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In this article, we study the heart of a cotorsion pairs on an exact category and a triangulated category in a unified meathod, by means of the notion of an extriangulated category. We prove that the heart is abelian, and construct a cohomological functor to the heart. If the extriangulated category has enough projectives, this functor gives an equivalence between the heart and the category of coherent functors over the coheart modulo projectives. We also show how an n-cluster tilting subcategory of an extriangulated category gives rise to a family of cotorsion pairs with equivalent hearts.

In the paper of Keller and Reiten, it was shown that the quotient of a triangulated category (with some conditions) by a cluster tilting subcategory becomes an abelian category. After that, Koenig and Zhu showed in detail, how the abelian structure is given on this quotient category, in a more abstract setting. On the other hand, as is well known since 1980s, the heart of any t-structure is abelian. We unify these two constructions by using the notion of a cotorsion pair. To any cotorsion pair in a triangulated category, we can naturally associate an abelian category, which gives back each of the above two abelian categories, when the cotorsion pair comes from a cluster tilting subcategory, or a t-structure, respectively.

For each positive integer n we introduce the notion of n-exangulated categories as higher dimensional analogues of extriangulated categories defined by Nakaoka-Palu. We characterize which n-exangulated categories are n-exact in the sense of Jasso and which are (n + 2)-angulated in the sense of Geiss-Keller-Oppermann. For extriangulated categories with enough projectives and injectives we introduce the notion of n-cluster tilting subcategories and show that under certain conditions such n-cluster tilting subcategories are n-exangulated. (A,C) 7 2.4. Realization of extensions 9 2.5. Definition of n-exangulated categories 11 3. Fundamental properties 14 3.1. Fundamental properties of n-exangulated category 14 3.2. Relative theory 17 4. Typical cases 22 4.1. Extriangulated categories 22 4.2. (n + 2)-angulated categories 24 4.3. n-exact categories 29 5. n-cluster tilting subcategories as n-exangulated categories 41 5.1. Higher extensions in an extriangulated category 41 5.2. The cup product and long exact sequences 44 5.3. From n-cluster tilting subcategories to n-exangulated categories 47 6. Examples 63 6.1. Examples of n-abelian and n-exact categories 64 6.2. Examples of (n + 2)-angulated categories 65 6.3. Examples which are neither n-exact nor (n + 2)-angulated 66 References 70

We give a simultaneous generalization of exact categories and triangulated categories, which is suitable for considering cotorsion pairs, and which we call extriangulated categories. Extension-closed, full subcategories of triangulated categories are examples of extriangulated categories. We give a bijective correspondence between some pairs of cotorsion pairs which we call Hovey twin cotorsion pairs, and admissible model structures. As a consequence, these model structures relate certain localizations with certain ideal quotients, via the homotopy category which can be given a triangulated structure. This gives a natural framework to formulate reduction and mutation of cotorsion pairs, applicable to both exact categories and triangulated categories. These results can be thought of as arguments towards the view that extriangulated categories are a convenient setup for writing down proofs which apply to both exact categories and (extension-closed subcategories of) triangulated categories.