re: semigroups with many objects

re: semigroups with many objects

[Peter Freyd]

Jacobson coined "rng" for a ring without identity and Bill returned
the favor by proposing "rig" for a ring without negation (at least
Bill's proposal can be pronounced). Alas, "catgory" is the only
approximation for the instant case -- the only one, that is, if you
refuse to count "ctegory" (category without automorphisms).

Seriously though, "semi-category" is the one proposal not needing
explanation. May I suggest that its very obviousness is why it was
avoided.

Re: Semigroups with many objects

[Lutz Schroeder]

Dear all,

> 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.

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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Re: Semigroups with many objects

[Topos8]

Well, it does seem that "semigroupoids" is the preferred terminology.
Searching on this term through the math xxx archive pulls up a number of papers
which study various kinds of C star algebras built on top of  semigroupoids.
The idea is to use the objects of the semigroupoid to index  a basis in a
(separable) Hilbert space and to use the arrows to define partial  isometries of
this Hilbert space in the obvious way. The algebra closure in the  weak operator
topology then defines the "semigroupoid" C star algbera. Of course  this
algebra does contain one idempotent for each object, but this is a  consequence of
taking the algebra- closure of the set of patial isometries  defined by the
arrows.

Carl Futia

Re: Semigroups with many objects

[Joachim Kock]

>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 ?

Perhaps 'semi-category' is the most widely used term. 

The word 'taxonomy' has also been used (Paré, Wood, Ageron), but
Koslowski has used that word for something a bit more complicated
('interpolads in SPAN').

On the other hand, Schroeder has used the word 'semi-category'
for the 'multiplicative graphs' of Ehresmann (some structure where
composition of arrows is not always defined even if their source
and target match).  (Curiously, in a preliminary version of the paper
by Moens, Berni-Canani, and Borceux, 'On regular presheaves and
regular semi-categories', the term 'multiplicative graph' was used
for 'semi-category' -- the final version uses 'semi-category'.)


I would also like to advogate 'semi-monoid' instead of 'semi-group',
and 'semi-monoidal category' for 'monoidal category without unit'.
It seems to be too late at this point to convince operadists to say
'semi-operad' for operads without unit.

In the same spirit I find it convenient to use 'semi-simplicial set'
for presheaves on Delta-mono, but I am told that this is confusing,
since apparently 'semi-simplicial set' meant something else fifty
years ago...

Cheers,
Joachim.

----------------------------------------------------------------
Joachim Kock <kock@mat.uab.es>
Departament de Matemàtiques -- Universitat Autònoma de Barcelona
Edifici C -- 08193 Bellaterra (Barcelona) -- ESPANYA
Phone: +34 93 581 25 34        Fax: +34 93 581 27 90
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Re: Semigroups with many objects

[Miles Gould]

On Thu, Nov 24, 2005 at 10:34:08PM -0500, Topos8@aol.com wrote:
> I don't like the term "semigroupoids" because it evokes (for me) the notion
> of invertibility which I want to avoid.

Google's not a perfect metric for popularity, but it returns about 350
hits for "semigroupoid", about 10 for "fair category", and none for
"near category" (the one hit it returns is spurious). Wikipedia has an
entry for "semigroupoid" (with the definition you're thinking of) and
nothing on any of the others. Looking at MathSciNet, we find 180 hits
for "semigroupoid", none for "fair category", and one for "near
category". It's worse than that, though, because that paper uses "near
category" to mean something different, namely a category-like object
with identities but without associativity!

All this suggests to me that "semigroupoid" is the standard term, and
certainly it's the only one I've ever heard before. I don't think you
need to worry about implied invertibility: if you know what both a
groupoid and a semigroup are, the term "semigroupoid" strongly suggests
a multi-object structure with associatively-composable arrows, but not
necessarily with identities. At least, it suggests that to me :-)

Hope that helps,

Miles

-- 
If you want to see your plays performed the way you wrote them,
become President.
  -- Vaclav Havel

Re: Semigroups with many objects

[duraid]

> 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

Re: Semigroups with many objects

[Topos8]

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

Semigroups with many objects

[Topos8]

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 ?

Thanks

Carl Futia