I mean groups and not abelian groups. In Grp, categorically, he product is the cartesian product and the coproduct is the free product. So what place takes the direct sum? Can it be defined categorically in Grp or any similar category? I would be surprised if can't be defined using limits of co/cones or something similar. If not, are they really useful (the direct sum of groups) for something?
2026-03-26 19:06:44.1774552004
Direct sum of groups categorically
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The direct sum $\bigoplus_i G_i$ is the "commutative coproduct": it's the universal group admitting a map from each of the $G_i$ whose images commute. The same construction for a finite collection of $k$-algebras produces the tensor product of $k$-algebras.