Haskell as category

Let us analyze Haskell from a category theory point of view. We will consider two categories:

\begin{align} C = & {I n t, B o o l, C h a r} & (3.5) \\ D = & {D o u b l e, B o o l} & (3.6) \end{align}

where the objects are standard Haskell types. We can then write

\begin{align} E n d (C h a r) = H o m (C h a r, C h a r) = & {t o U p p e r, m i r r o r, ∖ x \to ’a’, \dots} & (3.7) \\ A u t (C h a r) = & {m i r r o r, \dots} & (3.8) \\ H o m (I n t, B o o l) = & {i s L e a p Y e a r, ∖ x \to x > 2, \dots} & (3.9) \end{align}

For instance, the function $f = (∖ x \to x > 2) \in H o m (I n t, B o o l)$ is not an monomorphism, because

\begin{align} α = & ∖ x \to if x is before k in alphabet 5 else 10 \in H o m (C h a r, I n t) & (3.10) \\ α^{'} = & ∖ x \to if x is before k in alphabet 7 else 12 \in H o m (C h a r, I n t) & (3.11) \end{align}

even though $α \neq α^{'}$ , we have

\begin{align} f \cdot α = f \cdot α^{'} = ∖ x \to T r u e & (3.12) \end{align}

however it is an epimorphism as for any $β_{i} \in H o m (B o o l, C h a r)$ with

\begin{align} β (T r u e) = b 1, β (F a l s e) = b 2 & (3.13) \\ β^{'} (T r u e) = b 1^{'}, β^{'} (F a l s e) = b 2^{'} & (3.14) \\ (3.15) \end{align}

$b 1, b 2, b 1^{'}, b 2^{'} \in C h a r$ , we have

\begin{align} β \cdot f = & ∖ x \to if x > 2 then b1, else b2 & (3.16) \\ β^{'} \cdot f = & ∖ x \to if x > 2 then b1’, else b2’ & (3.17) \end{align}

hence $β \neq β^{'} \to β . f \neq β^{'} \cdot f$ .

1 asd::Vector

\to

Vector
2    asd v = let sdf = asd^2
3    in v \= 4 if True −− asd
4    then asd = 2
5    else f k
6    where f x = x 2 "asdasd"
7    % asd