Vandenbossche, G.
Our aim is to generalize to the non-commutative case, the generic representation of commutative rings by sheaves on their quantales of ideals. As the quantale of two-sided ideals is not a sufficiently rich structure, we define and work on a quantaloid of left and right ideals. A workable notion of sheaf is introduced using matrices with values in a quantaloid. For a given ring R, we obtain a category of sheaves where the terminal object is endowed with a special subobject. There exists a representing sheaf for R in the sense that the elements of R correspond to the sections from the special subobject and the global sections correspond to the center.
- Benabou, J.:Théorie des ensembles empiriques (1), Séminaire 1987?1988, Mezura 17, Publications Langues 'O, Services de la recherche de l'Inalco, Paris.
- Betti, R., Carboni, A., Street, R., and Walters, R. F. C.: Variation through enrichment,J. Pure Appl. Algebra 29 (1983), 109?127.
- Borceux, F. and Cruciani, R.: A generic sheaf representation theorem for non commutative rings,J. of Algebra 167(2) (1994), 291?308.
- Borceux, F. and Van den Bossche, G.: A generic sheaf representation for rings, inCategory Theory, Proceedings Como 1990, Springer LNM 1488, pp. 30?42.
- Higgs, D.:A Category Approach to Boolean-Valued Set Theory, Lecture Notes, University of Waterloo, 1973.
- Mulvey, C.: &,Rend. Circ. Mat. Palermo 12 (1986), 99?104.
- Rosenthal, K. I.: Free quantaloids,J. Pure Appl. Algebra 72(1) (1991), 67?82.
- Walters, R. F. C.: Sheaves on sites as Cauchy complete categories,J. Pure Appl. Algebra 24 (1982), 95?102.
| Bibliographic reference |
Vandenbossche, G.. Quantaloids and non-commutative ring representations. In: Applied Categorical Structures : a journal devoted to applications of categorical methods in algebra, analysis, order, topology and computer science, Vol. 3, no. 4, p. 305-320 (1995) |
| Permanent URL |
http://hdl.handle.net/2078.1/47445 |
