Figure 2.

Universal construction and comma category. The diagram in (A) shows the general form of a universal arrow, which is the pair (A, ϕ), from an object X in a category C to a functor F: D → C from a category D to C. To be a universal arrow, as such, for every object Y in D and every arrow f: X → F(Y) in C there must exist a unique arrow u: A → Y, in D, such that f = F(u) ○ ϕ. The diagram in (B) shows the corresponding comma category, denoted (X ↓ F), whose objects are the pairs (Y, f) and arrows are the arrows u that uniquely satsify the “triangle” equation, f = F(u) ○ ϕ. The collection of objects in the comma category includes the universal arrow, (A, ϕ), because for Y set to A and f set to ϕ the triangle equation is uniquely satisfied by setting u to the identity arrow 1_A. The universal arrow is the initial object in the comma category, which is straightforward to prove. Dashed lines indicate that the arrows are unique. An example universal construction (C) and corresponding comma category (D) is the universal arrow from the object (real number) 2.9 to the inclusion functor (function) from the integers to the real numbers (regarded as posets with the usual order ≤, hence categories) is the pair (3, ≤). The number 3 corresponds to the smallest integer greater than or equal to 2.9. In general, for a real number x ∈ ℝ, the object component of the universal arrow (a, ≤) from x to the inclusion function/functor is obtained by rounding up to the nearest integer a ∈ ℤ greater than or equal to x, i.e., obtained by the ceiling function, a = ⌈x⌉, and the corresponding comma category consists of all the integers y greater than or equal to a, i.e., the set {y ∈ ℤ|⌈x⌉ ≤ y}.