Proposition 6.162.
Let \(C\) be a category admitting small weakly contractible colimits.
The inclusion \(C \hookrightarrow \int_{\An}C\) preserves weakly contractible colimits.
The inclusion \(C \hookrightarrow \int_{\An} C\) uniquely extends to a left adjoint \(L\colon \PSh^{\mathrm{small}}(C) \to \int_{\An} C\).
The right adjoint \(R\colon \int_{\An} C \hookrightarrow \PSh^{\mathrm{small}}(C)\) is fully faithful, exhibiting \(\int_{\An}C\) as a Bousfield localization of \(\PSh^{\mathrm{small}}(C)\). In particular, \(\int_{\An} C\) admits small colimits.
For a category \(D\) with small colimits, a colimit-preserving functor \(F\colon \PSh^{\mathrm{small}}(C) \to D\) inverts \(L\)-local maps if and only if its restriction \(C \to D\) preserves weakly contractible colimits.
Proof
(1) Consider a functor \(X_{\bullet}\colon I \to C\), where \(I\) is weakly contractible, and set \(X := \colim_{i \in I} X_i\). We have to show that the colimit of this diagram in \(\int_{\An}C\) is \((*, x)\). This amounts to the claim that for every object \((B,Y) \in \int_{\An} C\), the canonical map
\[\Hom_{\int_{\An}C}((*,X), (B,Y)) \to \lim_{i \in I\catop} \Hom_{\int_{\An}C}((*,X_i),(B,Y))\]
is an isomorphism. Since this map lives over \(\Hom_{\An}(*,B) = B\), it suffices to show that the induced map on fibers over each \(b \in B\) is an isomorphism. But this map on fibers is the map \(\Hom_C(X,Y_b) \to \lim_{i \in I\catop} \Hom_C(X_i, Y_b)\), which is an isomorphism by construction of \(X\).(2) We will show that the left Kan extension of the inclusion \(C \hookrightarrow \int_{\An} C\) along the Yoneda embedding \(C \hookrightarrow \PSh^{\mathrm{small}}(C)\) exists. Given a small presheaf \(\Ff\), the pointwise formula for left Kan extensions tells us that it suffices to show that in \(\int_{\An}C\) we may form the colimit of the functor \[\El(\Ff) = C_{/\Ff} \xrightarrow{\fgt} C \hookrightarrow \int_{\An} C.\]
By assumption on \(\Ff\), the category \(\El(\Ff)\) comes with a final functor \(I \to \El(\Ff)\) from some small category \(I\). It will thus suffice to show that for any small category \(I\) and any functor \(X\colon I \to C\) the colimit of \(I \to \int_{\An}C\) exists. We do this in two steps: - Consider the map \(p\colon I \to \abs{I}\) to the geometric realization of \(I\). We will show that the Kan extension of \(X\) along \(p\) exists in \(\int_{\An}C\). Since \(p\) is a cofinal functor, its relative slices are weakly contractible by Quillen's Theorem A. By part (1), this means that the Kan extension of \(X\) along \(p\) exists in \(C\), and that it is preserved by the inclusion \(C \hookrightarrow \int_{\An}C\).
- The colimit \(\colim_i X_i\) in \(\int_{\An}C\) is now computed as the colimit of the left Kan extension \(p_!X\colon \abs{I} \to \int_{\An}C\), which exists since \(\abs{I}\) is an anima and we established the existence of anima-indexed colimits in Lemma 6.158.
\[R\colon \int_{\An} C \to \PSh^{\mathrm{small}}(C), \qquad R((A,X))(Y) := \Hom_{\int_{\An}C}((*,Y), (A,X)).\]
The functor \(\pi\colon \int_{\An} C \to \An\) induces a map \(\Hom_{\int_{\An}C}((*,Y), (A,X)) \to \Hom_{\An}(*,A) \cong A\), whose fiber over \(a \in A\) is \(\Hom_C(Y,X_a)\), so we obtain isomorphisms \[R((A,X))(Y) \; \cong \; \colim_{a \in A} \Hom_C(Y,X_a) \; \cong \; \colim_{a \in A} y(X_a)(Y).\]
In particular, we see that \(R((A,X)) \simeq \colim_{a \in A} y(X_a)\) is a small colimit of representables, so that it is indeed contained in \(\PSh^{\mathrm{small}}(C)\).(3) To show that \(R\) is fully faithful, we must show that for every object \((A,X) \in \int_{\An}C\), the counit \(LR(A,X) \to (A,X)\) is an isomorphism. But this is clear from the computation \(R(A,X) \cong \colim_{a \in A} y(X_a)\) from before, and the fact that \((A,X) \cong \colim_{a \in A} (*,X_a)\) by the explicit description of anima indexed colimits from Lemma 6.158.(4) If \(F\) inverts all \(L\)-local maps, then \(F\) in particular inverts the canonical map \(\colim_{i \in I} y(X_i) \to y(\colim_{i \in I} X_i)\) for every weakly contractible diagram \(X\colon I \to C\), so that \(F(y(\colim_i X_i)) \simeq \colim_i F(y(X_i))\). Conversely, assume \(F \circ y\colon C \to D\) preserves weakly contractible colimits. It will suffice to show that \(F\) inverts the unit map \(\Ff \to RL(\Ff)\) for all \(\Ff \in \PSh^{\mathrm{small}}(C)\). Choose a final map \(I \to \El(\Ff)\) from a small category. Writing \(\Ff\) as the associated colimit of representables and left Kan extending along \(p\colon I \to \abs{I}\) expresses \(\Ff\) as an \(\abs{I}\)-indexed colimit of objects \(\colim_{i \in I_x}y(X_i)\), where \(I_x=\fib_x(I\to\abs I)\) is weakly contractible. The same construction in \(C\) expresses \(RL(\Ff)\) as the corresponding \(\abs I\)-indexed colimit of \(y(\colim_{i\in I_x}X_i)\). Thus the unit \(\Ff\to RL(\Ff)\) is an \(\abs I\)-indexed colimit of the canonical maps which \(F\) inverts by assumption. Since \(F\) preserves colimits, it inverts the unit.