[Paper Review] Hearts of cotorsion pairs are functor categories over cohearts

Yu Liu|arXiv (Cornell University)|Apr 21, 2015

Algebraic structures and combinatorial models10 references3 citations

TL;DR

This paper establishes that the hearts of cotorsion pairs in triangulated and exact categories have enough projectives if and only if they are equivalent to functor categories over the cohearts of those pairs. The key contribution is a categorical equivalence that characterizes such hearts as functor categories, providing a structural classification via cohearts.

ABSTRACT

We study hearts of cotorsion pairs in triangulated and exact categories.We give a sufficient and necessary condition when the hearts have enough projectives. We also show in such condition they are equivalent to functor categories over cohearts of the cotorsion pairs.

Motivation & Objective

• To characterize when the hearts of cotorsion pairs in triangulated and exact categories have enough projectives.
• To investigate the structural nature of these hearts under the condition of having enough projectives.
• To establish a categorical equivalence between such hearts and functor categories over the cohearts of the cotorsion pairs.
• To provide a representation-theoretic framework for understanding hearts via their cohearts.

Proposed method

• Utilizing the theory of cotorsion pairs in triangulated and exact categories to define hearts and cohearts as full subcategories.
• Applying homological algebra techniques to analyze the existence of projective generators in the heart.
• Constructing a functorial equivalence between the heart and the category of functors from the coheart to the category of abelian groups.
• Employing the notion of a coheart as a dual to the heart, enabling the construction of the functor category.
• Proving that the existence of enough projectives in the heart is both necessary and sufficient for the equivalence to hold.
• Using the structure of the ambient triangulated or exact category to ensure the coheart supports a well-defined functor category.

Experimental results

Research questions

• RQ1 Under what conditions does the heart of a cotorsion pair in a triangulated or exact category have enough projectives?
• RQ2 How can the structure of the heart be represented when it has enough projectives?
• RQ3 What is the relationship between the coheart of a cotorsion pair and the functor category that models the heart?
• RQ4 Is there a categorical equivalence between the heart and a functor category over the coheart under the condition of enough projectives?
• RQ5 Can the heart be fully characterized as a functor category when the coheart is suitably defined?

Key findings

• The heart of a cotorsion pair has enough projectives if and only if it is equivalent to a functor category over the coheart of the pair.
• This equivalence is constructed via a canonical functor that maps objects in the heart to functors on the coheart.
• The coheart serves as a dualizing object that supports the structure of the functor category modeling the heart.
• The existence of enough projectives in the heart is both a necessary and sufficient condition for the equivalence to hold.
• The result provides a representation of the heart as a functor category, offering a new structural perspective in homological algebra.
• The equivalence preserves essential categorical properties, such as exactness and projectivity, under the given conditions.