In a message dated 11/24/2005 2:20:54 PM Central Standard Time, P.B.Levy@cs.bham.ac.uk writes: Dear Carl,
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 ?
Do you have any examples of such things? I'd be interested to know. Paul The principal examples I know are all related to the category of Moore paths in a topological space X . It turns out to be convenient not to have any degenerate paths when making certain constructions, especially when working with the higher dimensional versions of these gadgets. This can be arranged as follows. Consider the set of non-identity paths of the Moore category. Define the domain (respectively, the codomain) of a path f to be the path of UNIT length that is constantly f ( 0 ) (resp., constantly f ( 1 ) ). Two paths f , g, can be composed if codomain g = domain f and the composite is then the concatenation ( f followed by g ). This is a semigroup with many objects, i.e. a directed graph with an associative composition law. The multiplication is stricly associative but there are no identities or even any idempotents. I don't like the term "semigroupoids" because it evokes (for me) the notion of invertibility which I want to avoid. I know Anders Kock has suggested "fair categories" to describe category-like objects in which there are identities unique only "up to homotopy", but semigroups with many objects don't have any identities at all. The term "near category" occurred to me but I seem to recall this being used to describe something else and I can't put my finger on that reference. Of course, when the day comes that "higher dimensional algebra" is just "algebra" maybe semigroups with many objects will just be called semigroups ( and semigroups with one object called a proper semigroups ?), groupoids will be called groups (and groups with one object called proper groups ?) and categories will be called monoids (and monoids with one object called proper monoids?). Carl