What does the term mean and what is the best way to use it? And, I also wanted to know if there is any information about where it comes from. And by the way, how do we pronounce it?
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You seem to be asking about the origin of the term as used in category theory. The history of the term there is somewhat unclear, but it can at least be traced back a little ways:
The term is sometimes attributed to Mac Lane, but this seems to be inaccurate; however, the widespread use of the term is probably due to his influential "Categories for the Working Mathematician", replacing the remarkably terrible term "triple".
The frequent but unfortunate use of the word "triple" in this sense has achieved a maximum of needless confusion, what with the conflict with ordered triple, plus the use of associated terms such as "triple derived functors" for functors which are not three times derived from anything in the world. Hence the term monad.
Mac Lane's use of the term was apparently prompted by J. P. May:
The name "operad" is a word that I coined myself, spending a week thinking about nothing else. Besides having a nice ring to it, the name is meant to bring to mind both operations and monads. Incidentally, I persuaded MacLane to discard the term "triple" in favor of "monad" in his book "Categories for the working mathematician", which was being written about the same time. I was convinced that the notion of an operad was an important one, and I wanted the names to mesh.
Elsewhere, Ross Street attributes the term to Jean Bénabou:
Meanwhile Jean Bénabou had invented weak 2-categories, calling them bicategories. (...) He pointed out that a lax functor from the terminal category 1 to Cat was a category A equipped with a "standard construction" or "triple" (that is, a monoid in the monoidal category [A, A] of endofunctors of A where the tensor product is composition); he introduced the term monad for this concept.
The attribution to Bénabou is also mentioned here.
The motivation for the term is to suggest a relationship with monoids, as can be deduced from the construction given in the quote above, and the Greek root "monos" comes second-hand. The connection to philosophy in general, or Leibniz in particular, is often asserted but never to my knowledge supported in any way. More likely if anything would be a connection to the term "monad" used in non-standard analysis, also related to Leibniz, but I'm not sure what the conceptual link there would be. An anecdote from Michael Barr relates the first use of the term:
(...) The attendance consisted of practically everyone in the world who had any interest in categories, with the notable exception of Charles Ehresmann. (...) One day at lunch or dinner I happened to be sitting next to Jean Benabou and he turned to me and said something like "How about 'monad'?" I thought about and said it sounded pretty good to me. (Yes, I did.) So Jean proposed it to the general audience and there was general agreement.
The off-the-cuff nature of the suggestion, and immediate positive response from a large audience, suggests that there's probably no written record of the term being introduced formally. It's certainly possible that the word was borrowed from use in philosophy or elsewhere, but in any case there appears to be no connection more meaningful than the level of "cheap pun".
As far as I know, the only way you're going to get a better answer than that is by asking Bénabou himself.
Joyce is famous for his neologisms and Kalipedia is one such. The entire sentence reads:
Kalipedia, he prophesied, would soon be generally adopted and all the graces of life, genuinely good music, agreeable literature, light philosophy, instructive pictures, plastercast reproductions of the classical statues such as Venus and Apollo, artistic coloured photographs of prize babies, all these little attentions would enable ladies who were in a particular condition to pass the intervening months in a most enjoyable manner.
The word is constructed from the Greek kali (meaning beauty, as in calligraphy) and paedia (meaning education) and could be construed to mean education through beauty or perhaps enrichment through beauty. Joyce (or his character, Mr. Mulligan) very likely derived the word from the Greek phrase, Kalos kagathos:
Kalos kagathos (Ancient Greek καλὸς κἀγαθός [kalos kaːɡatʰǒs]), of which kalokagathia (καλοκαγαθία) is the derived noun, is a phrase used by classical Greek writers to describe an ideal of personal conduct, especially in a military context. Its use is attested since Herodotus and the classical period. The phrase is adjectival, composed of two adjectives, καλός ("beautiful") and ἀγαθός ("good" or "virtuous"), the second of which is combined by crasis with καί "and" to form κἀγαθός. Werner Jaeger summarizes it as ”the chivalrous ideal of the complete human personality, harmonious in mind and body, foursquare in battle and speech, song and action”.
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Best Answer
It's an onomatopoeic term for an explosion or perhaps something collapsing in an horrendous cascade of messiness.
It use is normally half-joking, certainly informal.
A variant is kablooie as in "Horrendous Space Kablooie", a term used in a comic for the Big Bang that some scientists prefer. Mostly because they have a sense of humour, but also because "Big Bang" was coined to ridicule the theory, not to name it.