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I don’t think the quote of Rutten is so much a hard truth as it is a general piece of wisdom. It reminds me very much of Freyd’s quip, “Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial,” which is glossed here at the nLab. But I can think of cases where the soft methods of category theory and universal algebra lead directly to solutions of problems that some people have apparently found hard.
Anyway, you might expand your search terms to include things like “Hopf algebras” or “comonoids”.
In all such discussions, a key question is the background in which we define comonoids and cogroups: what is the monoidal product relative to which we define them? For example, if we use cartesian product as the monoidal product, then a famous exercise is that every object carries exactly one comonoid structure (which makes them uninteresting in some sense). Often, when people speak of cogroups, they dualize the notion or group (which uses cartesian product) so that the relevant monoidal product is the coproduct. However, in the category of sets (or other categories where the initial object is strict, e.g., a cartesian closed category or an extensive category such as $Set$ or $Top$, the only cogroup is the initial object! For this reason, you basically never hear about cogroups except in the context of cogroup objects! For example, the spheres famously have cogroup object structures in the category of pointed topological spaces.
I don’t have a great answer why cogroup objects etc. are less studied than their dual counterparts, other than to say the collective consciousness has a lot more experience with induction than co-induction, or that some concepts as applied to concrete structures seem to be harder for human brains than their dual counterparts, e.g., subobjects are usually easier in concrete settings than quotient objects.
Perhaps one idea that is relevant is to look at the coalgebras that arise as models of modal logics. In that area one has also algebra type structures since on has an extended Stone duality theory. The algebras are Boolean algebras with operators, and the coalgebras various types of relational structure. The different relational structures do get studied in different ways, using different tools. For instance, equivalence relations are usually not thought of from a coalgebraic viewpoint, and they tend to be used rather than studied! They are examples of groupoids and that can be useful, and the underlying graph is a good intuitive way of handling them. Perhaps one needs to look at the covarieties of the relational structures / coalgebras rather than all of them.
Finally groups are useful because they correspond to symmetries of objects. Cogroups need not be so useful, unless they arise in some area as ???cosymmetries???.
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