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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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