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?
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? 


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_{n1} \to \dots $ is a gluing of monads $0\to Z_{n} \to P_n \to Z_{n1} \to 0$, where $Z_{k} $ is the kernel of the map $P_{k} \to P_{k1} $. 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). 


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 "unitone". In a later development, Archimedes and Apollonius used monads as building blocks for larger units , for example "myriad" was the "unittenthousand" and "chiliad" the "unitthousand". 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). 

