Tag
Categories & Functors
8 resources tagged “Categories & Functors” across the knowledge library.
8 items
Guide6 Aug 2026Abelian CategoriesAdditive and abelian categories, the axioms, exactness in a general abelian category, the Freyd-Mitchell embedding theorem, and the standard examples.4 min readRead MoreGuide6 Aug 2026Adjoint FunctorsAdjoint pairs, unit and counit, the tensor-hom adjunction, preservation of limits and colimits, and the exactness consequences that make adjointness central to homological algebra.4 min readRead MoreGuide6 Aug 2026Categories and MorphismsCategories, objects and morphisms, monomorphisms and epimorphisms defined by cancellation, isomorphisms, and why the arrow-theoretic definitions differ from the element-based ones.3 min readRead MoreGuide6 Aug 2026Duality and Opposite CategoriesThe opposite category, the duality principle, dual pairs of notions in homological algebra, and the limits of formal duality.3 min readRead MoreGuide6 Aug 2026Functors and Their Exactness PropertiesCovariant and contravariant functors, composition, additive functors, faithful and full functors, and the exactness hierarchy that governs derived functors.3 min readRead MoreGuide6 Aug 2026Natural TransformationsNatural transformations and naturality squares, natural isomorphisms, functor categories, and the role of naturality in the long exact sequences.4 min readRead MoreGuide6 Aug 2026Products, Coproducts and Universal ConstructionsUniversal properties, products and coproducts in a general category, initial and terminal objects, and uniqueness up to unique isomorphism.3 min readRead MoreGuide6 Aug 2026Pullbacks, Pushouts and General LimitsPullbacks and pushouts, their construction in module categories, limits and colimits over a diagram, and exactness of filtered colimits.4 min readRead More