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For an exact category    $${{\mathcal {E}}}$$  E with enough projectives and with a    $$d\mathbb {Z}$$  d Z -cluster tilting subcategory, we show that the singularity category of    $${{\mathcal {E}}}$$  E admits a    $$d\mathbb {Z}$$  d Z -cluster tilting subcategory. To do this we introduce cluster tilting subcategories of left triangulated categories, and we show that there is a correspondence between cluster tilting subcategories of    $${{\mathcal {E}}}$$  E and    $${\underline{{{\mathcal {E$$  E ̲ . We also deduce that the Gorenstein projectives of    $${{\mathcal {E}}}$$  E admit a    $$d\mathbb {Z}$$  d Z -cluster tilting subcategory under some assumptions. Finally, we compute the    $$d\mathbb {Z}$$  d Z -cluster tilting subcategory of the singularity category for a finite-dimensional algebra which is not Iwanaga–Gorenstein.

$$d\mathbb {Z}$$-Cluster tilting subcategories of singularity categories