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In this article, we initiate the study of hereditary extriangulatedcategories. Many important categories arising in representation theory inconnection with various theories of mutation are hereditary extriangulated.Special cases include homotopy categories of 2-term complexes with projectivecomponents, which are related to silting mutation, and cluster categories (withrelevant relative extriangulated structures) where cluster tilting mutationtake place.  We prove that there is a theory of irreducible mutation for maximal rigidobjects and subcategories in hereditary extriangulated categories of dominantdimension 1. Applied to the examples above, this recovers 2-term siltingmutation in triangulated categories and cluster tilting mutation. Byconstructing suitable extriangulated categories, we also recover tau-tiltingmutation for gentle algebras and flips for their non-kissing facets. Combinedwith results by Adachi-Tsukamoto and Pauksztello-Zvonareva, our mutation alsoprovides mutation for intermediate co-t-structures.  In spirit of our earlier work, we study negative extensions in hereditaryextriangulated categories. We give sufficient conditions for the existence ofuniversal balanced negative extensions. We explicitly compute certain, a priorinon-universal, versions of negative extensions in hereditary categoriesconstructed from triangulated categories with rigid subcategories. We discussexamples where these two constructions give the same delta-functors and wherethey disagree.

Hereditary extriangulated categories: Silting objects, mutation, negative extensions