Haskell/Solutions/Category theory

Which law implies transitivity?

Formally, transitivity of a partial order is defined as: for any $a$, $b$, and $c$ if $a<b$ and $b<c$ then $a<c$. In the category defined by a partial order this translates to: for any objects $a$, $b$, and $c$ if there exist morphisms $a\to b$ and $b\to c$ then there exists a morphism $a\to c$. This is guaranteed by the second law of categories, i.e., that they are closed under composition. Indeed, the last morphism is a composition of the first two.

Why does adding an arrow break category laws?

Let's consider what must be the result of the compositions $h\circ g$ and $g\circ f$. Notice that since $h:B\to A$ and $g:A\to B$ then $h\circ g:A\to A$ and given that there is only one morphism from $A$ to $A$, namely $\operatorname {id} _{A}$, it follows that $h\circ g=\operatorname {id} _{A}$. Using similar reasoning we can show that $g\circ f=\operatorname {id} _{B}$.

Now, let us consider the composition $(h\circ g)\circ f$; substituting $h\circ g=\operatorname {id} _{A}$ into it yields $\operatorname {id} _{A}\circ f$, which can be simplified to $f$ using the third law of categories. However, the first law of categories states that this composition should be equal to $h\circ (g\circ f)=h\circ \operatorname {id} _{B}=h$, yet $f\neq h$.