A colleague of mine is wondering if anyone has studied "partial categories," by which she means directed graphs with identities but with only some compositions (including all identity compositions) defined. A partial category can be thought of as a category enriched in pointed sets (with smash product as tensor and S^0 as unit). The slogan is that the basepoint in each hom-set stands in for "does not exist". But enriched functors don't give the right notion of maps; these should preserve identities and all specified compositions. Enriched functors behave appropriately with regards to the identites but may "forget" extant arrows and in particular need not preserve composites. So perhaps this perspective is not useful. I'll happily pass along any suggestions. Thanks, Emily Riehl [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I have a reference question. Who first coined the term ``chaotic category'' for a groupoid with a unique morphism between each pair of object, and in what context? It is a ridiculously elementary concept, but one that is extremely useful in work on equivariant bundle theory that is needed for equivariant infinite loop space theory and equivariant algebraic K-theory. Peter May [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Why not use the term `indiscrete groupoid' for the functor that gives a right adjoint to the functor Ob: Groupoids \to Sets? The left adjoint is then of course the `discrete groupoid'. This agrees with the terminology for discrete and indiscrete topologies. I confess to have used different terminology in various places. Of course one use of these notions is to show that the functor Ob preserves limits and colimits, which is a start on constructing them. It is not surprising that this concept occurs widely. In groupoids there is a notion of covering morphism and the universal cover of a group G is of course an indiscrete groupoid G'; this groupoid is by no means `trivial' since it comes equipped with a covering morphism p: G' \to G. This approach to covering space theory is given in my book `Topology and groupoids'. Ronnie Ronnie On 29/09/2011 02:35, Peter May wrote:
I have a reference question. Who first coined the term ``chaotic category'' for a groupoid with a unique morphism between each pair of object, and in what context? It is a ridiculously elementary concept, but one that is extremely useful in work on equivariant bundle theory that is needed for equivariant infinite loop space theory and equivariant algebraic K-theory.
Peter May
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Thu, 29 Sep 2011 14:41:03 +0100, Ronnie Brown <ronnie.profbrown@btinternet.com> wrote:
Why not use the term `indiscrete groupoid' for the functor that gives a right adjoint to the functor Ob: Groupoids \to Sets? The left adjoint is then of course the `discrete groupoid'. This agrees with the terminology for discrete and indiscrete topologies.
I have also heard the indiscrete topology refered to as the chaotic topology. Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Peter From memory, Bourbaki uses the term "chaotic" for the right adjoint to the forgetful Top --> Set. Since the left adjoint assigns the discrete topology, John Kelley used the term "indiscrete" (as a bit of a joke I thought; maybe because it indiscreetly defies separation properties) for the right adjoint. It is natural to use the same terms (discrete, and chaotic or indiscrete) for the adjoint to ob : Cat --> Set. I consider it one of those cultural things: the French use "chaotic", the Americans use "indiscrete", and I am happy with and have used both with a slight preference for "chaotic". Who used it first for categories would be hard to trace since the Top / Cat analogy runs long and deep. Regards, Ross On 29/09/2011, at 11:35 AM, Peter May wrote:
I have a reference question. Who first coined the term ``chaotic category'' for a groupoid with a unique morphism between each pair of object, and in what context? It is a ridiculously elementary concept, but one that is extremely useful in work on equivariant bundle theory that is needed for equivariant infinite loop space theory and equivariant algebraic K-theory.
Peter May
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
The use of a lot of terminologies stemming from topology for describing purely algebraic properties seems to be widespread and fashionable among an important part of the community of categorists. This may be a convenient source of intuition and analogies by giving a topological or geometrical fragrance to such algebraic concepts. However the considerable drawback is that this habit is a source of unsolvable clashes for people who are currently using topological or more specially Lie groupoids, more generally structured (in Ehresmann's sense), i. e. internal, groupoids, who are obliged to create alternative terminologies. For the special case of the duet discrete/undiscrete (or indiscrete, or sometimes coarse) I'm personally using presently null/banal (there are a lot of different terminologies used by various authors). (As to the term "chaotic", I prefer to avoid comments, being afraid to perturb the beautifully non chaotic weather we are presently enjoying in our region). Jean Pradines ----- Message d'origine ----- De : "Ronnie Brown" <ronnie.profbrown@btinternet.com> À : "Peter May" <may@math.uchicago.edu> Cc : <categories@mta.ca> Envoyé : jeudi 29 septembre 2011 15:41 Objet : categories: Re: Reference requested
Why not use the term `indiscrete groupoid' for the functor that gives a right adjoint to the functor Ob: Groupoids \to Sets? The left adjoint is then of course the `discrete groupoid'. This agrees with the terminology for discrete and indiscrete topologies.
I confess to have used different terminology in various places.
