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
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
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.
participants (3)
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Francis Borceux -
Philippe Gaucher -
wlawvere＠buffalo.edu