Problem Set 10 Jason Starr Fall 2015 MAT 536 Problem Set 10

Problem 1.(The Mapping Cone is a Homotopy Limit and a Homotopy Colimit) Let A be an Abelian category. Let A● and B● be objects in Ch●(A). Let f ● ∶ A● → B● be a morphism in Ch●(A). For every object T ● of Ch●(A), a left homotopy annihilator to T ● is a pair (Q●, σ●) of a morphism Q● ∶ B● → T ● in Ch●(A) and a nullhomotopy (σn ∶ An → T n−1)n∈Z of Q● ○ f ●, i.e., for every n ∈ Z, Q ○ f = dn−1 T ○ σ + σn+1 ○ dA. For every object S● of Ch●(A), a right homotopy annihilator from S● is a pair (∆●, τ ●) of a morphism ∆● ∶ S● → A●[+1] and a nullhomotopy (τn ∶ Sn → B[+1]n = B)n∈Z of f ●[+1] ○ ∆●, i.e., for every n ∈ Z, fn+1 ○∆ = dn−1 B[+1] ○ τ + τn+1 ○ dS = −dB ○ τ + τn+1 ○ dS.