Mapping Cones

An important construction is that of the mapping cone C of a map of complexes w A —> B in A. (For this construction we need only assume that the category A is additive.) C7 is the complex whose degree n component is... 

Most of the basic properties of standard triangles involve homotopy, and so are best stated in K( ). For example, the mapping cone C of the identity map A —> A is homotopically equivalent to zero, a homotopy between the identity map of C and the zero map being as indicated ... 

First we prove that I is /F-injective. It suffices to show that for any exact complex F, any chain map F —> I is null-homotopic. Let C = Cone(95), where Cone denotes the mapping cone. Consider the exact sequence... 

Proof. Let G —> I be a K-injective resolution in Mod(X,) such that I is bounded below. Let C be the mapping cone of this. Since ( )o has an exact left adjoint, Gq Iq is a A -injective resolution in A (Mod(Xo)). So it suffices to show that Homy (Fo, Cq) is exact. As each term of F is equivariant, this complex is isomorphic to Homy, (F, C)o, which is exact by the lemma. ... 

Note that the mapping cone and the mapping cylinder, which will be defined in Section 6.3, are examples of the space attachment constructions. Attaching a cell along its boimdary is another such example, in this case X = S ", A = dW, and the attachment map is an arbitrary continuous map / dB Y. 

Definition 6.10. Let f X — Y be a continuous map between two topological spaces. The mapping cone of f is the quotient space... 

Fig. 6.1. A mapping cylinder and a mapping cone of a map taking everything to...

For the diagram from Example 15.9(3), we see that pb is the canonical projection of the mapping cone onto an interval, whereas pf collapses the actual cone inside the mapping cone to a point, which is the same as to collapse the image of X in E to a point. 

Define the pseudo chain complex t(Yj, p ) to be the algebraic mapping cone of the composite pseudo chain map... 

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