Questions: Limits and Colimits

Question 1

The universal property of a limit L over a diagram D: J → C says that for any other cone (N, f) over D:

• A. There exists at least one morphism from N to L compatible with the cone morphisms
• B. L is the smallest object in C that receives morphisms from every object in the image of D
• C. There exists a unique morphism from N to L that commutes with all the cone morphisms to D(j)
• D. There exists a unique morphism from L to N that commutes with all the cone morphisms to D(j)

Answer: (C)

The limit is a terminal cone: any other cone N must factor through L via a *unique* morphism N → L making all the relevant triangles commute. Option A is too weak (merely 'at least one' omits the essential uniqueness). Option B is the 'smallest object' misconception — size has no direct role; uniqueness of factorization is what matters. Option D reverses the direction.

Question 2

The limit of a diagram is the smallest object that maps to nearly every node in the diagram.

• T. True
• F. False

Answer: False

This is the most common misconception about limits. The limit is not characterized by size but by the universal property: every other cone factors through it *uniquely*. There may be many objects that map to all nodes; what makes the limit special is the unique factorization, not minimality. Trying to think of limits as 'smallest' breaks down quickly — in Set, the product A×B is not 'small' in any obvious sense, but it is the limit of the two-object discrete diagram.

Question 3

How does the Cartesian product A × B of two sets arise as a limit in the category Set?

Model answer: Consider the discrete index category J with two objects and no non-identity morphisms, and the diagram D: J → Set sending the two objects to A and B respectively. A cone over D is a set C equipped with functions f: C → A and g: C → B. The limit is the terminal such cone: the set A × B with projections π₁: A×B → A and π₂: A×B → B, such that any C with maps to A and B factors uniquely through A×B via the pairing ⟨f, g⟩.

The key is recognizing that 'maps to both A and B consistently' is exactly what pairs (a, b) capture. The universal property says A×B is the most economical such object: every cone C → A, C → B compresses uniquely into a map C → A×B. This pattern generalizes to any diagram shape: the limit is always the 'most efficient' cone.