2025/02/28 by Kvamme, Sondre
#18E20 #18E40 #Category Theory (math.CT) #FOS: Mathematics #Primary 18E99 #Representation Theory (math.RT) #Secondary 18G80
paper · doi:10.48550/arxiv.2502.21064
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.