Accessibility and Gorenstein injective envelopes

Let $\mathcal{G}$ be a Grothendieck category. We prove completeness of the Gorenstein injective cotorsion pair whenever $\mathcal{G}$ admits a set of Tate trivial generators, and show that having such generators is necessary for completeness. In this case it must be a perfect cotorsion pair, cogenerated by a set, and equivalent to an injective abelian model structure on $\mathcal{G}$. Examples include Grothendieck categories (possibly without enough projectives) that admit a generating set consisting of objects of finite projective dimension, such as the category of quasi-coherent sheaves on a quasi-compact and semi-separated scheme. More generally, for a given set $\mathcal{S}$, we characterize the completeness of the Gorenstein $\mathcal{B}$-injective cotorsion pair, where $\mathcal{B} = \mathcal{S}^\perp$, in terms of the existence of a set of $\mathcal{B}$-Tate trivial generators for $\mathcal{G}$. The key ingredient to our proof is the fact that any class of the form $\mathcal{B} :=\mathcal{S}^\perp$ is an accessibly embedded, accessible subcategory of $\mathcal{G}$. The general approach allows for further applications such as the existence of Ding injective envelopes and other relative Gorenstein injective envelopes without imposing additional assumptions on $\mathcal{G}$.