In the category Set, what plays the role of morphisms?
ASubsets of a given set
BFunctions between sets
CElements belonging to a set
DEquivalence relations on a set
In Set, objects are sets and morphisms are functions between sets. Composition is function composition (which is associative), and identity morphisms are identity functions (which send every element to itself). This is the canonical example of a category and the one that most closely matches intuitions from prior mathematics.
Question 2 True / False
In most category, nearly every morphism should be a function that maps elements from one object to elements of another.
TTrue
FFalse
Answer: False
Morphisms need not be functions. In a poset (partially ordered set) viewed as a category, objects are elements of the poset and there is at most one morphism from a to b — it exists if and only if a ≤ b. This morphism records an ordering relationship, not a mapping of elements. Category theory deliberately abstracts away internal structure, so 'morphism' means only 'arrow satisfying the composition and identity axioms,' not 'function.'
Question 3 Short Answer
State the two axioms that composition of morphisms must satisfy in any category.
Think about your answer, then reveal below.
Model answer: Associativity: (h ∘ g) ∘ f = h ∘ (g ∘ f) whenever defined. Identity: for every object A there is id_A such that f ∘ id_A = f and id_B ∘ f = f for any morphism f: A → B.
Associativity ensures that chaining multiple morphisms is unambiguous — parenthesization does not matter. The identity axiom ensures every object has a 'do-nothing' arrow that acts as a neutral element under composition, analogous to 0 under addition or 1 under multiplication. These two axioms are the entire algebraic content of the definition of a category.