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On principally generated Q-modules in general, and skew local homeomorphisms in particular | | Hans Heymans
; Isar Stubbe
; | | Date: |
1 Feb 2008 | | Abstract: | Ordered sheaves on a small quantaloid Q have been defined in terms of
Q-enriched categorical structures; they form a locally ordered category Ord(Q).
It has previously been shown by the second author that the "free-cocompletion’’
KZ-doctrine on Ord(Q) has Mod(Q), the quantaloid of Q-modules, as category of
Eilenberg-Moore algebras. In the first part of this paper we apply Q-enriched
category theory, particularly the theory of totally algebraic cocomplete
Q-categories as developed by the second author, to give an intrinsic
description of the Kleisli algebras: we call them the ’locally principally
generated Q-modules’. We deduce that Ord(Q) is biequivalent to the 2-category
of locally principally generated Q-modules and left adjoint module morphisms
between them, and thus provide a rephrasing of the notion of ordered sheaf on Q
in terms of the possibly more familiar Q-modules. Several examples are briefly
discussed, but one particularly important example is worked out in full detail
in the second part of this paper: the locally principally generated Q-modules
in the case where Q is (the one-object suspension of) a locale X. By relating
X-modules to objects of the slice category Loc/X we obtain an account of
ordered sheaves on X as ’skew local homeomorphisms into X’ as asymmetrical
analogues of local homeomorphisms. | | Source: | arXiv, 0802.0097 | | Services: | Forum | Review | PDF | Favorites | |
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