By Deo S.

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**Extra resources for Algebraic topology: a primer**

**Example text**

Let J′ be the full subcategory of J whose objects are all the complexes in J admitting a quasi-isomorphism to a right-F -acyclic complex. Then J′ is a Δ-subcategory of J. (d) X is right-F -acyclic iﬀ every map C → X in J with C exact factors as C → C ′ → X with C ′ exact and F (C ′ ) ∼ = 0. 52 2 Derived Functors (e) X is said to be “unfolded for F ” if for every Z ∈ E the natural map HomE (Z, F (X)) → limHomE (Z, F (Y )) −→ X→Y is an isomorphism, where the lim is taken over the category of all quasi-isomorphisms −→ X → Y in J [De, p.

I n and I = (0) for n ≪ 0); and such that moreover with the usual translation functor T we have • • ITB • = T IB • , ϕTB • = T (ϕB • ) . 1) To see this, ﬁrst construct quasi-isomorphisms ϕB • as in [H, p. 6, 1)] for those B • such that H 0 (B • ) = 0 and B m = 0 for m < 0. 1)). 1) . + Now let K+ I be the full subcategory of K whose objects are the boundedbelow I-complexes. Since the additive subcategory I ⊂ A is closed under + ﬁnite direct sums, one sees that K+ I is a Δ-subcategory of K . 2) which is, together with θ = identity, a Δ-equivalence of categories, quasiinverse to the inclusion DI+ ֒→ D+.

We say then that j˜ and ρ—or more precisely (˜ j, identity) and (ρ, θ)—are Δ-equivalences of categories, quasi-inverse to each other. 2)op . 2), with A• → X • replaced by X • → A• . 2)op , for example, suppose that we have a family of quasiisomorphisms (“right L′ -resolutions”) ϕX • : X • → A•X • ∈ L′ (X • ∈ L′′ ) . 1)op , L′ is localizing in L′′ . 1) is equivalent to ﬁnding for each object X • of D′′ an isomorphism to an object in D′ ⊂ D′′ (see [M, p. 92, (iii)⇒(ii)]). But Q′′ (ϕX • ) is such an isomorphism.