Request for references about abelian properties of functor categories and exact sequences in direct systems of modules | Step-by-Step Solution

What We're Solving:

You're investigating the abelian properties of functor categories and how exact sequences behave in these categories, particularly whether exactness can be checked "componentwise" (pointwise) and how this relates to direct systems of modules.

The Approach:

We'll build your understanding systematically by exploring the key concepts and their relationships. This will help you identify the right references and understand what questions to ask.

Step-by-Step Solution:

Step 1: Understanding Functor Categories A functor category [C, A] consists of functors from category C to category A, with natural transformations as morphisms. When A is abelian (like modules over a ring), we want to know: is [C, A] also abelian?

Step 2: The Pointwise Structure Operations in [C, A] are typically defined "pointwise" - for functors F, G and object c in C, we define (F + G)(c) = F(c) + G(c). This suggests exactness might also work pointwise!

Step 3: When Functor Categories Are Abelian If A is abelian, then [C, A] is abelian. Kernels, cokernels, and other abelian structures are computed pointwise.

Step 4: Exact Sequences and Pointwise Exactness A sequence of functors is exact if and only if it's exact at each object of C. This connects to your interest in direct systems - colimits preserve exactness under certain conditions.

Step 5: Key References to Explore Look into these foundational sources:

• Mac Lane's "Categories for the Working Mathematician" - Chapter V covers functor categories
• Weibel's "An Introduction to Homological Algebra" - Excellent for exact sequences
• Mitchell's "Theory of Categories" - Deep treatment of abelian categories
• Kashiwara & Schapira's "Categories and Sheaves" - Modern perspective

The Framework:

Conceptual Outline:

• 1. Foundation: Establish when [C, A] is abelian (it is, when A is!)
• 2. Exactness: Prove the pointwise characterization of exact sequences
• 3. Applications: Connect to direct/inverse systems and their limits
• 4. Examples: Work with specific cases (modules, sheaves, etc.)

Research Strategy:

• Start with Mac Lane for the big picture
• Use Weibel for homological algebra techniques
• Look for papers on "exactness of colimits" for direct system applications
• Search for "pointwise exact sequences" in functor categories

Memory Tip:

Think "Local to Global" - in functor categories, global properties (like exactness) are determined by local behavior at each object. Just like how sheaf theory works - check the property everywhere locally, and you get it globally!

This is a rich area where category theory, algebra, and topology intersect beautifully. The key insight is that functor categories inherit the "nice" properties of their target categories in a very natural, pointwise way.

Would you like me to elaborate on any of these steps or help you dive deeper into specific aspects?