Early undergraduateCategory TheoryPlanned
Categories
Categories as objects and morphisms with associative composition and identities; the categories of sets, groups, vector spaces, topological spaces and posets; monoids and preorders as categories; isomorphisms; and opposite categories with the duality principle.
math.cat.categories
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Structure-preserving maps: homomorphisms and isomorphismsAlgebraic structures: monoids, groups and rings
Required prerequisites, all the way down: 14
You will learn to
- State the axioms of a category and verify them for Set, Grp, Vect_k, Top and a poset regarded as a category.
- Recognise a monoid as a one-object category, and a preorder as a category with at most one morphism between any two objects.
- Define isomorphisms in a category and prove that inverses are unique.
- Form the opposite category and dualise simple statements, for example exchanging monomorphisms and epimorphisms.
Definition
- A category C consists of objects; morphisms f : A → B between them; a composite g ∘ f for each pair of composable morphisms; and an identity morphism id_A for each object, such that composition is associative and identities are two-sided units.
- A morphism f : A → B is an isomorphism if there is g : B → A with g ∘ f = id_A and f ∘ g = id_B.
- The opposite category Cᵒᵖ has the same objects as C and one morphism B → A for each morphism A → B of C, with composition reversed.
- f is a monomorphism if f ∘ g = f ∘ h implies g = h; epimorphisms are defined dually.
Theorem
- Inverses in a category are unique when they exist.
- In Set, the monomorphisms are the injections and the epimorphisms are the surjections.
- In the category of rings with identity, the inclusion ℤ → ℚ is an epimorphism but is not surjective.
- Duality principle: if a statement holds in every category, so does the dual statement obtained by reversing all arrows.
Common misconceptions
- Thinking morphisms must be functions; in a poset regarded as a category, a morphism is just the fact that x ≤ y.
- Assuming epimorphisms are always surjective.
- Believing the objects of a category must have elements.
Leads to
Optional discoveries
- Natural transformations Once there are maps between categories (functors), one can ask for maps between functors: natural transformations.Wonder Trail →
- Higher categories Categories themselves, with functors and natural transformations, form a 2-category, the starting point of higher category theory.Wonder Trail →
- Categorical logic Categories can also model logic and computation, with types or propositions as objects and programs or proofs as morphisms.
- ← Diagrams where paths agree
- ← Composition
- ← Orders, lattices and well-orders
- ← Linear algebra
- ← Transformation geometry and the Erlangen programme
- ← Quivers and path algebras
Learning pathways
Further reading
- Tom Leinster, Basic Category Theory (2014)
- Steve Awodey, Category Theory (2006)
- Saunders Mac Lane, Categories for the Working Mathematician (1971)