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
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