Functors Main article: functor Functors are structure-preserving maps between categories. They can be thought of as morphisms in the category of al...

Preservation of limits Representable functors are naturally isomorphic to Hom functors and therefore share their properties. In particular, (covari...

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...

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

Universal cones Limits and colimits are defined as universal cones . That is, cones through which all other cones factor. A cone ? from L to F is a...

A bimorphism is a morphism that is both an epimorphism and a monomorphism. Isomorphism : f  : X ? Y is called an isomorphism if there exists a morp...

It follows that if coproducts exists in a given category (they need not) they are unique up to a unique isomorphism that respects the injections. I...

X and is the operation on Y . Each type of algebraic structure has its own type of homomorphism. For specific definitions see: group homomorphism r...

R -Mod, an injective object is an injective module . R -Mod has injective hulls (as a consequence, R-Mod has enough injectives). In the category of...

X i , f ij ) be a direct system of objects and morphisms in a category C (same definition as above). The direct limit of this system is an object X...

The functor category of all additive functors from this category to the category of abelian groups is isomorphic to the category of left modules ov...

Among other useful concepts are regular epimorphism , extremal epimorphism , strong epimorphism , and split epimorphism . A regular epimorphism coe...