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If we are speaking about HAs then the presence of $1$ makes the connected and the irreducibles the same thing. Since then the trivial (1-dim) comodule always makes sense.
@Spyros, a connected coalgebra has a unique simple comodule, which must be 1-dim. An irreducible one has a unique simple comodule. So at the level of coalgebras, connected are irreducibles (but not necessarily the other way around). So the irreducibles are a wider class.
It is true that during the first decades of the developments of the HA theory, the definitions were often assuming that we were talking about graded objects. This stems from the historical development of the topic and is still reflected in some modern references (especially those coming from the algebraic topology and the cohomology point of view). However, i think that most contemporary textbooks (i.e. those developed after the 80's), including Sweedler's textbook, have dropped this requirement and view graded HAs as a separate topic on its own.
