Subject: categories: Re: Semigroups with many objects

From: [email-masked]
Date: Thu, 24 Nov 2005 22:34:08 EST

In a message dated 11/24/2005 2:20:54 PM Central Standard Time,
[email-masked] 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

---

From: Miles Gould <[email-masked]>
Date: Fri, 25 Nov 2005 17:51:45 +0000

> 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

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