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Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C1, the second term of the coradical filtration of C. If C = C1, we associate with C a K-linear functor HC : C-Comod → HC-Comod that restricts to a representation equivalence HC : C-comod → HC-comod • sp, where HC is a coradical square complete hereditary bipartite K-coalgebra such that every simple HC comodule is injective or projective. Here HC-comod • sp is the full subcategory of HC-comod whose objects are finite-dimensional HC-comodules with projective socle having no injective summands of the form [ S(i) 0 ] (see Theorem 5.11). Hence, we conclude that a coalgebra C with C = C1 is left pure semisimple if and only if HC is left pure semisimple. In Section 6 we get a diagrammatic characterisation of coradical square complete coalgebras C that are left pure semisimple. Tameness and wildness of such coalgebras C is also discussed.

Bipartite coalgebras and a reduction functor for coradical square complete coalgebras