semi-categories

semi-categories

[Francis Borceux]

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

Re: semi-categories

[wlawvere]

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
>
>
>
>

Re: semi-categories

[Philippe Gaucher]

Le mercredi 30 Novembre 2005 23:24, vous avez écrit :
> 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). 

Indeed, at least in my case, a flow must not be viewed as a generalization of 
the notion of small categories. Let me explain a little bit what I am doing 
with these objects. I was not very explicit in my previous post. And so the 
terminology I use is not in "competition".

I want to model HDA, at least those coming from precubical sets. I use a set 
of states X^0 and between each state A and B of the HDA, there is a 
topological space P_{A,B}X whose elements represent the non-constant 
execution paths from A to B. The topology of this space models the 
concurrency of the situation between A and B. And execution paths can be 
composed with a strictly asssociative law. There does not necessarily exist a 
loop from a given state A to itself : so P_{A,A}X can be empty. This fact is 
one reason among several other ones why I remove the identity maps.

Inside this model, I am able to define what is a dihomotopy equivalence. The 
main problem to define dihomotopy is that some contractible parts of "the 
directed spaces of execution paths" must not be contracted. Otherwise in the 
categorical localization, the relevant geometric information is lost. In 
particular, initial and final states must be unchanged by a dihomotopy 
equivalence. A very simple example : take two execution paths going from one 
initial state 0 to one final state 1. If contractions in the direction of time 
are allowed, one finds in the same equivalence class a loop. Some examples of 
unwanted final states are deadlocks of concurrent systems : a deadlock is 
nothing else but a final state from a geometric viewpoint. Flows allow to 
propose a solution of this problem : in fact I introduced this notion of 
flows on purpose, to make the following solution work. 

The first kind of dihomotopy equivalence is a morphism f:X->Y such that 
f^0:X^0->Y^0 is a bijection and such that Pf:PX->PY is a weak homotopy 
equivalence. It turns out that there is a model structure on Flows whose weak 
equivalences are exactly the preceding kind of morphisms. By imposing the 
condition f^0:X^0->Y^0 bijective, we do not take any risk : nothing is 
contracted in the direction of time. So no geometric information is lost. But 
this kind of identification is too rigide ! The second kind of dihomotopy 
equivalence is generated by taking a representative set of inclusions of 
posets P1\subset P2, where P1 and P2 are finite bounded posets and where the 
inclusions preserve the bottom  element and the top element (which are 
different by hypothesis in a bounded poset). For example, the inclusion of 
posets {0<1}\subset{0<A<1} represents the directed segment (going from the 
initial state 0 to the final state 1) identified with the composition of two 
directed segments. This second kind of dihomotopy equivalence models 
"refinement of observation". Of course, initial and final states are still 
preserved.

The category of flows up to dihomotopy equivalences is between the homotopy 
category of the model structure associated to the first kind of dihomotopy 
equivalence and the homotopy category of the Bousfield localization of the 
same model structures with respect to the set of Q(P_1\subset P_2) where Q is 
the cofibrant replacement functor. I call the weak equivalences of the 
Bousfield localization "quasi-dihomotopy". Morally speaking, quasi-dihomotopy 
is like dihomotopy except in non-observable areas of the time flow where the 
topological configuration can changed.

pg.