Subject: categories: Semigroups with many objects

From: [email-masked]
Date: Thu, 24 Nov 2005 10:55:17 EST

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

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From: [email-masked]
Date: Fri, 25 Nov 2005 14:56:24 +1100 (EST)

> which refer to identities ?

Koslowski calls these "taxonomies", see e.g. "Monads and interpolads in
bicategories" (TAC vol 3, no 8 (1997)).

     Duraid

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From: Joachim Kock <[email-masked]>
Date: Fri, 25 Nov 2005 22:24:37 +0100

>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.

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Joachim Kock <[email-masked]>
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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