Question on dual category.

Question

Question 1: Let $X$ be a set and consider the power set category $P(X)$ where the objects are subsets of $X$ and an arrow $A \to B$ is a subset inclusion $A \subseteq B$. I have difficulties to understand the dual category $P(X)^{op}$, this is the category with same objects but the arrows are reversed, so that the arrow $A \to B$ is now the arrow $B \to A$, but then the arrows are infact inclusions of the form $B \subseteq A$. Now, I understand the arrows in the dual category, but my problem is what if $B$ is not a subset of $A$? In order to have a category, do we care if the arrows actually are true statements? Because $B \subseteq A$ might be a false statement.

Question 2: In proving that two groups or not isomorphic I often look at the elements of the groups, for example does one group have a nilpotent element when the other doesn't? Or does one group have a element of order 5 when the other doesn't? Or is there an element that has itself as inverse while such element does not exist in the other group? I have different ways to check that groups or not isomorphic but when it comes to categories I have no idea what to do. How can i check that two categories are not isomorphic?

Answer 1

Since question 1 was addressed in comments, let me take a stab at question 2. First off, when looking at categories we are almost never interested in isomorphism; instead, we care about equivalence, which is something like "isomorphism up to isomorphism". If you haven't yet learned what an equivalence is, you will soon.

Because categories come in so many different shapes and sizes, there is no "standard" way to tell if two are not equivalent; but just like in groups we look at group-theoretic properties that one group may have while the other does not, in category theory we look at category-theoretic properties that one category may have while the other does not. Such properties can include:

• Being thin: having at most one morphism between objects.
  • If it is, then these are basically preorders, and you can use any technique you like to show nonisomorphism for preorders.
• Having all or some (co)limits of a certain type; notably:
  • Initial object
  • Terminal object
  • Products
  • Coproducts
• How these behave; e.g., are the initial and terminal objects the same objects?
• The cardinality of the isomorphism class of objects (if it is a set).
• Cardinalities of sets of morphisms. In particular: are there objects without arrows between them?
• The monoid structure on endomorphism monoids of various objects.
• The monoid $\mathrm{End}(\mathrm{Id}_C)$ of all natural transformations from the identity functor on a category $C$ to itself often has a lot of information.

And so on, and so forth. As you become more familiar with category theory it will become obvious just like with group theory that there is an essentially endless supply of "categorical properties" that you can use to differentiate between categories.