* semi-categories
@ 2005-11-29 13:55 Francis Borceux
2005-11-30 22:24 ` semi-categories wlawvere
0 siblings, 1 reply; 3+ messages in thread
From: Francis Borceux @ 2005-11-29 13:55 UTC (permalink / raw)
To: categories
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
^ permalink raw reply [flat|nested] 3+ messages in thread
* Re: semi-categories
2005-11-29 13:55 semi-categories Francis Borceux
@ 2005-11-30 22:24 ` wlawvere
2005-12-02 12:25 ` semi-categories Philippe Gaucher
0 siblings, 1 reply; 3+ messages in thread
From: wlawvere @ 2005-11-30 22:24 UTC (permalink / raw)
To: categories
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
>
>
>
>
^ permalink raw reply [flat|nested] 3+ messages in thread
* Re: semi-categories
2005-11-30 22:24 ` semi-categories wlawvere
@ 2005-12-02 12:25 ` Philippe Gaucher
0 siblings, 0 replies; 3+ messages in thread
From: Philippe Gaucher @ 2005-12-02 12:25 UTC (permalink / raw)
To: categories
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.
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