Geodesic
On the dearth of coproducts in the category of locally compact groups
Theory and applications of categories, Tome 38 (2022), pp. 791-810

Voir la notice de l'article provenant de la source Theory and Applications of Categories website

We prove that a family of at least two non-trivial, almost-connected locally compact groups cannot have a coproduct in the category of locally compact groups if at least one of the groups is connected; this confirms the intuition that coproducts in said category are rather hard to come by, save for the usual ones in the category of discrete groups. Along the way we also prove a number of auxiliary results on characteristic indices of locally compact or Lie groups as defined by Iwasawa: that characteristic indices can only decrease when passing to semisimple closed Lie subgroups, and also along dense-image morphisms.

Publié le :
Classification : 22D05, 22E46, 22D12, 18A30
Keywords: locally compact group, Lie group, almost-connected, pro-Lie, coproduct, representation 
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     author = {Alexandru Chirvasitu},
     title = {On the dearth of coproducts in the category of locally compact groups},
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     pages = {791--810},
     publisher = {mathdoc},
     volume = {38},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2022_38_a19/}
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Alexandru Chirvasitu. On the dearth of coproducts in the category of locally compact groups. Theory and applications of categories, Tome 38 (2022), pp. 791-810. http://geodesic.mathdoc.fr/item/TAC_2022_38_a19/