Lemma 18.4.4. Let \(T\colon (\An _*^{\fin })\catop \times \An _*^{\fin } \to \An \) be a functor with the following properties:
- (1)
-
For every \(L \in \An _*^{\fin }\), the functor \(T(-,L)\colon (\An _*^{\fin })\catop \to \An \) sends finite colimits in \(\An _*^{\fin }\) to limits.
- (2)
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For every \(K \in \An _*^{\fin }\), the functor \(T(K,-)\colon \An _*^{\fin } \to \An \) is reduced and excisive.
Then the projection from the end to the component at \(S^0\), \[ \int _{K \in \An _*^{\fin }} T(K,K) \to T(S^0,S^0), \] is an equivalence.
Proof. Let \(A := \An _*^{\fin }\) and let \(j\colon A \hookrightarrow \An _*\) be the inclusion. Also let \[ W\colon \An _* \to \PSh (A), \qquad X \mapsto \Hom _{\An _*}(-,X)\vert _A \] be the restricted Yoneda embedding from Lemma 22.3.9. By currying, the functor \(T\) determines a functor \[ \widetilde {T}\colon A \to \PSh (A), \qquad L \mapsto T(-,L). \] By assumption (1) and Lemma 22.3.9, the functor \(\widetilde {T}\) factors uniquely as \(W\circ T'\) for a functor \(T'\colon A \to \An _*\). Since \(W\) is fully faithful and preserves and reflects limits, and since limits in presheaf categories are computed pointwise, assumption (2) implies that \(T'\) is reduced and excisive.
Let \(Y_A\colon A \to \PSh (A)\) denote the Yoneda embedding. Using the end formula for natural transformations, the Yoneda lemma, and the identity \(W\circ j\simeq Y_A\), we obtain equivalences \[ \int _{K \in A}T(K,K) \simeq \Nat (Y_A,\widetilde {T}) \simeq \Nat (j,T'). \] Since \(T'\) is reduced and excisive, Lemma 16.5.14 gives \[ \Nat (j,T') \simeq \Nat (P_1j,T') \simeq \Hom _{\An _*}(S^0,T'(S^0)). \] Finally, by the definition of \(W\) we have \[ \Hom _{\An _*}(S^0,T'(S^0)) \simeq W(T'(S^0))(S^0) \simeq T(S^0,S^0). \] Tracing through the construction identifies this composite with projection from the end to the component at \(S^0\). □
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