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I'm a developer and have become fixated on functional programming due to its expressivity. I have begun learning Haskell but have reached a very significant wall when trying to comprehend functors, monads, monoids and applicative functors. Currying, lambdas, and function composition all make sense to me, but the concept of a Monad is not that intuitive. I have taken a semester's worth of Discrete mathematics: this book, which covers some set theory (which from some reading I guess a set is a category, and a morphism is a mapping function). I have also read Learn you a Haskell.

My question is simply, from a programmer's / pseudo-mathematician perspective, what is the best introductory text to category theory (or to the topics I mentioned above) that will allow me to understand these concepts and expound upon my understanding if need be later on?

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Maybe this q/a could be helpful for understanding monads. –  Petr Pudlák Nov 14 '12 at 7:06
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This isn't an answer, just a warning: if you look up monads in a book on category theory it won't be obvious which part corresponds to how monads are used in computer science. Historically monads were used to talk about their algebras (en.wikipedia.org/wiki/…) but as far as I can tell, in computer science what's more important is actually their Kleisli categories (en.wikipedia.org/wiki/Kleisli_category). This point tripped me up for awhile when I was trying to learn something about Haskell. –  Qiaochu Yuan Nov 14 '12 at 7:06
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Maybe Barr and Wells, Category theory for computer science is useful. However, they only treat the very basics of monads (called triples in that book). Wadler wrote extensively about monads, the two papers Comprehending monads and Monads for functional programming look quite promising from their abstracts. Finally, the nlab page dedicated to monads in Computer Science. –  commenter Nov 14 '12 at 12:37
    
My advice is category theory is really hard and wont help you program haskell at all. IF you just want to learn a bit of category theory having some haskell will help and you can watch TheCatsters on youtube. –  user58512 Jan 20 '13 at 18:33

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