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
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
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
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
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 ----------------------------------------------------------------
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.
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
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.
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 -- ----------------------------------------------------------------------------- Lutz Schroeder Phone +49-421-218-4683 Dept. of Computer Science Fax +49-421-218-3054 University of Bremen lschrode@informatik.uni-bremen.de P.O.Box 330440, D-28334 Bremen http://www.informatik.uni-bremen.de/~lschrode -----------------------------------------------------------------------------
participants (7)
-
duraid＠octopus.com.au -
Joachim Kock -
Lutz Schroeder -
Miles Gould -
Peter Freyd -
Philippe Gaucher -
Topos8＠aol.com