What morphisms must exist in a category if the morphism between two hom-sets relative to that category's objects exist?

Question

I'm asking this and other questions in an attempt to tackle something I'm having a hard time to understand from another point of view.

Let's consider a category $C$ with 4 objects, $a$, $b$, $c$, and $d$, and some morphisms such that

• $|Hom(a,c)| \ge 1$,
• $|Hom(b,d)| \ge 1$,
• there is a morphism $m$ in $Set$ such that $m: Hom(a,c) \to Hom(b,d)$.

What do the 3 hypothesis above imply as regards the other hom-sets in $C$?

My understanding is that, at the minumum, $|Hom(b,a)| \ge 1\ \wedge\ |Hom(c,d)| \ge 1$ must be true, but it's not required that the strict inequality holds, nor that the other hom-sets are not empty.

Furthermore, I believe that the concept of dual/opposite category need not be brought in to answer this question.

Is all the above correct?

Answer 1

Let's consider a category $C$ with $4$ objects, $a$, $b$, $c$, and $d$, and some morphisms such that

• $|\mathrm{Hom}(a,c)| \ge 1$,
• $|\mathrm{Hom}(b,d)| \ge 1$,
• there is a morphism $m$ in $\mathsf{Set}$ such that $m: \mathrm{Hom}(a,c) \to \mathrm{Hom}(b,d)$.

Let's pause and observe that the third hypothesis follows from the second hypothesis (so it doesn't add anything). Assuming $\mathrm{Hom}(b,d)$ is non-empty, we can pick some $f\in \mathrm{Hom}(b,d)$ and define $m: \mathrm{Hom}(a,c) \to \mathrm{Hom}(b,d)$ to be the constant function with value $f$, $m(x) = f$.

What do the 3 hypothesis above imply as regards the other hom-sets in $C$?

Absolutely nothing.

My understanding is that, at the minumum, $|\mathrm{Hom}(b,a)| \ge 1\ \wedge\ |\mathrm{Hom}(c,d)| \ge 1$ must be true.

Nope, that's not the case. And I'm very curious what your reasoning is - this seems like a total non sequitur!

For example, $C$ could have exactly six arrows: $\mathrm{id}_a$, $\mathrm{id}_b$, $\mathrm{id}_c$, $\mathrm{id}_d$, $f\colon a\to c$, and $g\colon b\to d$. Then $\mathrm{Hom}(b,a)$ and $\mathrm{Hom}(c,d)$ are both empty.