Questions: Slice and Coslice Categories

Question 1

In the slice category Set/S where S = {0, 1, 2}, which of the following correctly describes an object of Set/S?

• A. A subset A ⊆ S
• B. A function f: A → S, which partitions the set A into three labeled fibers f⁻¹(0), f⁻¹(1), f⁻¹(2)
• C. A set A with no additional structure, because S is the reference and provides all the structure
• D. A pair of sets (A, B) where A and B are both subsets of S

Answer: (B)

Objects of C/X are not objects of C — they are morphisms f: Y → X in C. In Set/S, an object is a function f: A → S, which assigns each element of A a label in S. This is equivalent to an S-indexed family of sets: the fiber f⁻¹(s) for each s ∈ S. Option A (subset of S) confuses the objects of C with objects of C/X. Option C misses that the object is the function, not just the domain set. The key shift is that in a slice category, *morphisms become objects*.

Question 2

A morphism in the slice category C/X from (Y, f: Y → X) to (Z, g: Z → X) is a morphism h: Y → Z in C. What additional condition must h satisfy?

• A. h must be an isomorphism in C
• B. h must factor through X — there must exist a morphism X → Z
• C. The triangle must commute: g ∘ h = f, so h preserves the relationship of Y and Z to X
• D. No additional condition — any morphism h: Y → Z in C is automatically a morphism in C/X

Answer: (C)

A morphism in C/X is not just any morphism between the underlying objects — it must make the triangle over X commute. The condition g ∘ h = f says that h is compatible with both objects' maps to X: following h and then g gives the same result as following f directly. Without this condition, C/X would just be C (ignoring X entirely), not a genuinely new 'relative' category. Option D is the most tempting wrong answer and captures the misconception that the slice category is just a full subcategory of C.

Question 3

The slice category C/X is a full subcategory of C, with objects being those Y for which there exists at least one morphism Y → X.

• T. True
• F. False

Answer: False

This is a precise and important distinction. First, objects of C/X are morphisms f: Y → X — not just objects Y. Multiple distinct objects of C/X can have the same underlying object Y (if there are multiple morphisms Y → X). Second, C/X is not full: a morphism in C/X between (Y, f) and (Z, g) is not every morphism Y → Z in C, only those h satisfying g ∘ h = f. C/X imposes an additional constraint that makes it a *relative* structure, not a restriction of C.

Question 4

In Set/S, a morphism h from (A, f: A → S) to (B, g: B → S) maps each fiber f⁻¹(s) into the corresponding fiber g⁻¹(s) for every s ∈ S.

• T. True
• F. False

Answer: True

The morphism condition in Set/S is g ∘ h = f. If a ∈ A and f(a) = s, then g(h(a)) = f(a) = s, so h(a) ∈ g⁻¹(s). This means h maps elements labeled s by f to elements labeled s by g — exactly fiber-preservation. This is the concrete meaning of the abstract commutativity condition in Set/S: morphisms of S-indexed families must respect the indexing, sending the s-fiber to the s-fiber for every s.

Question 5

Explain what makes a universal property in a slice category C/X 'relative' to X, and in what sense is this weaker than an absolute universal property in C itself?

Model answer: A universal property in C/X is defined within the world of objects-over-X: it asserts that some object (A, f: A → X) is initial or terminal among all objects equipped with a morphism to X, with morphisms being commuting triangles. The entire notion of 'unique' morphism is relative to this constrained universe — uniqueness among morphisms h: (Y, g) → (A, f) in C/X, not among all morphisms Y → A in C. Changing X produces an entirely different slice category with different universal objects. An absolute universal property in C holds with no reference to a fixed base object. The slice property is weaker because the uniqueness is asserted only within the restricted context where everything comes with a map to X; outside that context, the object may have no universal character at all.

This question targets the conceptual core of why slice categories matter: they relativize categorical notions, making them context-dependent. Understanding that universal properties in C/X depend on the choice of X — and fail to hold absolutely in C — is the key to grasping how slice categories generalize the idea of 'structure over a base,' which is the foundation for fibrations, dependent types, and parameterized constructions throughout mathematics.