Lie Local Subgroupoids
by
Ilhan Içen
Inönü University, Malatya
Coauthors: Osman Mucuk (Erciyes University, Kayseri)

The notion of the local equivalence relation on a topological space is generalised to that of local subgroupoid. The main results are construction of the holonomy groupoid and the notion of s-sheaf for the local subgroupoids s.

The concept of local equivalence relations, which was introduced by Grothendieck and Verdier in a series of exercises presented as open problems concerning the construction of a certain kind of topos was investigated further by Rosenthal and more recently by Kock and Moerdijk.

A local equivalence relation is a global section of the sheaf E which is defined by the presheaf

E = { E(U), EUV, X },

where E(U) is the set of all equivalence relations on the open subset U of X and E_UV is the restriction map from E(U) to E(V), for V subset of U.

The recent idea of a local subgroupoid of a groupoid G on a topological space X as a global section of the sheaf L associated to the presheaf

LG = { L(U), LUV, X }

where L(U) is the set of all wide subgroupoids of G|U and L_UV is the restriction map from L(U) to L(V) for V subset of U.

We define the holonomy groupoid of certain Lie local subgroupoid by using the idea of locally Lie groupoid. We define a strictly regular Lie local subgroupoid s and prove that if s is a strictly regular Lie local subgroupoid of Lie groupoid G on X and

glob(s) = H,     W = È x in X  Hx,

then (H, W) admits the structure of a locally Lie groupoid. So we obtain a holonomy groupoid H^s of the strictly regular Lie local subgroupoid s.

We introduce the concept of s-sheaves for strictly regular Lie local subgroupoids s. Corresponding concept for local equivalence relation r was extensively investigated by Rosental and Kock and Moerdjik. where they show that the r-sheaves form an étendue. This still leaves as an open problem that of describing the kind of topos formed by the category of s-sheaves.

REFERENCES

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Date received: July 20, 2001