Which is the origin and the reason for the choice of the term "monad" in the sense of homological algebra?
Does this concept have any relation whatsoever to the "monads" from category theory?
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3$\begingroup$ I don't know why they're called that, but the Wikipedia article you link to dates this usage to 1964, whereas I'm pretty sure the category theory usage was coined later than that (late 1960s, I guess). You weren't asking this, but I might as well add: in category theory, the word "monad" is intentionally like "monoid", since from an appropriately elevated viewpoint, monads and monoids are the same thing. $\endgroup$– Tom LeinsterCommented Jan 21, 2014 at 23:24
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$\begingroup$ One other mathematical usage of monad comes to mind: in logic, unary predicates (i.e. “X is a dog”, but not relations like “X loves Y”) are called monadic predicates, and predicate logic based just on monadic predicates is called monadic logic. The earliest use of this terminology I can find is 1956, when Halmos wrote a paper titled monadic Boolean algebras, but the study of such logics goes back a couple of decades further, and I suspect the terminology does as well, though my google-fu is failing me. $\endgroup$– Peter LeFanu LumsdaineCommented Jan 22, 2014 at 0:02
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3$\begingroup$ Well, I learned something new. There seems to be a reasonable consensus that the term "monad" in category theory is due to Benabou (with apologies to Peter May): english.stackexchange.com/a/30661. $\endgroup$– Todd TrimbleCommented Jan 22, 2014 at 0:50
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$\begingroup$ To my knowledge, the term monad has been 'invented' by Leibniz; maybe knowing his understanding of the term gives some clues, why it is used in certain mathematical contexts. $\endgroup$– Manfred WeisCommented Jan 22, 2014 at 5:23
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2$\begingroup$ Todd, apologies to me? For what? As correctly reported in the link you give, I persuaded MacLane to use the (preexisting) term monad in place of triple in his book. No claim that either Saunders or I introduced the term (I'm guilty of the name ``operad'', part of the point of which is that it goes so nicely with monad.) $\endgroup$– Peter MayCommented Jan 24, 2014 at 3:04
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2 Answers
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A projective resolution "splits" into monads in the following way. Each exact complex of projectives $\dots \to P_{n+1} \to P_n \to P_{n-1} \to \dots $ is a gluing of monads $0\to Z_{n} \to P_n \to Z_{n-1} \to 0$, where $Z_{k} $ is the kernel of the map $P_{k} \to P_{k-1} $. So, a monad is an elementary piece of any projective resolution, like Leibniz's monads are the elementary pieces of more complicated entities (as far as I know).
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To appreciate the usage of "monad" as a concept in mathematics (rather than philosophy), it might help to go back nearly two millennia to the first use of this term in algebra [*]: Diophantus of Alexandria used monad (denoted $\dot{M}$) in his book Arithmetika to indicate the zeroth power of the unknown variable. "All numbers consist of a certain multitude of monads." Think of "monad" as "unit-one". In a later development, Archimedes and Apollonius used monads as building blocks for larger units , for example "myriad" was the "unit-ten-thousand" and "chiliad" the "unit-thousand". Archimedes called these "numbers formed analogously to those based on monads". In this way large numbers could be manipulated with ease.
It would seem the modern usage of "monad" follows this ancient line of thought, to indicate "a single entity that generates all other entities".
[*] Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (Dover, 1992).
answered Jan 22, 2014 at 9:05
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2$\begingroup$ That's informative, but I'm still puzzled. I can see that monads in category theory have something "single" about them (just as monoids do). But how does a three-term complex with certain properties have a feeling of "singleness" about it? $\endgroup$ Commented Jan 23, 2014 at 15:00
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$\begingroup$ OK, I think user42369 has answered my question. $\endgroup$ Commented Jan 24, 2014 at 1:28