Tilting Preenvelopes and Cotilting Precovers in General Abelian Categories

We consider an arbitrary Abelian category $\mathcal {A}$ and a subcategory {T}$ closed under extensions direct summands, characterize those that are (semi-)special preenveloping in ; as byproduct, we generalize to this setting several classical results for categories of modules. For instance, get the special subcategories summands precisely which $({}^{\perp _{1}}\mathcal {T},\mathcal {T})$ is right complete cotorsion pair, where ${}^{\perp {T} := \text {Ker} (\text {Ext}_{\mathcal {A}}^{1}(-,\mathcal {T}))$ . Particular cases appear when {T}=V^{\perp _{1}}:=\text {A}}^{1}(V,-))$ , Ext1-universal object V such $\text {A}}^{1}(V,-)$ vanishes on all (existing) coproducts copies V. many choices show these latter examples exhaust possibilities. then that, has epi-generator, torsion classes given by (quasi-)tilting objects exactly any $T\in \mathcal epimorphic image some (and ${\mathscr{B}}:=\text {Sub}(\mathcal subobjects reflective) they are, turn, constituents pairs (resp., ${\mathscr{B}}$ ). In final section, apply {A}=\text {mod-}R$ finitely presented modules over coherent ring R, something gives new raises questions even at level tilting theory