Thank you to all people who contributed to the discussion about Cantor- Bernstein. I learnt many interesting things, although what I had I mind was rather the general categorical structure underlying the proof. A similar question could be asked about the "elementary" proof of the following theorem : if there is an injection from A to B, then there is injection from 2A to 2B. Another question. The construction of the holomorph of a group is well-known : Hol(G) = G x Aut(G) . Similar constructions can be done for graphs, categories, sketches, ... Has anyone seen anything about that ? Pierre Ageron
In reply to Pierre Ageron's question: A nice way to regard the holomorph is as follows. The category (groups) is embedded in the category Gd=(groupoids), which is cartesian closed. So Gd has an internal endomorphism and also automorphism object AUT(-) (the maximal internal subgroup object of END(-)). So if G is a group, then AUT(G) is a group-groupoid, also known as a cat^1-group. As a group it is isomorphic to the holomorph of G. (See for example, J Shrimpton, JPAA 72 (1991) 303-318 for background.) Under the correspondence between group-groupoids and crossed modules, the holomorph of G corresponds to the crossed module G--> Aut(G) given by the inner automorphism map. Shrimpton gets an analogous crossed module Q(\Gamma) --> Aut(\Gamma) when \Gamma is a directed graph, and this has an associated group-groupoid, which can be called the holomorph of \Gamma. So the holomorph should be regarded as simply the internal automorphism object (in the appropriate category). If the objects don't form a cartesian closed category to start with, then something must be done about it (like turning groups into groupoids). Maybe you can only form a monoidal closed category. Then you have problems with group objects. See Brown and Gilbert Proc London Math Soc (3) 59 (1989) 51-73 for an example of what can be done (in this case with the category of crossed modules, where the cartesian closed structure is not the really interesting one). The area is interesting, in view of the widespread use of groups in maths and science. What are and what should be "groups"? If S is a structure, how can and should Aut(S) be structured? Ronnie Brown Prof R. Brown Tel: +44 248 382474 School of Mathematics Fax: +44 248 355881 Dean St email: mas010@uk.ac.bangor University of Wales Bangor Gwynedd LL57 1UT UK
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Pierre Ageron -
Prof R. Brown