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Extension categories and their homotopy. (English) Zbl 0852.18011
Let $$\mathcal C$$ be a Waldhausen category with an invertible suspension functor, e.g., the category of chain complexes of objects of a fixed abelian category where the suspension is the shifting to the left. If $$X$$ is a chain complex in $$\mathcal C$$ then the “associated total complex” $$T(X)$$ is inductively defined in the Waldhausen setting. Then $$X$$ is called acyclic if $$0 \to T(X)$$ is a weak equivalence. Then define $$\text{Ext}^n (A,B)$$ for $$n \geq 1$$ to be the category whose objects are diagrams $$B \to X^1 \to \cdot \to X^{n + 1}$$ and $$A \to X^{n + 1}$$ where the chain complex $$B \to \cdot \to X^{n + 1}$$ is acyclic and the map $$A \to X^{n + 1}$$ is a weak equivalence and a cofibration; if $$n = 0$$ then $$A \to X^1$$ will be any morphism but $$B \to X^1$$ should be a weak equivalence and a cofibration. Morphisms in $$\text{Ext}^n (A,B)$$ are morphisms of diagrams which are the identity on $$A$$ and $$B$$. It is then nicely proved that the loop space of the category $$\text{Ext}^n (A,B)$$ is naturally homotopy equivalent to $$\text{Ext}^{n - 1} (A,B)$$ $$(n \geq 1)$$. In this way the $$\Omega$$-spectrum $$\text{Ext} (A,B)$$ is defined and the $$\pi_0$$ of $$\text{Ext}^0$$ is identified by the $$\operatorname{Hom}$$ of the triangulated category associated to $$\mathcal C$$ where the weak equivalences have been inverted. Previous results by A. Robinson and the second author are obtained as immediate consequences from this general Waldhausen setting. In particular if $$\mathcal C$$ is additive then the spectra $$\text{Ext} (A,B)$$ are all wedges of suspensions of Eilenberg-MacLane spectra.

##### MSC:
 18G35 Chain complexes (category-theoretic aspects), dg categories 55U15 Chain complexes in algebraic topology 19D10 Algebraic $$K$$-theory of spaces 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 18E10 Abelian categories, Grothendieck categories 18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects) 18E30 Derived categories, triangulated categories (MSC2010)
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