At the moment I am writing a chapter on groupoid spaces. In this chapter I introduce the notion of a “group space” and the notion of a “groupoid space”. Of course, most of the theory is exactly the same as for groupoid schemes, so in fact I am simply editing a copy of the chapter on groupoid schemes. What I am wondering is whether it is OK to use “groupoid space”‘ and “group space”, or if I should use the longer and perhaps more correct “algebraic groupoid space” and “algebraic group space”? Or, is it better to use “groupoid in algebraic spaces” and “group algebraic space”?
For now I’ll stick to my first choice, but if you object please leave a comment.
I have often agonized over this. At the moment I feel that “closed subspace” is acceptable (for a closed immersion of algebraic spaces) but that “group space” is too vague. I lean towards:
1) groupoid in algebraic spaces
2) group algebraic space
For 2), algebraic group space might be acceptable but could possibly be misleading. However, if one uses the natural definition
space = étale/fppf-sheaf
as Laumon Moret-Baillet do, then “group space” = group object in category of spaces, and “algebraic group space” = “group algebraic space” both seem adequate. Similarly, groupoid space is too vague.
OK, I think I will change this. Both Bob Friedman and Jarod Alper agreed that “group algebraic space” is better. Also Bob jokingly suggested we abbreviate it “gas”.