I'm just getting started in category theory, and I'm not understanding the basic definitions. For example, a common example of a category is a poset. So suppose I have a trivial poset $P$ of 10 elements:

  1. Can I take the 10 elements of $P$ to be the objects of a category $A$?
  2. If so, does $A$ have one morphism representing the partial order, or are there many morphisms, one between each pair of qualifying elements?
  3. Can I alternatively take $P$ to be the only object of a category $B$?
  4. If so, does $B$ just have one morphism $P \rightarrow P$?
  5. Given a second poset $Q$, can I take $P$ and $Q$ to be the only two objects of a category $C$?
  6. If so, are there now 3 morphisms: $P \rightarrow P, Q \rightarrow Q, P \rightarrow Q$?

So far I find it easy to work with concepts like functors, subcategories, isomorphism, equivalence, etc. What I can't figure out is how to correlate these concepts with concrete data. Apologies if these questions are extremely trivial, but I have spent a lot of time reading and searching, and have yet to encounter a thorough explanation of these terms.

  1. Yes.

  2. The Poset structure on $P$ gives just a single morphism between objects: We say we have an arrow $a \to b$ if and only if $a\leq b$ (or, depending on your preferences, if and only if $a\geq b$). You could add aditional arrows to your category, but you would have to provide a rule for composition. (In the poset case, composition is just given by the transitive property).

  3. Yes. A category with one object is called a Monoid. In general, you can define a category of Posets, where the arrows are functions such that $f(a)\leq f(b)$ whenever $a \leq b$. You can view the category whose only object is $P$ as a subcategory of the category of Posets with the arrows as above. If you include all such arrows, we say that it is a full subcategory.

  4. This works too. A class with only one object and a single arrow from that object to itself is a category, although not a very interesting one. But if you just specify the objects of the category without saying what the arrows are, there's limitless possibilities for what the set of arrows might look like.

  5. Sure. You can make the maps between them be the maps I described in #3 (this also includes many maps from each poset to itself), for example, or you could come up with your own rule for what counts as an arrow, perhaps only including the requisite arrow from each object to itself.

  6. That's one possibility. As I mentioned in #5, there are other possibilities as well.

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    $\begingroup$ Great, so it sounds like categories are very flexible, and I can define them in any way that the definition allows. One thing I still don't get about morphisms: the arrow is described as $a \rightarrow b$, which makes perfect sense as a mapping--but as a function it appears to have 2 arguments and return a boolean. Are morphisms not regarded as functions in the latter sense? $\endgroup$ – Byron Hawkins Apr 18 '12 at 4:42
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    $\begingroup$ A morphism is what we get when we abstract away from functions the notion of composability. Morphisms aren't always functions, but functions are often morphisms. $\endgroup$ – Thomas Belulovich Apr 18 '12 at 7:12
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    $\begingroup$ Indeed, you may define a category in any way you like, so long as it satisfies the definition of category. But categories appear naturally as well, which is why we study them. Some common examples: groups/rings/modules with homomorphisms, sets with functions, topological spaces with continuous maps, smooth manifolds with smooth maps, and posets with $\leq$. Note that in all examples except for the last, the arrows are given by some type of function. $\endgroup$ – Brett Frankel Apr 18 '12 at 17:02

There are two things that could be going on here: each is a categorical perspective on posets. One models a poset as a category, while the second talks about the family of posets as a category. Usually when posets are brought up early on, you'll at see at least the first of these.

1) A poset $P$ can be considered a category, with a morphism from $x \to y$ whenever $x \le y$. The transitive law becomes the composition law, and the reflexive law becomes the identity morphism. You need some third axiom about there being at most one morphism between any two objects to give you the antisymmetric property.

2) There is also a category whose objects are posets, and whose morphisms are order-preserving maps. (A morphism $f : P \to Q$ would be something where $x \le y \implies f(x) \le f(y)$.) This setup is a bit closer to the more familiar categories (sets, groups, spaces, etc...) where object = set of things and morphism = structure preserving set map.


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