Transport of structure in higher homological algebra

Towards a higher reasoning level in formalized Homological Algebra

We present a possible solution to some problems to mechanize proofs in Homological Algebra: how to deal with partial functions in a logic of total functions and how to get a level of abstraction that allows the prover to work with morphisms in an equational way.

TORSION CLASSES AND t-STRUCTURES IN HIGHER HOMOLOGICAL ALGEBRA

Higher homological algebra was introduced by Iyama. It is also known as n-homological algebra where n > 2 is a fixed integer, and it deals with n-cluster tilting subcategories of abelian categories. All short exact sequences in such a subcategory are split, but it has nice exact sequences with n + 2 objects. This was recently formalised by Jasso in his theory of n-abelian categories. There is a...

Homological Algebra

Categories and Homological Algebra

The aim of these Notes is to introduce the reader to the language of categories with emphazis on homological algebra. We treat with some details basic homological algebra, that is, categories of complexes in additive and abelian categories and construct with some care the derived functors. We also introduce the reader to the more sophisticated concepts of triangulated and derived categories. Ou...

Some homological algebra

1.1 Adjoint functors Let C and D be categories and let f∗ : C → D and f∗ : D → C be functors. Then f∗ is a right adjoint to f∗ and f∗ is a left adjoint to f∗ if, for each D ∈ D and C ∈ C there is a natural vector space isomorphism HomC(fD,C) Φ −→HomD(D, f∗C). For C ∈ C and D ∈ D define τC = Φ(idf∗C) ∈ HomC(ff∗C,C) and φD = Φ(idf∗D) ∈ HomD(D, f∗fD). In general τC and φD are neither injective nor...