From: Francis Borceux <borceux@math.ucl.ac.be>
To: categories@mta.ca
Subject: semi-categories
Date: Tue, 29 Nov 2005 14:55:58 +0100 [thread overview]
Message-ID: <p06020402bfb20185ec95@[130.104.3.128]> (raw)
There has been a long discussion on the list
about "categories without identities", whatever
you decide to call them. And the attention has
been brought to axioms which could -- in this
more general context -- replace the identity
axiom.
I would like to focus on a very striking categorical aspect of this problem.
A (right) module M on a ring R with unit must satisfy the axiom m1=m
... but what about the case when R does not have a unit ?
Simply dropping the axiom m1=m leaves you with
the unpleasant situation where you have two
different notions of module, in the case where R
has a unit.
Therefore people working in linear algebra have considered the axiom
the scalar multiplication M@R ---> M is an isomorphism
(@=tensor product sign)
which is equivalent to the axiom m1=m, when the
ring has a unit ... but makes perfect sense when
the ring does not have a unit. Such modules are
generally called "Taylor regular".
A ring R with unit is simply a one-objet additive
category and a right module M on R is simply an
additive presheaf M ---> Ab (=the category of
abelian groups).
A ring without unit is thus a "one-object
additive category without identity", again
whatever you decide to call this.
But what is the analogue of the axiom
M@R ---> M is an isomorphism
when R is now an arbitrary small (enriched)
"category without identities" and M is an
arbitrary (enriched) presheaf on it ?
All of us know that to define a (co)limit, we do
not need at all to start with an indexing
category: an arbitrary graph with arbitrary
commutativity conditions works perfectly well. In
particular, a "category without identities" is
all right. And the same holds in the enriched
case, with (co)limits replaced by "weighted
(co)limits".
Now every presheaf on a small category is
canonically a colimit of representable ones ...
but this result depends heavily on the existence
of identities ! When you work with a presheaf M
on a "category R without identities", you still
have a canonical morphism
canonical colimit of representables ---> R
and you can call M "Taylor regular" when this is
an isomorphism. Again in the enriched case,
"colimit" means "weighted colimit". This
recaptures exactly the case of "Taylor regular
modules", when working with Ab-enriched
categories.
A sensible axiom to put on a "category R without
identities" is the fact that the representable
functors are "Taylor regular". (We should
certainly call this something else than "Taylor
regular", but let me keep this terminology in
this message.)
And when R is a "Taylor regular category without identities", the construction
presheaf on R |---> corresponding canonical colimit of representables
yields a reflection for the inclusion of Taylor
regular presheaves in all presheaves.
A very striking property is the existence of a
further (necessarily full and faithful) left
adjoint to this reflection. This second inclusion
provides in fact an equivalence with the full
subcategory of those presheaves which satisfy the
Yoneda isomorphism.
This yields thus a nice example of what Bill
Lawvere calls the "unity of opposites": the two
inclusions identify the category of Taylor
regular presheaves with
* on one side, those presheaves which are colimits of representables;
* on the other side, those presheaves which satisfy the Yoneda lemma.
This underlines the pertinence of these "Taylor
regular categories without identities".
To my knowledge, the best treatment of these
questions is to be found in various papers by
Marie-Anne Moens and by Isar Stubbe, in
particular in the "Cahiers" and in "TAC".
And very interesting examples occur in functional
analysis (the identity on a Hilbert space is a
compact operator ... if and only if the space is
finite dimensional) and also in the theory of
quantales.
Francis Borceux
--
Francis BORCEUX
Département de Mathématique
Université Catholique de Louvain
2 chemin du Cyclotron
1348 Louvain-la-Neuve (Belgique)
tél. +32(0)10473170, fax. +32(0)10472530
borceux@math.ucl.ac.be
next reply other threads:[~2005-11-29 13:55 UTC|newest]
Thread overview: 3+ messages / expand[flat|nested] mbox.gz Atom feed top
2005-11-29 13:55 Francis Borceux [this message]
2005-11-30 22:24 ` semi-categories wlawvere
2005-12-02 12:25 ` semi-categories Philippe Gaucher
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