Hearts of t-structures are abelian

Proof. For (1), first observe that \(C^{\heartsuit }\) is a 1-category, in the sense that the hom animae \(\Hom _{C^{\heartsuit }}(X,Y)\) are sets. Indeed, for \(k \geq 1\), we have \[ \pi _k\Hom _{C^{\heartsuit }}(X,Y) \cong \pi _0 \Hom _{C}(X,Y[-k]) = 0 \] since \(X \in C_{\geq 0}\) and \(Y[-k] \in C_{\leq -k} \subseteq C_{\leq -1}\). Further, since both \(C_{\leq 0}\) and \(C_{\geq 0}\) are closed under direct sums in \(C\), it is clear \(C^{\heartsuit }\) is additive. We also observe that \(C^{\heartsuit }\) admits kernels and cokernels, which are computed as \[ \ker (Y \to Z) = \tau _{\geq 0}(\fib (Y \to Z)) \qquadtext { and } \coker (X \to Y) = \tau _{\leq 0}(\cofib (X \to Y)). \] Before finishing the proof of (1), let us first address the ‘only if’-directions in (2) and (3). Since replacing \(C\) by \(C\catop \) translates (2) into (3) and vice versa, it suffices to prove the ‘only if’-direction in (2). To this end, let \(g\colon Y \twoheadrightarrow Z\) be an epimorphism in \(C^{\heartsuit }\). We need to show that \(\fib (g) \in C_{\geq 0}\). (Since \(C_{\leq 0}\) is closed under limits, the condition \(\fib (g) \in C_{\leq 0}\) is automatic.) The assumption on \(g\) implies that the commutative square

is a pushout square in \(C^{\heartsuit }\). Since pushouts in \(C^{\heartsuit }\) are computed by first forming the pushout in \(C_{\geq 0}\) and then applying \(\tau _{\leq 0} \colon C_{\geq 0} \to C^{\heartsuit }\), it follows that the map \(Z \to Z \sqcup _Y Z\) obtained by forming the pushout in \(C\) induces an isomorphism \(Z = \tau _{\leq 0} Z \iso \tau _{\leq 0}(Z \sqcup _Y Z)\). Passing to cofibers then gives \(\tau _{\leq 0}(\cofib (g)) = 0\), or equivalently \(\cofib (g) \in C_{\geq 1}\). But then the relation \(\fib (g) \simeq \cofib (g)[-1]\) implies \(\fib (g) \in C_{\geq 0}\), which is what we needed to show.

We now prove (4). Again, (a) translates to (b) under replacing \(C\) by \(C\catop \), so it suffices to prove (a). By full faithfulness of the inclusion \(C^{\heartsuit } \hookrightarrow C\), it is clear that if the square is a pullback in \(C\) then it is also a pullback in \(C^{\heartsuit }\). Conversely, if the square is a pullback in \(C^{\heartsuit }\), then it in particular induces an isomorphism \(\ker (h) \iso \ker (g)\) on kernels in \(C^{\heartsuit }\). But by the direction of (2) that we just proved, these kernels agree with the fibers in \(C\). We deduce that the induced map \(\fib (h) \to \fib (g)\) is an isomorphism in \(C\), which implies that the square is also a pullback square in \(C\). This finishes the proof of (4).

We now return to (1), finishing the proof that \(C^{\heartsuit }\) is abelian. Consider a commutative square

in \(C^{\heartsuit }\), and assume that \(f\) is a monomorphism in \(C^{\heartsuit }\) and that \(g\) is an epimorphism in \(C^{\heartsuit }\). We need to show that this square is a pullback square in \(C^{\heartsuit }\) if and only if it is a pushout square in \(C^{\heartsuit }\). But by (4) this is immediate from stability of \(C\).

Finally, we prove the ‘if’-directions in (2) and (3). As before, it suffices to do this for (2). Given a morphism \(g\colon Y \to Z\) in \(C^{\heartsuit }\) whose fiber lies in \(C^{\heartsuit }\), we must show \(g\) is an epimorphism. Since \(C^{\heartsuit }\) is abelian, it suffices to show that its cokernel \(\coker (g) = \tau _{\leq 0}(\cofib (g))\) is zero, i.e., that \(\cofib (g) \in C_{\geq 1}\). But this follows from the relation \(\cofib (g) \simeq \fib (g)[1]\) and the assumption \(\fib (g) \in C_{\geq 0}\). □