Truncation functors for derived ∞-categories

Proof. (1) Given a chain complex \(C\) and an integer \(k \in \Z \), we construct a new chain complex \(\tau _{\geq k}(C)\) as follows: \[ \tau _{\geq k}(C)_n := \begin {cases} C_n & n > k \\ \ker (d_k\colon C_k \to C_{k-1}) & n = k \\ 0 & n < k. \end {cases} \] The structure maps \(d_i\colon \tau _{\geq k}(C)_n \to \tau _{\geq k}(C)_{n-1}\) are the ones from \(C_{\bullet }\) when \(n > k\), are zero when \(n < k\), and for \(n = k\) is the canonical map \(C_{k+1} \to \ker (d_{k})\) induced by \(d_{k+1}\colon C_{k+1} \to C_k\). This construction is functorial in \(C\), defining a functor \[ \tau _{\geq k}\colon \Ch (\Aa ) \to \Ch (\Aa )_{\geq k}. \] There is a canonical chain map \(\epsilon \colon \tau _{\geq k}(C) \to C\) given by the identity in degree \(n > k\), by the zero map in degree \(n < k\), and in degree \(n = k\) by the inclusion \(\ker (d_k) \hookrightarrow C_k\). It is clear that this chain map is a monomorphism. Furthermore, if \(D\) is a chain complex such that \(D_n\) vanishes for \(n < k\), any chain map \(f\colon D \to C\) uniquely factors through \(\tau _{\geq k}(C)\): the commutative square

guarantees that \(f_k\) factors through \(\ker (d_k) \hookrightarrow C_k\). This shows that the transformation \(\epsilon \colon \tau _{\geq k} \to \id \) exhibits the functor \(\tau _{\geq k}\colon \Ch (\Aa ) \to \Ch (\Aa )_{\geq k}\) as right adjoint to the inclusion.

(2) Observe that the counit \(\epsilon \colon \tau _{\geq k}(C) \to C\) of the adjunction from part (1) induces isomorphisms \(H_n(\tau _{\geq k}(C)) \xrightarrow {\cong } H_n(C)\) for \(n \geq k\). Since we have \(H_n(\tau _{\geq k}(C)) = 0\) for \(n < k\), it follows that the functor \(\tau _{\geq k}\) sends quasi-isomorphisms to quasi-isomorphisms, hence induces a functor \[ \tau _{\geq k}\colon \D (\Aa ) \to \D (\Aa )_{\geq k}. \] In a similar way, the unit and counit of the adjunction from part (1) induce natural transformations at the level of derived \(\infty \)-categories, exhibiting \(\tau _{\geq k}\) as a right adjoint of the functor \(\D (\Aa )_{\geq k} \to \D (\Aa )\) induced by the inclusion. Since the unit of the adjunction is a natural isomorphism, it follows that this inclusion is fully faithful, showing (2).

Parts (3) and (4) are similar, and can be seen as instances of (1) and (2) applied to \(\Aa \catop \). The chain complex \(\tau _{\leq k}(C)_{\bullet }\) may be explicitly given as follows: \[ \tau _{\leq k}(C)_n := \begin {cases} 0 & n > k \\ \coker (d_{k+1}\colon C_{k+1} \to C_{k}) & n = k \\ C_n & n < k. \end {cases} \qedhere \] □