Re: Categories and functors, query

["Nikita Danilov" <danilov@gmail.com>]

Michael Barr writes:
 >
 > This reminds me of a speculation I have often had (although Saunders
 > denied and he knew Birkhoff pretty well).  In the 30s and 40s, the word
 > "homomorphism" was regularly used but always meant surjective.  By the
 > late 40s and 50s people were talking about "homomorphism into" meaning
not
 > necessarily surjective.  So groups had lattices of subgroups and lattices
 > of quotient groups and Birkhoff invented lattice theory at least partly
in
 > the hope that the structure of those two lattices would tell you a lot
 > about the structure of the group.  I don't think this actually happened
to
 > any great extent.  But I have wondered whether Birkhoff might instead
have

Noether's `set theoretic foundations of group theory', where group
axioms are based on a notion of coset decomposition rather than
multiplication, seems to be much earlier (20s) attempt to the same:

    http://www.math.jussieu.fr/~leila/grothendieckcircle/mclarty2.pdf

 >
 > Michael

Nikita.