I've been taking a close look at the paper "Arrows and Computation" by Ross Paterson. It's not relevant to my question, but I'm doing this because I think that Arrows may be a useful abstraction for a problem I have (I need to both construct a computational pipeline AND perform a computation over the pipeline itself). The paper is published as Chapter 10 in "The fun of Programming" and is also available here: http://www.soi.city.ac.uk/~ross/papers/fop.html My question originates with the statement at the bottom of page 205 (the 5th page of the paper) regarding the behavior of the function 'first': " ... while 'first' routes the state through f:". My problem is that this statement appears to be erroneous. My reading of the code would indicate that 'first' does not do this at all. The state portion of the Arrow remains unaltered. Nor does it even seem desirable for 'first' to have an impact on the state portion of the Arrow, as I would think that this would preclude using 'first' and 'second' to lift binary operators into the Arrow. So what is it that am I missing? The relevant code is as follows. Paterson used Greek letters where I have used Latin. I represented his product operator with the letter x, requiring surrounding backquotes. If there are other changes to his code I am unaware of them and they are an error on my part. class Arrow a where pure :: (b -> c) -> a b c (>>>) :: a b c -> a c d -> a b d first :: a b c -> a (b,d) (c,d) x :: (a -> a') -> (b -> b') -> (a,b) -> (a',b') (f `x` g) ~(a,b) = (f a, g b) assoc :: ((a,b),c) -> (a,(b,c)) assoc ~(~(a,b),c) = (a,(b,c)) unassoc :: (a,(b,c)) -> ((a,b),c) unassoc ~(a,~(b,c)) = ((a,b),c) newtype State s i o = ST ((s,i) -> (s,o)) instance Arrow (State s) where pure f = ST (id `x` f) (ST f) >>> (ST g) = ST (g . f) first (ST f) = ST (assoc . (f `x` id) . unassoc) -Reilly Hayes