Timeline for Functors on the category of abelian groups which satisfy $F(G\times H) \cong F(G)\otimes_{\mathbb{Z}} F(H)$

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Jun 4, 2019 at 13:14	history	made wiki			Post Made Community Wiki by Todd Trimble
Jun 4, 2019 at 8:27	comment	added	Simon Henry		Also, any functor $F: Ab \rightarrow Ab$ with the expected property is automatically a functor from $Ab$ to $CRing$, indeed due to the fact that the product in $Ab$ is also the coproduct, every object in $Ab$ has a unique monoid structure for the cartesian monoidal structure, and every morphism is a morphism of monoids, so for any functor $F$ from Ab to Ab which is monoidal from the cartesian to the tensor monoidal structure, $F(X)$ has a canonical ring structure for all $X$, a functoriality in $X$ is compatible to the ring structure. A similar argument show they are in fact Hopf algebras.
Jun 3, 2019 at 12:05	comment	added	Denis Nardin		It might be worth mentioning that every right adjoint functor $\mathrm{CRing}→\mathrm{Ab}$ is of the form you describe (this is just because a right adjoint functor $\mathrm{CRing}→\mathrm{Set}$ is necessarily corepresentable plus some Yoneda)
Jun 3, 2019 at 2:56	history	answered	Tim Campion	CC BY-SA 4.0