Category
Objects, morphisms with specified source and target, identity morphisms, and associative composition with the identity laws.
Find a concept
REFERENCE SHEET
Categories & morphisms
Describe structure through composable maps, without assuming the objects are sets with elements.
Before this Sets & maps, Groups & subgroups
(h∘g)∘f = h∘(g∘f)
Definitions
Isomorphism
A morphism f:X→Y with a two-sided inverse g:Y→X. The relevant category determines what structure is preserved.
Monomorphism and epimorphism
A monomorphism is left-cancellable: f∘g=f∘h implies g=h. An epimorphism is right-cancellable: g∘f=h∘f implies g=h.
Key theorems
Inverse uniqueness
A morphism with a two-sided inverse has exactly one such inverse.
Isomorphisms cancel
Every isomorphism is both a monomorphism and an epimorphism; the converse need not hold in a general category.
Set-specific characterization
In Set, monomorphisms are injections and epimorphisms are surjections. This characterization is not a definition valid in all categories.
Notable applications
Shared language
Groups and homomorphisms, spaces and continuous maps, and vector spaces and linear maps each form categories.
Orders as categories
A poset defines a category with one arrow x→y when x≤y and no arrow otherwise.
Work it out
Try before revealing01 / Prove
If g and h are both two-sided inverses of f, show g=h.
Hint
Insert the identity as f∘h.
Solution
g=g∘id=g∘(f∘h)=(g∘f)∘h=id∘h=h.
02 / Identify
In the category associated to a poset, when are two objects isomorphic?
Hint
An inverse requires arrows in both directions.
Solution
Exactly when they are equal: arrows both ways mean x≤y and y≤x, and antisymmetry gives x=y.
Connect the ideas
Build on this
These sheets use this topic as a prerequisite.
Go deeper
Further reading and full-length treatments: