Data Warehouse - Slowly Changing Dimensions with Many to Many Relationships
Does DimCategory have any Type 2 attributes that you are tracking?

If it is not, then you would only need to add a new record to your bridge table based on whenever a Type 2 attribute in DimPerson changes since you will be inserting a new record in DimPerson with a new surrogate key. You would then need to add or update your bridge table with this new surrogate key for whichever categories are assigned. 

However, if an attribute on DimCategory changes, you would just update that record and there would be no change in the surrogate key so the bridge table would be unchanged.

Now if you want to track things like when a Category assignment to a Person changes as a Type 2 attribute, then you would need to build the Effective Start and End Dates in your bridge table to be able to tell at what point in time a Person's Categories change.