "B. Banaschewski and H. Herrlich",
title = "Subcategories Defined by Implications",
journal = "Houston Journal of Mathematics",
year = "1976",
volume = "2",
number = "2",
pages = "149--171"
}
@Article{
adam:vari03,
author = {Ji{\v{r}}{\'\i} Ad{\'a}mek and Hans-E. Porst},
title = "On Varieties and Covarieties in a Category",
journal = "Mathematical Structures in Computer Science",
year = "2003",
volume = {13},
pages = "201-232"
}
@Techreport{
awod:coal00,
author = "Steve Awodey and Jesse Hughes",
title = "The Coalgebraic Dual of {B}irkhoff's Variety
Theorem",
institution = "Department of Philosophy, Carnegie Mellon
University",
year = "2000",
number = "CMU-PHIL-109",
note = "\url{http://phiwumbda.org/~jesse/papers/
index.html}"
}
On 8/08/2006, at 1:36 AM, Tom Leinster wrote:
> Dear category theorists,
>
> Here's something that I don't understand. People sometimes talk about
> algebraic structures "satisfying laws". E.g. let's take groups.
> Being
> abelian is a law; it says that the equation xy = yx holds. A group G
> "satisfies no laws" if
>
> whenever X is a set and w, w' are distinct elements of the free
> group F(X) on X, there exists a homomorphism f: F(X) ---> G
> such that f(w) and f(w') are distinct.
>
> For example, an abelian group cannot satisfy no laws, since you
> could take
> X = {x, y}, w = xy, and w' = yx. There are various interesting
> examples
> of groups that satisfy no laws.
>
> To be rather concrete about it, you could define a "law satisfied
> by G" to
> be a triple (X, w, w') consisting of a set X and elements w, w' of F
> (X),
> such that every homomorphism F(X) ---> G sends w and w' to the same
> thing.
> A law is "trivial" if w = w'. Then "satisfies no laws" means
> "satisfies
> only trivial laws".
>
> You could then say: given a group G, consider the groups that
> satisfy all
> the laws satisfied by G. (E.g. if G is abelian then all such
> groups will
> be abelian.) This is going to be a new algebraic theory.
>
> What bothers me is that I feel there must be some categorical story
> I'm
> missing here. Everything above is very concrete; for instance, it's
> heavily set-based. What's known about all this? In particular,
> what's
> known about the process described in the previous paragraph,
> whereby any
> theory T and T-algebra G give rise to a new theory?
>
> Thanks,
> Tom
>
>
>
>
next reply other threads:[~2006-08-09 2:24 UTC|newest]
Thread overview: 12+ messages / expand[flat|nested] mbox.gz Atom feed top
2006-08-09 2:24 Rob Goldblatt [this message]
-- strict thread matches above, loose matches on Subject: below --
2006-08-25 8:49 Laws Jiri Adamek
2006-08-12 15:37 Laws F W Lawvere
2006-08-11 23:18 Laws Tom Leinster
2006-08-10 0:19 Laws George Janelidze
2006-08-08 23:31 Laws Jon Cohen
2006-08-08 19:19 Laws F W Lawvere
2006-08-08 11:28 Laws George Janelidze
2006-08-08 8:38 Laws Prof. Peter Johnstone
2006-08-08 6:30 Laws flinton
2006-08-08 5:08 Laws Peter Selinger
2006-08-07 13:36 Laws Tom Leinster
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=E1GAud2-00021R-PN@mailserv.mta.ca \
--to=rob.goldblatt@mcs.vuw.ac.nz \
--cc=categories@mta.ca \
/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).