Hi Andrew,
You can definitely have some fun analyzing your crime data You will have to convert the categorical data to counts, or proportions, but you can still learn a lot about how the different types of crimes relate to each other. Here are some ideas:
1) Use Optimized Hot Spot Analysis. It will overlay your study area with a fishnet grid and count the number of crimes (of all types) that fall within each grid cell, then it will perform hot spot analysis to show you the hot and cold spot areas. This answers the question: where are the hot and cold spots of (all) crime? However, you can also run Optimized Hot Spot Analysis on different types of crime, then visually compare the hot spot maps.
2) You can use the fishnet grid cells from (1) or else use census tracts and count the number of unique crime types within each polygon to see where you have the highest crime diversity (you might find that some places only have robberies, for example, while other places experience a wide variety of different crime types)... you can run hot spot analysis on the diversity counts.
3) For fishnet grid cells (output from Optimized Hot Spot Analysis, then use Spatial Join) or census tracts, count the number of assaults, the number of robberies, the number of auto thefts, etc. within each polygon, then convert those counts to a percentage of all crime. You can then run grouping analysis to find polygons with similar challenges ... one group might be high for assault and narcotics, for example, but low for robbery... knowing the profiles -- the specific challenges -- of each group can help you identify effective prevention strategies.
4) If you have distinct clusters of particular crimes, you can create standard deviational ellipses around each cluster then overlay the ellipses for two different types of crimes to see how spatially integrated they are. (I'm doing an analysis right now that looks at violent crime in relation to alcohol establishments... to see how integrated those two "activity" spaces are).
5) Using the output from (1) you can find spatial outliers for all crime or for specific crime types: a high crime count area surrounded by low crime count areas, or a low crime count area surrounded by high crime count areas. These anomalies are often very interesting (what is that one neighborhood doing right... it is has no problem at all with narcotics while surrounding areas are high for drug related crimes? ... or why is this one neighborhood so high in relation to surrounding neighborhoods?)
6) Be sure to check out the new space time pattern mining tools in the 10.3 release as well: An overview of the Space Time Pattern Mining toolbox—ArcGIS Help | ArcGIS for Professionals
I'm sure others will have ideas as well. I hope this is helpful!
Best wishes,
Lauren Scott
Esri