Subject: Re: categories: Re: Semigroups with many objects

From: Philippe Gaucher <[email-masked]>
Date: Sat, 26 Nov 2005 05:30:06 +0100

Le vendredi 25 Novembre 2005 04:56, [email-masked] 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.

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From: Lutz Schroeder <[email-masked]>
Date: Wed, 30 Nov 2005 16:51:03 +0100

Dear all,

> thing I remember is that that was a computer-scientific work.

That would have been my paper with Paulo Mateus "Universal aspects of
probabilistic automata" in MSCS (and also "Monads on composition graphs"
in APCS). We do indeed use the word "precategory" for strucures with
identities, and with a partially defined composition law satisfying the
identity laws (strongly) and the associative law in the sense that
f(gh)=(fg)h holds strongly (or Kleene) provided that both gh and fg are
defined.

Moreover, as pointed out in a previous message, I have used the word
"semicategory" for similar structures, but with a stronger associative
law, requiring that f(gh)=(fg)h are both defined whenever fg and gh are
defined (or slight variations of this). Ehresmann used the term
"multiplicative graph" (and also sometimes "neocategory", I believe) for
structures satisfying the identity law, with no associativity imposed at
all.

-- Lutz

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