Explore alternative definitions for the initial object in category theory and determine their equivalence | Step-by-Step Solution

What We're Solving:

We're exploring two different ways to define an initial object in category theory using colimits, and checking if they're equivalent to each other and to the standard definition.

The Approach:

This is an exercise in understanding how fundamental concepts in category theory can be expressed through the unifying language of limits and colimits! We'll work through each definition carefully, then compare them to see why they give us the same result.

Step-by-Step Solution:

Step 1: Recall what an initial object is An initial object in a category C is an object I such that for every object X in C, there exists exactly one morphism I → X.

Step 2: Understand the first proposed definition "Take entire category with identity morphisms as diagram" means we're looking at a diagram that includes:

• Every object in C as a vertex
• Only the identity morphisms (id_X : X → X for each object X)
• No morphisms between different objects

Step 3: What would the colimit of this diagram be? For a colimit, we need an object L with morphisms from every object in our diagram to L, satisfying the universal property. Since our diagram has no morphisms between different objects, the cocone condition is automatically satisfied for any choice of morphisms X → L.

The universal property says: L is initial in the category! Here's why: given any other object Y, we can form a cocone from our diagram to Y, and by universality, there's a unique morphism L → Y.

Step 4: Compare with nLab's definition The nLab defines an initial object as the colimit over the empty diagram. An empty diagram has no objects and no morphisms.

Step 5: Why are these equivalent? Here's the key insight: Both approaches force the colimit to be initial, but for different reasons:

• Empty diagram: Any object can serve as a "colimit" of nothing, but the universal property forces it to map uniquely to every other object
• Discrete diagram of whole category: The object must receive morphisms from everywhere, and universality again forces it to map uniquely to every other object

The Answer:

Both definitions are equivalent and both correctly characterize the initial object! The first definition (colimit over discrete diagram of whole category) works because it captures the idea that an initial object must be "reachable from" or "comparable to" every object. The nLab definition (colimit over empty diagram) works because it directly encodes the universal mapping property.

Memory Tip:

Think of the initial object as the "starting point" of your category. Whether you define it as "the place everything can reach" (discrete diagram) or "the place that reaches everything" (empty diagram + universal property), you're capturing the same fundamental idea that it connects to all other objects in a unique way!

The beauty here is seeing how category theory's abstract machinery (colimits) can express the same concept through different but equivalent perspectives. Great question for building intuition about these universal constructions!