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I'd like to create a Categorized style in QGIS, classify with it data in a shapefile and then save the shapefile with the new categorized style. Is it possible?
Shapefiles are not designed to contain style information - they contain data.
While you can construct attributes for storing things like "polygon color", it's better to save the styles separately as QML (QGIS style files) or SLD (standardized style files supported in QGIS 1.8).
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