3
$\begingroup$

Consider a (somehow naturally) specified mapping from one class of structures into another (or the same) class of structures, e.g. the mapping of a

Is there a somehow standarized way to detect whether such a mapping can be made a functor?

A prerequisite is to equip both sides (classes) of the mapping with morphisms to make them categories. If one has achieved this - usually in one of many possible ways - one has to look for specific mappings of morphisms in the source category to morphisms in the target category that fulfill the functor conditions.

Given such morphisms and a mapping of morphisms one may be able to check whether they fulfill the functor conditions, but - given only the mapping between objects -

how to find appropriate morphisms and their mapping, resp. how to show that you cannot find them?

Only some of the examples above give rise to functors, but I am especially insecure about the last two examples, so I'd like to focus the question:

How do you show that a mapping between structures can not be made a functor?

  • 3
    An example is in http://math.stackexchange.com/questions/158438/ -- find objects whose morphisms are very restricted2012-08-13
  • 0
    @Jack: Thanks for the example, I wouldn't have found it without your help!2012-08-13
  • 0
    The automorphism group "assignment" is almost never a functor.2012-08-14
  • 0
    @Zhen Lin: Does that mean, that the relation between a graph and its automorphism group cannot be treated categorically?2012-08-14
  • 0
    If you delete all non-invertible morphisms in the category you start with, then you get a functor. Otherwise what are you going to do with the morphisms?2012-08-14

1 Answers 1