Modern pure sciences (including algebra) are getting so advanced in theoretical point of view and practitioners (engineers, biologists, technologists, ... ) are keep asking us the reason of studying pure sciences. As a pure science researcher we are clear about our research, goals and motivations, but sometimes would be difficult to answer for those who are more concerned about applications.
According to Zwanenburg, Poll, like algebras provide the basis for algebraic specification, coalgebras have been used as the basis for coalgebraic specification, notably in the experimental specification language CCSL. A notion of dialgebraic specification, defined in exactly the same way. Because of lack of space, many details of definitions and proofs have to be omited.
Thus some coallgebras real life application can be provided, fo example:
https://core.ac.uk/download/pdf/82357754.pdf
As an example of a Leibniz algebra we have the Dorfman bracket associated with a Courant algebroid (really a symplectic Lie 2-algebroid). This and related things come up in mechanics and quantisation with constraints and quite recently double field theory. I would say that geometric mechanics with constraints is quite "real life".
While I do think that my research should be grounded in applications in physics and geometric mechanics, you are quite free to study abstract notions simply for the intellectual challenge. You could ask a poet why he writes poems that seem to be little use to anyone. The reason is culture. Mathematics is part of intellectual culture.
PURE SCIENCE IS ALSO IMPORTANT IN PROVING THEORIES
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