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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.

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Category theorists have no animosity toward the empty set. In particular, the set of morphisms between two objects may be empty. –  Zhen Lin Jan 8 at 5:42
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As you will soon discover (see @Aaron answer below, for example), "the" category $\mathbf 2$ has different meanings for different authors. You mean the category with no arrows between different objects while, for example, MacLane means the category with 2 objects and exactly one arrow joining the first to the second object. –  magma Jan 8 at 9:29

4 Answers 4

up vote 7 down vote accepted

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.

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A category need not have any morphisms other than the identities. Such a category is called discrete.

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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.

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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.

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your category $\mathbf 2$ is different from the category mentioned by the OP. The category described by the OP is simply the discrete category with 2 objects, while yours is the linear order with three objects (corresponding to the ordinal number 3). –  magma Jan 8 at 9:21
    
@magma When I was reading the question, somehow I didn't make the connection between the category $\mathbf 2$ and the final paragraph. My choice of naming convention here comes from looking at the category as categorical representation of the geometric $2$-simplex. Usually, when I see discrete categories, they are explicitly labeled as such. –  Aaron Jan 8 at 20:11

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