(i) Prove that every isomorphism in an additive category is...

(i) Prove that every isomorphism in an additive category is both monic and epic. (ii) Prove that a morphism in an abelian category is an isomorphism if and only if it is both monic and epic. (iii) Prove, in ComRings, that $\varphi: R \rightarrow \operatorname{Frac}(R)$ is both monic and epic, but that $\varphi$ is not an isomorphism.

(i) Prove that every isomorphism in an additive category is both monic and epic.
(ii) Prove that a morphism in an abelian category is an isomorphism if and only if it is both monic and epic.
(iii) Prove, in ComRings, that $\varphi: R \rightarrow \operatorname{Frac}(R)$ is both monic and epic, but that $\varphi$ is not an isomorphism.

Key Concepts

Isomorphism

An isomorphism in a category is a morphism that has an inverse; that is, for morphism f, there exists a morphism g such that f ? g and g ? f are the respective identity morphisms. This concept captures the idea of structural equivalence between objects in a category, meaning that they are indistinguishable from the categorical perspective.

Monomorphism

A monomorphism is a morphism that is left-cancellable; formally, a morphism f is monic if whenever f ? g1 = f ? g2, then g1 must equal g2. This property is analogous to injectivity in familiar algebraic settings and ensures that the morphism does not ‘identify’ distinct elements in its domain.

Epimorphism

An epimorphism is a morphism that is right-cancellable; that is, f is epic if whenever g1 ? f = g2 ? f, then g1 must equal g2. In many cases, especially within concrete categories, epimorphisms play a role similar to surjections, although the categorical definition may differ from the set-theoretic notion of ‘onto’.

Additive Category

An additive category is a category in which every hom-set is abelian and where finite biproducts (acting as both products and coproducts) exist. In this setting, the structure allows the use of addition and ‘zero’ morphisms, and ensures that constructions like kernels and cokernels have additive properties that are essential for homological considerations.

Abelian Category

An abelian category is an additive category that is especially well-behaved; it has all kernels and cokernels, and every monomorphism and epimorphism is normal (meaning every monomorphism is a kernel of some morphism and every epimorphism is a cokernel of some morphism). Abelian categories form the framework for homological algebra, providing an environment where exact sequences and other concepts can be rigorously studied.

Epimorphisms in Commutative Rings

In the category of commutative rings (ComRings), the categorical notions of monomorphisms and epimorphisms do not always align with the familiar injective or surjective functions. For instance, a ring homomorphism may be an epimorphism in ComRings even if it is not surjective in the set-theoretic sense. This phenomenon, illustrated by mappings such as the canonical inclusion of a ring into its field of fractions, highlights the subtle difference between categorical epimorphisms and the usual notion of surjectivity, and underscores the need for careful categorical reasoning in algebra.

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