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