Question about the category $\textbf{2}$ I am reading about category theory. So a category has a collection of objects and a collection of morphisms. Denote the set of morphisms from $A$ to $B$ by $Mor(A,B)$ so $f:A\rightarrow B$ has $f \in Mor(A,B)$. I haven't found it stated anywhere but do we take $Mor(A,B)$ to always be a non-empty set?
I wonder because I wanted to try to understand the small categories like $\textbf{2}$.
I was wondering if I take the collection of objects to be two vertices $a$ and $b$ and the morphisms to just be the identity morphism for $a$ and the identity morphism for $b$ so $Mor(a,b)=Mor(b,a)=\varnothing$ and $Mor(a,a)=\{1_a\}$ and $Mor(b,b)=\{1_b\}$, if this would form a category or if I would need to throw in a morphism $a\rightarrow b$ to form the category.
 A: A category need not have any morphisms other than the identities. Such a category is called discrete.
A: Every set gives rise to a category. In particular, let $S$ be a set. Then we may form a a category which we may call $disc(S)$ which is called the discrete category of $S$. The objects of $S$ are the elements, and we set $$Mor(s,s')=\emptyset\mbox{ if } s\neq s',$$ and $$Mor(s,s)=\{Id_s\}.$$ A category satisfying the above is called a discrete category. Likewise given any (small) discrete category, we may recover the underlying set by taking the set of objects.
Moreover, their is a one to one correspondence between the functors between two discrete categories and the set of functions between the two underlying sets.
A: There are two different categories sometimes known as $\mathbb{2}$.
The first option you describe — with only identity morphisms, so $\newcommand{Hom}{\mathrm{Hom}}\Hom(a,b) = \Hom(b,a) = \phi$ — would be less ambiguously called the discrete category on the set 2.  Generally, one can construct the discrete category on any set $X$, by taking $X$ as the objects, and just identities as the morphisms.
The second option you describe — with $\Hom(b,a) = \phi$ but with a single morphism $a \to b$ thrown in — is less ambiguously known as the arrow category, the walking arrow, or the poset category on $\{0 < 1\}$.  Generally, any poset or pre-order gives a category, with objects the elements of the pre-order, and with $\Hom(a,b) \cong 1$ if $a \leq b$ and $\Hom(a,b) = 0$ otherwise. Indeed, pre-orders can be seen in this way as a class of categories: categories such that each $\Hom(a,b)$ contains at most one arrow.
In any case: both of the walking arrow and the discrete category on 2 are perfectly good categories, and indeed arise quite often in practice.
A: While there is no problem with a $\operatorname{Hom}(A,B)=\emptyset$, when $A\neq B$, there are non-identity morphisms in the category $\mathbf 2$.  Assuming we are using the same notation, we have three objects, $[0], [1], [2]$, and $\operatorname{Hom}_{\mathbf 2}([i],[j])$ is empty if $i>j$ and has exactly one morphism otherwise (which is the identity when $i=j$).
More generally, any partially ordered set gives rise to a category, and this gives a particularly nice collection of small categories to study.
