Problems formulated with adjoint functors 1.3 Adjoint functors as solving optimization problems 1.4 The case of partial orders 2 Formal definitions...

Diagonal functor : The diagonal functor is defined as the functor from D to the functor category D C which sends each object in D to the constant f...

Yoneda's lemma asserts that every natural transformation between Hom functors is of this form. In other words, the Hom functors give rise to a full...

C ) can be made precise in several ways; the most succinct formulation uses the language of adjoint functors . Every functor F  : D ? E induces a f...

It follows that any functor which preserves limits will take terminal objects to terminal objects, and any functor which preserves colimits will ta...

contravariant functor is right exact if and only if it turns finite limits into colimits. A functor is exact if and only if it is both left exact a...

Ab is injective if and only if it is divisible ; it is projective if and only if it is a free abelian group. The category has a projective generato...

Indeed, the term "zero object" originated in the study of preadditive categories like Ab , where the zero object is the zero group . A preadditive ...

As mentioned above, the category of all abelian groups is an abelian category. The category of all finitely generated abelian groups is also an abe...

If there is a monoidal functor from a monoidal category M to a monoidal category N , then any category enriched over M can be reinterpreted as a ca...

C . (Recall that a category C is preadditive if all its morphism sets are Abelian groups and morphism composition is bilinear , i.e. if C is enrich...

Limits and universal morphisms Colimits in comma categories may be "inherited". If and are cocomplete, is a cocontinuous functor, and another funct...