Slicer for value that could fall into multiple categories

Probably a better way to do this, but here is one way.

Create a disconnected table with your categories:

Category

Next, create the following measures:

CategoryBlue = IF(SEARCH("Blue",MAXX(VALUES(Items[Category]),[Category]),1,BLANK()),1,0)

CategoryGreen = IF(SEARCH("Green",MAXX(VALUES(Items[Category]),[Category]),1,BLANK()),1,0)

CategoryRed = IF(SEARCH("Red",MAXX(VALUES(Items[Category]),[Category]),1,BLANK()),1,0)

CategoryPurple = IF(SEARCH("Purple",MAXX(VALUES(Items[Category]),[Category]),1,BLANK()),1,0)

CategoryYellow = IF(SEARCH("Yellow",MAXX(VALUES(Items[Category]),[Category]),1,BLANK()),1,0)

MeasureSelected = 
IF (
    HASONEVALUE ( Categories[Category] ),
    SWITCH (
        FIRSTNONBLANK ( Categories[Category], Categories[Category] ),
        "Blue", [CategoryBlue],
        "Green", [CategoryGreen],
        "Purple", [CategoryPurple],
	"Red", [CategoryRed],
        "Yellow", [CategoryYellow]
    ),
    BLANK ()
)

Put a Table or Matrix visualization with some column from your items, your Category column from your categories table and MeasureSelected. Filter the visual for MeasureSelected = 1. Add a slicer for your Category column from your categories table.