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Extra resources for Higher Algebra
Example text
Here S −1 C denotes the full subcategory of C spanned by S-local objects, where an object X ∈ C is said to be S-local if and only if, for each f : Y → Y in S, composition with f induces a homotopy equivalence MapC (Y, X) → MapC (Y , X). 2. STABLE ∞-CATEGORIES AND HOMOLOGICAL ALGEBRA 33 If C is stable, then we extend the morphism f to a fiber sequence Y →Y →Y , and we have an associated long exact sequence θ . . → ExtiC (Y , X) → ExtiC (Y, X) →i ExtiC (Y , X) → Exti+1 C (Y , X) → . . The requirement that X be {f }-local amounts to the condition that θi be an isomorphism for i ≤ 0.
For each 0 ≤ i ≤ n, we let di : An → An−1 denote the associated face map (determined by the unique injective map [n − 1] → [n] whose image does not contain i ∈ [n]). Let d(n) : An → An−1 denote the alternating sum 0≤i≤n (−1)i di . An easy calculation shows that d(n − 1) ◦ d(n) 0 for n > 0, so that d(2) d(1) · · · → A2 → A1 → A0 → 0 → · · · is a chain complex with values in A. We will denote this chain complex by C∗ (A), and refer to it as the unnormalized chain complex associated to A• . If A• is a simplicial object of A, we let C∗ (A) denote the unnormalized chain complex of the underlying semisimplicial object of A• .
We refer the reader to [30] for a gentle account of this spectral sequence, and to [108] for a general introduction to spectral sequences. CHAPTER 1. STABLE ∞-CATEGORIES 38 Our first step is to construct some auxiliary objects in C. 2. Let C be a pointed ∞-category, and let I be a linearly ordered set. We let I[1] denote the partially ordered set of pairs of elements i ≤ j of I, where (i, j) ≤ (i , j ) if i ≤ i and j ≤ j . An I-complex in C is a functor F : N(I[1] ) → C with the following properties: (1) For each i ∈ I, F (i, i) is a zero object of C.