$d$-abelian quotients of $(d+2)$-angulated categories

Let $\mathcal{T}$ be a triangulated category. If $T$ is a cluster tilting object and $I=[\mathrm{add}T]$ is the ideal of morphisms factoring through an object of $\mathrm{add}T$, then the quotient category $\mathcal{T}\mathrm{/}I$ is abelian. This is an important result of cluster theory, due to Keller-Reiten and K"{o}nig-Zhu. More general conditions which imply that $\mathcal{T}\mathrm{/}I$ is abelian were determined by Grimeland and the first author. Now let $\mathcal{T}$ be a suitable $(d+2)$-angulated category for an integer $d⩾1$. If $T$ is a cluster tilting object in the sense of Oppermann-Thomas and $I=[\mathrm{add}T]$ is the ideal of morphisms factoring through an object of $\mathrm{add}T$, then we show that $\mathcal{T}\mathrm{/}I$ is $d$-abelian. The notions of $(d+2)$-angulated and $d$-abelian categories are due to Geiss-Keller-Oppermann and Jasso. They are higher homological generalisations of triangulated and abelian categories, which are recovered in the special case $d=1$. We actually show that if $\mathrm{\Gamma }={\mathrm{End}}_{\mathcal{T}}T$ is the endomorphism algebra of $T$, then $\mathcal{T}\mathrm{/}I$ is equivalent to a $d$-cluster tilting subcategory of $\mathrm{mod}\mathrm{\Gamma }$ in the sense of Iyama; this implies that $\mathcal{T}\mathrm{/}I$ is $d$-abelian. Moreover, we show that $\mathrm{\Gamma }$ is a $d$-Gorenstein algebra. More general conditions which imply that $\mathcal{T}\mathrm{/}I$ is $d$-abelian will also be determined, generalising the triangulated results of Grimeland and the first author.