Re: semi-categories

[wlawvere@buffalo.edu]

Perhaps it has not been sufficiently emphasized that semi-categories and
the like are not really "generalizations" of categories (though formally
they may appear so). Actually they present possibly-useful SPECIAL classes
of categories. That is because we represent one ultimately in an actual
large category (such as sets or abelian groups) and those representations
are indeed representations of a certain ordinary (V-) category, namely the
one freely generated by the given semicategory. The forgetful 2-functor
has a left adjoint, just as does the one from categories to directed
graphs etc. To be a value of such a left adjoint means that the large
category of representations may have special properties, for example it
may unite by a bicontinuous quotient p a pair of subcategories i, j whose
domains are identical but where i, j are themselves opposite in that they
are the respective adjoints to the same p. This is the kind of UIAO that
Francis refers to.

Is there a convincing example showing that it can be useful mathematically
to treat operator ideals (such as compact, nuclear, etc) as
semicategories?

I always believed that Jacobson invented rngs because algebraic practice
(not the dreaded categorists) had convinced him to grudgingly conclude
that after all ideals in rings are ideals but not subrings, whereas the
opposite view is not a convenience but a confusion which denies ideals
their dignity.

Bill Lawvere

Quoting Francis Borceux <borceux@math.ucl.ac.be>:

>
> 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=3Dm
> ... but what about the case when R does not have a unit ?
>
> Simply dropping the axiom m1=3Dm 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
>       (@=3Dtensor product sign)
>
> which is equivalent to the axiom m1=3Dm, 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 (=3Dthe 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
> constructi> on
>
>       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
> D=E9partement de Math=E9matique
> Universit=E9 Catholique de Louvain
> 2 chemin du Cyclotron
> 1348 Louvain-la-Neuve (Belgique)
> t=E9l. +32(0)10473170, fax. +32(0)10472530
> borceux@math.ucl.ac.be
>
>
>
>