Re: Re: Semigroups with many objects

[Philippe Gaucher <gaucher@pps.jussieu.fr>]

Le vendredi 25 Novembre 2005 04:56, duraid@octopus.com.au a écrit :
> > 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 ?
>
> Koslowski calls these "taxonomies", see e.g. "Monads and interpolads in
> bicategories" (TAC vol 3, no 8 (1997)).
>
>      Duraid

Dear all,

I call a "small semigroup with many objects enriched over the model category 
of compactly generated topological spaces" a "flow" in my work (these objects 
are interesting for me only if they are enriched over very particular model 
categories satisfying particular properties). The terminology comes from the 
fact that I use them to study the time flow of a higher dimensional automaton 
(up to directed homotopy). 

For "taxonomy", I would be very curious to know the origin of the terminology. 
What does it mean exactly ?

In the paper q-alg/9608025 "Flexible sheaves", Carlos Simpson calls a "(not 
necessarily small) semigroup with many objects enriched over the category of 
topological spaces" a continuous semicategory. 

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.

The word "non-unital category" is also used sometime in mathematical papers. 

pg.