Exact Categories

By Theo Buehler. 2008

697 citations recorded by the source corpus.

We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3 x 3-lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any embedding into abelian categories and we sketch Deligne's approach to derived functors. The construction of classical derived functors with values in an abelian category painlessly translates to exact categories, i.e., we give proofs of the comparison theorem for projective resolutions and the horseshoe lemma. After discussing some examples we elaborate on Thomason's proof of the Gabriel-Quillen embedding theorem in an appendix.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e2569dace7ea72f805abc984e89ce2e152147140

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Citations in the corpus

Gorenstein Homological Aspects of Monomorphism Categories via Morita Rings (2015, methodology): “Our approach follows the appendix of Keller [28, Appendix A], see also [16].”
$F$-zips with additional structure (2012, background): “Every admissible homomorphism in an exact category has a kernel and a cokernel, and they satisfy a number of axioms: see [Bühler 2010].”
Monads of effective descent type and comonadicity (2006, positive): “(Co)monads on (co)exact categories in which every object is projective (injective) We use the notion of an exact category in the sense of Barr [2], with the convention that " exact " always includes the existence of finite limits.”
n-Abelian and n-exact categories (2014, background): “ sequences. Furthermore, if X0 d X1 Xn Xn+1 0 d1 dn 1 dn is an X-admissible n-exact sequence, we say that d0 is an X-admissible monomorphism and that dnis an X-admissible epimorphism. In analogy with [18], we depict X-admissible monomorphisms by ˆ and X-admissible epimorphisms by . A sequence ˆ!! o”
On exact categories and applications to triangulated adjoints and model structures (2010, background): “The concept of an exact category is originally due to Quillen [54], but the common reference for a simple axiomatic description is [33, Appendix A] and an extensive treatment of the concept is also given in [12].”
Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves (2013, background): “The central concept is that of an exact category, which is originally due to Quillen, but the common reference for a simple axiomatic description is [46, Appendix A] and an extensive treatment is given in [10].”