$t$-Structures with Grothendieck hearts via functor categories

We study when the heart of a t-structure in a triangulated category $\mathcal{D}$ with coproducts is AB5 or a Grothendieck category. If $\mathcal{D}$ satisfies Brown representability, a t-structure has an AB5 heart with an injective cogenerator and coproduct-preserving associated homological functor if, and only if, the coaisle has a pure-injective t-cogenerating object. If $\mathcal{D}$ is standard well generated, such a heart is automatically a Grothendieck category. For compactly generated t-structures (in any ambient triangulated category with coproducts), we prove that the heart is a locally finitely presented Grothendieck category. We use functor categories and the proofs rely on two main ingredients. Firstly, we express the heart of any t-structure in any triangulated category as a Serre quotient of the category of finitely presented additive functors for suitable choices of subcategories of the aisle or the co-aisle that we, respectively, call t-generating or t-cogenerating subcategories. Secondly, we study coproduct-preserving homological functors from $\mathcal{D}$ to complete AB5 abelian categories with injective cogenerators and classify them, up to a so-called computational equivalence, in terms of pure-injective objects in $\mathcal{D}$. This allows us to show that any standard well generated triangulated category $\mathcal{D}$ possesses a universal such coproduct-preserving homological functor, to develop a purity theory and to prove that pure-injective objects always cogenerate t-structures in such triangulated categories.