Infinity-tilting theory

We define the notion of an infinitely generated tilting object of infinite homological dimension in an abelian category. A one-to-one correspondence between $\mathrm{\infty }$-tilting objects in complete, cocomplete abelian categories with an injective cogenerator and $\mathrm{\infty }$-cotilting objects in complete, cocomplete abelian categories with a projective generator is constructed. We also introduce $\mathrm{\infty }$-tilting pairs, consisting of an $\mathrm{\infty }$-tilting object and its $\mathrm{\infty }$-tilting class, and obtain a bijective correspondence between $\mathrm{\infty }$-tilting and $\mathrm{\infty }$-cotilting pairs. Finally, we discuss the related derived equivalences and t-structures.