Bourbaki complained that in Arabic ‘al’ is an article so classical ‘bi-’ for two is incompatible with it. Thus Bourbaki and much of the French school makes it a point to say ‘cogebra, bigebra’ instead of ‘coalgebra, bialgebra’.
In modern algebra, dealing with co-operations instead of only with operations for some monoidal product , is a specialty of a group of people who are often centered around the study of Hopf algebras, but the subject is much wider including internal co(al)gebras, cogroups in cartesian categories, Tannaka reconstruction, PROPs (which have even operations taking arguments to arguments, hence generalizing both operations and co-operations), co-operads, bi(al)gebroids, Hopf algebroids, entwining structures, mixed distributive laws and so on. Common tools are graphical calculations (extending also to planar algebras and alike), Sweedler notation and so on.
We may call this area the theory of gebras as in the article
The entries about or closely related to gebras and co-operations in lab so far include bialgebra, Hopf algebra, comonoid, bimonoid, coalgebra, comonad, coring, Sweedler coring, cocategory, coaction, corepresentation, coinvariant, comodule algebra, cotensor product, induced comodule, module algebra, Hopf module, crossed product algebra, Hopf-Galois extension, measuring, cleft extension, quantum group, matrix Hopf algebra, Hopf envelope, bialgebroid, Hopf algebroid, distributive law, convolution algebra, Heisenberg double, noncommutative thin scheme, Lie bialgebra, Tannaka duality, quasi-Hopf algebra