Of course one use of these notions is to show that the functor Ob preserves limits and colimits, which is a start on constructing them.
It is not surprising that this concept occurs widely. In groupoids there is a notion of covering morphism and the universal cover of a group G is of course an indiscrete groupoid G'; this groupoid is by no means `trivial' since it comes equipped with a covering morphism p: G' \to G. This approach to covering space theory is given in my book `Topology and groupoids'.
Ronnie
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Or codiscrete groupoid? David On 29 September 2011 23:11, Ronnie Brown <ronnie.profbrown@btinternet.com> wrote:
Why not use the term `indiscrete groupoid' for the functor that gives a right adjoint to the functor Ob: Groupoids \to Sets? The left adjoint is then of course the `discrete groupoid'. This agrees with the terminology for discrete and indiscrete topologies.
I confess to have used different terminology in various places.
Of course one use of these notions is to show that the functor Ob preserves limits and colimits, which is a start on constructing them. ...
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Emily, one notion of "partial category" that has been studied is Freyd's notion of "paracategory". It differs from what you wrote below as follows: instead of taking compositions of two arrows as the primitive operation, one takes compositions of n arrows as primitive operations, for all n. So if f1, ..., fn are n arrows (so that the codomain of each is the domain of the next), then [f1, ..., fn] is their composition, which may be defined or undefined. In the case of (total) categories, adding n-ary compositions makes no difference, since they are already definable in terms of identities and binary composition. But in the partial case, it does make a difference, as it is possible, for example, that [f,g,h] is defined, but [f,g] and [g,h] are undefined. The axioms are: (a) [] : A -> A is defined (the composition of the empty path from A to A) (b) [f] is defined and equal to f, for all arrows f, (c) if ff, gg, hh are are (composable) paths, and if [gg] is defined, then [ff,[gg],hh] is defined if and only if [ff,gg,hh] is defined, and in this case, they are both equal. The main representation theorem is: Every reflexive subgraph of a category is a paracategory; conversely, every paracategory can be faithfully completed to a category. It is not the only possible notion of partial category, and may not always be the notion that one wants. -- Peter Emily Riehl wrote:
A colleague of mine is wondering if anyone has studied "partial categories," by which she means directed graphs with identities but with only some compositions (including all identity compositions) defined.
A partial category can be thought of as a category enriched in pointed sets (with smash product as tensor and S^0 as unit). The slogan is that the basepoint in each hom-set stands in for "does not exist". But enriched functors don't give the right notion of maps; these should preserve identities and all specified compositions. Enriched functors behave appropriately with regards to the identites but may "forget" extant arrows and in particular need not preserve composites. So perhaps this perspective is not useful.
I'll happily pass along any suggestions.
Thanks, Emily Riehl
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I studied these things in my PhD work. The thesis itself is in German, I'm afraid, but there's a series of papers based on it which you can find on my homepage. Maybe the most interesting one for you might be the paper with Mateus on probabilistic automata (which form a certain type of partial category) in MSCS 12 (2002), pp. 481–512. Regards, Lutz Am 28.09.2011 22:34, schrieb Emily Riehl:
A colleague of mine is wondering if anyone has studied "partial categories," by which she means directed graphs with identities but with only some compositions (including all identity compositions) defined.
A partial category can be thought of as a category enriched in pointed sets (with smash product as tensor and S^0 as unit). The slogan is that the basepoint in each hom-set stands in for "does not exist". But enriched functors don't give the right notion of maps; these should preserve identities and all specified compositions. Enriched functors behave appropriately with regards to the identites but may "forget" extant arrows and in particular need not preserve composites. So perhaps this perspective is not useful.
I'll happily pass along any suggestions.
Thanks, Emily Riehl
-- -------------------------------------- Prof. Dr. Lutz Schröder Senior Researcher DFKI Bremen Safe and Secure Cognitive Systems Cartesium, Enrique-Schmidt-Str. 5 D-28359 Bremen phone: (+49) 421-218-64216 Fax: (+49) 421-218-9864216 mail: Lutz.Schroeder@dfki.de www.dfki.de/sks/staff/lschrode -------------------------------------- ------------------------------------------------------------- Deutsches Forschungszentrum für Künstliche Intelligenz GmbH Firmensitz: Trippstadter Strasse 122, D-67663 Kaiserslautern Geschäftsführung: Prof. Dr. Dr. h.c. mult. Wolfgang Wahlster (Vorsitzender) Dr. Walter Olthoff Vorsitzender des Aufsichtsrats: Prof. Dr. h.c. Hans A. Aukes Amtsgericht Kaiserslautern, HRB 2313 ------------------------------------------------------------- [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (9)
-
David Roberts -
Emily Riehl -
jpradines -
Lutz Schröder -
Peter May -
Ronnie Brown -
Ross Street -
selinger＠mathstat.dal.ca -
Tom Prince