[I answered this on Friday, but haven't seen my reply so far.] =3D20
Is there an accepted terminology for semigroups with many objects, i.= =3D e. gadgets that satisfy the all the axioms satisfied by categories excep= =3D ting those which refer to identities ? =3D20 Koslowski calls these "taxonomies", see e.g. "Monads and interpolads in bicategories" (TAC vol 3, no 8 (1997)).
Well, not quite. In a taxonomy the identity axiom is not simply removed, but replaced by a weaker requirement. Essentially this says that every morphism factors. In the corresponding semigroups, which I would call "interpolative", this can also be formulated as follows: the multiplication * : S x S ---> S is a coequalizer of S x * and * x S . (This formulation can be lifted to taxonomies as well.) It would be interesting to know, whether the gadget that prompted this question is a taxonomy in this sense. [By the way, the term "taxonomy" resulted from a misunderstanding on my part of a remark by Robert Pare and Richard Wood. It has since been abbreviated to "taxon".] The notion of "category without units" also shows up in Azumaya=3D20 theory as studied by Francis Borceux and others. -- J=3DFCrgen --=3D20 Juergen Koslowski If I don't see you no more on this world ITI, TU Braunschweig I'll meet you on the next one koslowj@iti.cs.tu-bs.de and don't be late! http://www.iti.cs.tu-bs.de/~koslowj Jimi Hendrix (Voodoo Child, SR)