Lemma 2.4.5. Let \(W\in \An _*\) be a pointed anima.
- (1)
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Every cofiber sequence \(X\to Y\to Z\) induces a fiber sequence of pointed mapping animae \[ \Hom _{\An _*}(Z,W)\longrightarrow \Hom _{\An _*}(Y,W)\longrightarrow \Hom _{\An _*}(X,W), \] and hence an exact sequence of pointed sets \[ [Z,W]_*\longrightarrow [Y,W]_*\longrightarrow [X,W]_*. \]
- (2)
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Every fiber sequence \(X\to Y\to Z\) induces a fiber sequence of pointed mapping animae \[ \Hom _{\An _*}(W,X)\longrightarrow \Hom _{\An _*}(W,Y)\longrightarrow \Hom _{\An _*}(W,Z), \] and hence an exact sequence of pointed sets \[ [W,X]_*\longrightarrow [W,Y]_*\longrightarrow [W,Z]_*. \]
Proof. The functor \(\Hom _{\An _*}(-,W)\colon \An _*\catop \to \An \) sends pushout squares to pullback squares. Since \(\Hom _{\An _*}(0,W)\) is terminal, applying this functor to the pushout square defining a cofiber sequence shows that the first sequence of mapping animae is a fiber sequence. Dually, the functor \(\Hom _{\An _*}(W,-)\colon \An _*\to \An \) preserves pullbacks and sends \(0\) to a terminal anima, which proves the corresponding claim for fiber sequences.
It remains to pass to pointed homotopy classes. For any fiber sequence \(F\to E\to B\) of pointed animae, a component of \(E\) lies in the image of \(\pi _0(F)\to \pi _0(E)\) precisely when its image in \(B\) lies in the component of the basepoint. Thus \(\pi _0(F)\to \pi _0(E)\to \pi _0(B)\) is exact as a sequence of pointed sets, proving both claims. □
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