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 3×3-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 em...

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 3×3-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 em...

The categorical formulation of the Eilenberg-Watts calculus relates, for any pair finite categories M and N, Fun^{le}(N,M) Fun^{re}(N,M) linear left or right exact functors Deligne product \bar N \boxtimes by adjoint equivalences. We establish a variant this case that are module over tensor category. This provides in particular canonical explicitly computable equivalences between (or, more gene...

In this paper we introduce a new approach to determinant functors which allows us to extend Deligne’s determinant functors for exact categories to Waldhausen categories, (strongly) triangulated categories, and derivators. We construct universal determinant functors in all cases by original methods which are interesting even for the known cases. Moreover, we show that the target of each universa...

In the majority of languages, functional distinction between functors and content words correlates with lower-level, perceptually observable properties. Functors are generally more frequent prosodically minimal than words. Previous studies demonstrate that frequency distribution different acoustic realization infrequent guide infants in discovering their native word order. However, whether if y...

Two exact embedding theorems are proved for arbitrary interpolation functors on couples of global Morrey spaces. Since the calculation spaces is known only in certain cases, these can replace This fact demonstrated by examples real Peetre, complex Calderon’s, and a functor generated unconditionally convergent series.