Arrows first, elements never
A category is objects, morphisms between them, associative composition and identities. Nothing in that definition mentions elements, so every notion must be recast in terms of arrows: a monomorphism is left-cancellable rather than injective, an epimorphism right-cancellable rather than surjective. For modules these agree with the familiar meanings; in other categories they come apart, and knowing where is part of using the language safely.
Learning objectives
- State the axioms of a category and give the standard examples.
- Define mono, epi and iso by cancellation.
- Give a category where epi is not surjective.
- Explain why arrow-theoretic definitions are necessary.
Section 01The definition and examples
A category C consists of a class of objects, a set C(A, B) of morphisms for each ordered pair, an associative composition, and an identity at each object. That is the whole definition.
| Category | Objects | Morphisms |
|---|---|---|
| Set | Sets | Functions |
| Ab | Abelian groups | Group homomorphisms |
| ModΛ | Left Λ-modules | Λ-homomorphisms |
| Ch(C) | Chain complexes in C | Chain maps |
| Grp | Groups | Group homomorphisms |
| A group G | One object | The elements of G, composition = multiplication |
| A poset | Its elements | One arrow x → y when x ≤ y |
A single group is a category with one object, and a partially ordered set is a category with at most one arrow between objects. Keeping these in mind stops the intuition from collapsing into ‘sets with structure’, which is where the arrow-theoretic definitions start to matter.
Section 02Mono, epi and iso
Left-cancellable. In module categories this coincides with injective.
Right-cancellable. In module categories this coincides with surjective — but not in every category.
In the category of rings, the inclusion ℤ ↪ ℚ is an epimorphism: any two ring maps out of ℚ agreeing on ℤ are equal, because a ring map is determined on fractions. It is plainly not surjective. In Hausdorff spaces, epimorphisms are the maps with dense image. Assuming epi means surjective is the standard trap.
An isomorphism is a morphism with a two-sided inverse. Mono and epi together do not imply iso in general — the ring inclusion above is both and is not an isomorphism. In abelian categories they do, which is one of the properties that makes abelian categories comfortable.
Section 03Working without elements
Universal properties
Characterise an object by the maps into or out of it. Determines the object up to unique isomorphism, and transplants to any category.
Diagram chasing by embedding
Freyd–Mitchell embeds any small abelian category in a module category, so element-based proofs of diagram lemmas are valid in general.
Generalised elements
Treat a morphism X → A as an ‘element of A of shape X’. Recovers element-style reasoning intrinsically.
Because the theorems of homological algebra are wanted for modules, for sheaves of modules, for chain complexes and for functor categories. Proving them once in arrow language covers all of these; proving them with elements covers only the first.
ReferenceFrequently asked questions
Is a category the same as a class of structures?
No. Many categories — a group viewed as a one-object category, a poset, a category of paths — have objects that are not structured sets at all. The definition asks only for arrows and composition.
Why require Hom to be a set?
To avoid size paradoxes. Categories with this property are called locally small; nearly every category in this subject is. Without it, constructions like the functor category can fail to exist.
Do monomorphisms always have kernels?
Only in categories with enough structure. In an abelian category every morphism has a kernel and a cokernel, and a monomorphism is exactly a morphism with zero kernel — which restores the familiar picture.
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