Dear all,
I had also seen the word "precategory" but I cannot remember where. Beware of the fact that the word precategory is also used for categories *with identities* such that the composition law is partially defined : that is the fact that the codomain of F is equal to the domain of G is not sufficient for GoF to exist. Once again, I cannot remember where I read this word. The only thing I remember is that that was a computer-scientific work.
That would have been my paper with Paulo Mateus "Universal aspects of probabilistic automata" in MSCS (and also "Monads on composition graphs" in APCS). We do indeed use the word "precategory" for strucures with identities, and with a partially defined composition law satisfying the identity laws (strongly) and the associative law in the sense that f(gh)=(fg)h holds strongly (or Kleene) provided that both gh and fg are defined. Moreover, as pointed out in a previous message, I have used the word "semicategory" for similar structures, but with a stronger associative law, requiring that f(gh)=(fg)h are both defined whenever fg and gh are defined (or slight variations of this). Ehresmann used the term "multiplicative graph" (and also sometimes "neocategory", I believe) for structures satisfying the identity law, with no associativity imposed at all. -- Lutz -- ----------------------------------------------------------------------------- Lutz Schroeder Phone +49-421-218-4683 Dept. of Computer Science Fax +49-421-218-3054 University of Bremen lschrode@informatik.uni-bremen.de P.O.Box 330440, D-28334 Bremen http://www.informatik.uni-bremen.de/~lschrode -----------------------------------------------------------------------------