Subject: categories: Re: Semigroups with many objects (fwd)

From: J=FCrgen Koslowski <[email-masked]>
Date: Mon, 28 Nov 2005 00:38:02 +0100 (CET)

[I answered this on Friday, but haven't seen my reply so far.]
=20
> > Is there an accepted terminology for semigroups with many objects, i.=
e.
> > gadgets that satisfy the all the axioms satisfied by categories excep=
ting
> > those
> > which refer to identities ?
>=20
> 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=20
theory as studied by Francis Borceux and others.

-- J=FCrgen

--=20
Juergen Koslowski               If I don't see you no more on this world
ITI, TU Braunschweig               I'll meet you on the next one
[email-masked]               and don't be late!
http://www.iti.cs.tu-bs.de/~koslowj      Jimi Hendrix (Voodoo Child, SR)