# [Haskell-cafe] cat as banana In AOP

cutsea110 cutsea110 at gmail.com
Thu Aug 25 01:00:15 UTC 2016

```Hi,all

I'm studying `Algebra of Programming' written by Prof. Richard Bird.

My question is for section 3.5 "Concatenation and currying > Exponentials".
In page 74, he said

```
cat = (| curry ([outr, cons] . (id+id x apply) . (id+assocr) . distl) |)
```

where (| ... |) means banana operator.

But I guess this left hand side `cat' is typo of `ccat` or `curry cat`?
I think this is reasoned below.

-----------------------------
cat . (alpha x id) = [outr, cons] . (id + id x cat) . phi   <= page 73
start expr (maybe this is from equation (3.6))

= {- Theorem 3.1 precondition -}

cat = apply . ( (| curry (h . G apply . phi |) x id)           -----
(A)

= {- on the other hand, exponential's universal property : g = curry f ==
apply . (g x id) = f -}

cat = apply . ((curry cat) x id)
-----  (B)

= {- from (A) and (B) -}

(curry cat) = (| curry (h . G apply . phi |)

-----------------------------

This result is

curry cat = (| curry ( [outr, cons] . (id + id x apply) . (id + assocr)
. distl) |)

not cat = rhs(above).