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Morita equivalence for continuous groups. (English) Zbl 0648.22001

The author’s primary objective is a representation of the (“Morita”) equivalence relation between two topological groups, of having equivalent categories of actions on discrete spaces. [He likens this to Morita equivalence of commutative rings. It is very unlike, in that Morita equivalent commutative rings are isomorphic, while all connected groups are equivalent in the context of this paper.] He gets such a representation, by bimodules (torsors) over, not the two groups \(G_ 1\), \(G_ 2\), but two localic (“continuous”) monoids \(M(G_ i)\). M(G) is spatial if G is metrizable. [This is easy to see, and is mentioned in the Introduction. In the paper, it cannot even be stated in a reasonable way, since the author calls locales “spaces” (and calls frames “locales”).] The result is generalized from equivalence to geometric morphism and from sets to any base topos.

MSC:

22A05 Structure of general topological groups
18B25 Topoi
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References:

[1] Grothendieck, Rev?tements ?tales el groupe fondamentale 224 (1971)
[2] DOI: 10.1007/BFb0074299 · Zbl 1375.18001
[3] Bass, Algebraic K-theory (1968)
[4] DOI: 10.1016/0022-4049(80)90020-1 · Zbl 0429.18006
[5] Morita, Sci. Rep. Tokyo Kyokiu Daigaku 6 pp 83– (1958)
[6] DOI: 10.2307/2000067 · Zbl 0592.18003
[7] Isbell, Math. Scand. 31 pp 5– (1972) · Zbl 0246.54028
[8] Moerdijk, Prodiscrete groups
[9] Joyal, An extension of the Galois theory of Grothendieck. Memoirs Amer. Math. Soc. 309 (1984)
[10] Moerdijk, Compositio Math. 58 pp 45– (1986)
[11] Johnstone, Stone Spaces (1982)
[12] Johnstone, Cahiers Top. G?om. Diff. 22 pp 3– (1981)
[13] DOI: 10.1016/0022-4049(82)90076-7 · Zbl 0473.06005
[14] Moerdijk, Trans. Amer. Math. Soc.
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