Explore whether creating a limit of a diagram with only identity morphisms can define an initial object in a category. | Step-by-Step Solution

Hi there! This is a fascinating question that brings together two fundamental concepts in category theory. Let's work through this step by step!

What We're Solving:

We need to determine whether the limit of a diagram consisting only of identity morphisms can serve as an initial object in a category. This requires us to understand what limits and initial objects are, then see if they coincide in this special case.

The Approach:

Our strategy is to:

• 1. Clarify what our diagram looks like (only identity morphisms)
• 2. Find the limit of this diagram using the definition
• 3. Check if this limit satisfies the definition of an initial object
• 4. Consider if this works in general or only under certain conditions

This approach helps us see how different universal constructions in category theory can sometimes overlap!

Step-by-Step Solution:

Step 1: Understanding our diagram A diagram with "only identity morphisms" means we have some objects in our category, but the only morphisms in our diagram are the identity morphisms on each object. So our diagram looks like: A ← A, B ← B, C ← C, etc., with no morphisms between different objects.

Step 2: Finding the limit For a limit to exist, we need an object L with morphisms to each object in our diagram such that:

• There's a unique morphism from L to each object in the diagram
• Any other object with morphisms to all diagram objects factors uniquely through L

Since our diagram objects have no connections between them, a limit L must have exactly one morphism to each object A, B, C, ... in the diagram.

Step 3: What does this limit look like? The limit of disconnected objects (objects with only identity morphisms) is their product! So L = A × B × C × ... with projection morphisms π_A: L → A, π_B: L → B, etc.

Step 4: Checking if this is an initial object An initial object has exactly one morphism to every object in the entire category.

Here's the key insight: Our limit (the product A × B × C × ...) only has the required unique morphisms to the objects in our original diagram, not necessarily to every object in the whole category.

Step 5: When does this work? This limit will be an initial object if and only if:

• Our diagram includes ALL objects in the category, AND
• The category has a product of all its objects

The Answer:

No, in general the limit does not define an initial object. The limit of a diagram with only identity morphisms gives us the product of those objects, which only has the required "initial object property" with respect to the objects in the diagram, not the entire category.

However, if the diagram contains ALL objects in the category AND the product of all objects exists, then yes, this limit would be an initial object.

Memory Tip:

Think of it this way: "Limits are local, initial objects are global!" A limit only needs to work for the objects in your diagram, but an initial object must work for the entire category. The limit of identities gives you a "local initial object" - initial with respect to your chosen objects, but not necessarily the whole category!

Great question - it really shows how different universal properties can sometimes overlap in interesting ways! 🌟