* Re: Semigroups with many objects (fwd)
@ 2005-11-27 23:38 J=FCrgen Koslowski
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From: J=FCrgen Koslowski @ 2005-11-27 23:38 UTC (permalink / raw)
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[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)
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2005-11-27 23:38 Semigroups with many objects (fwd) J=FCrgen Koslowski
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