Is there an accepted terminology for semigroups with many objects, i.e. gadgets that satisfy the all the axioms satisfied by categories excepting those which refer to identities ?
Perhaps 'semi-category' is the most widely used term. The word 'taxonomy' has also been used (Paré, Wood, Ageron), but Koslowski has used that word for something a bit more complicated ('interpolads in SPAN'). On the other hand, Schroeder has used the word 'semi-category' for the 'multiplicative graphs' of Ehresmann (some structure where composition of arrows is not always defined even if their source and target match). (Curiously, in a preliminary version of the paper by Moens, Berni-Canani, and Borceux, 'On regular presheaves and regular semi-categories', the term 'multiplicative graph' was used for 'semi-category' -- the final version uses 'semi-category'.) I would also like to advogate 'semi-monoid' instead of 'semi-group', and 'semi-monoidal category' for 'monoidal category without unit'. It seems to be too late at this point to convince operadists to say 'semi-operad' for operads without unit. In the same spirit I find it convenient to use 'semi-simplicial set' for presheaves on Delta-mono, but I am told that this is confusing, since apparently 'semi-simplicial set' meant something else fifty years ago... Cheers, Joachim. ---------------------------------------------------------------- Joachim Kock <kock@mat.uab.es> Departament de Matemàtiques -- Universitat Autònoma de Barcelona Edifici C -- 08193 Bellaterra (Barcelona) -- ESPANYA Phone: +34 93 581 25 34 Fax: +34 93 581 27 90 ----------------------------------------------------------------