Higher extension closure and $d$-exact categories

We prove that any weakly idempotent complete $d$-exact category is equivalent to a $d$-cluster tilting subcategory of a weakly idempotent complete exact category, and that any weakly idempotent complete algebraic $(d+2)$-angulated category is equivalent to a $d$-cluster tilting subcategory of an algebraic triangulated category closed under $d$-shifts. Furthermore, we show that the ambient exact category of a $d$-cluster tilting subcategory is unique up to exact equivalence, assuming it is weakly idempotent complete. This follows from the inclusion of the $d$-cluster tilting subcategory satisfying a universal property. As a consequence of our theory we also get that any $d$-torsion class is $d$-cluster tilting in an extension-closed subcategory, and we recover the fact that any $d$-wide subcategory is $d$-cluster tilting in a unique wide subcategory. In the last part of the paper we rectify the description of the $d$-exact structure of a $d$-cluster tilting subcategory of a non-weakly idempotent complete exact category.