t-structures

Definition 6.3.1 ([Beilinson et al. (1982)]). A t-structure on a stable \(\infty \)-category \(C\) consists of a pair \((C_{\geq 0}, C_{\leq 0})\) of full subcategories of \(C\) satisfying the following conditions:

(1)

(Closure under shifts) We have \(C_{\geq 0}[1] \subseteq C_{\geq 0}\) and \(C_{\leq 0}[-1] \subseteq C_{\leq 0}\).

(2)

(Orthogonality) If \(X \in C_{\geq 0}\) and \(Y \in C_{\leq 0}\), then \(\Hom _{C}(X,Y[-1]) = 0\).

(3)

(Decomposition) Every \(X \in C\) sits in an exact sequence of the form \begin {align*} \tau _{\geq 0} X \to X \to \tau _{\leq -1} X, \end {align*}

with \(\tau _{\geq 0} X \in C_{\geq 0}\) and \(\tau _{\leq -1} X \in C_{\leq 0}[-1]\).

Objects in \(C_{\geq 0}\) are called connective (with respect to the t-structure), while objects in \(C_{\leq 0}\) are called coconnective. For \(n \in \Z \) we set \(C_{\geq n} := C_{\geq 0}[n]\) and \(C_{\leq n} := C_{\leq 0}[n]\).

If \(D\) is another stable \(\infty \)-category equipped with a t-structure, then an exact functor \(F\colon C \to D\) is called left t-exact if it sends \(C_{\leq 0}\) to \(D_{\leq 0}\), and hence by exactness sends \(C_{\leq n}\) to \(D_{\leq n}\) for all \(n \in \Z \). Similarly, \(F\) is called right t-exact if it sends \(C_{\geq 0}\) to \(D_{\geq 0}\), or equivalently sends \(C_{\geq n}\) to \(D_{\geq n}\) for all \(n\). We say \(F\) is t-exact if it is both left and right t-exact.