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?

  • $\begingroup$ An arrow $B\to A$ in the dual category still means $A\subseteq B$, not $B\subseteq A$. So if $A$ is not a subset of $B$, then there is no arrow in $Hom_{\mathcal{C}}(A,B)$ and therefore there is no arrow in $Hom_{\mathcal{C}^{op}}(B,A)$. $\endgroup$ – Janik Sep 24 '16 at 14:35
  • $\begingroup$ @Janik thank you for clearing this for me. More generally, in the opposite category does the reserved arrows always mean the same as the original arrow in the category? $\endgroup$ – category Sep 24 '16 at 14:54
  • $\begingroup$ Yes and no. Your category has the following property: For every $A,B\subseteq X$, there is at most one arrow $A\to B$. For every category with that property, you can define a reflexive and transitive relation $A\thicksim B \Leftrightarrow \text{There is an arrow } A\to B$. So yes, in such categories, the "meaning" $A\thicksim B$ is preserved under the functor $()^{op}$. But for general categories, the existence and number of arrows $A\to B$ doesn't necessarily have a "meaning". So in general there is not much more intuition behind $\mathcal{C}^{op}$ than "all arrows are reversed". $\endgroup$ – Janik Sep 24 '16 at 15:09
  • $\begingroup$ @Janik Could you explain in detail how the meaning is preserved only because we can define such a relation? $\endgroup$ – category Sep 24 '16 at 15:50
  • $\begingroup$ That was the only case in which I could think of some senseful meaning. There might be other categories with some nice interpretation of arrows, but I don't think this is true for a general category. Your example is indeed a very good one to sharpen one's intuition with the opposite category, but in general I would advise to keep in mind the abstract definition. Like with groups: There are good examples for symmetry groups, but if you think of a arbitrary group, you don't think of an object and its symmetry group, but just the abstract algebraic concept of a group. $\endgroup$ – Janik Sep 24 '16 at 16:35

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.


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