Connective objects and t-structures

Proof. Property (1) is immediate from the definition of \(\D (\Aa )_{\geq n}\) and \(\D (\Aa )_{\leq n}\): a complex with homology concentrated in degrees \(\geq 0\) has, after a positive shift, homology concentrated in degrees \(\geq 1\), so in particular \(\geq 0\).

Property (2) was established in Corollary 6.1.25: if \(A\) is connective and \(B\) is \((- 1)\)-coconnective, then \(\Hom _{\D (\Aa )}(A,B) = 0\).

Property (3) is Corollary 6.1.32: for every complex \(A\), there is an exact sequence \(\tau _{\geq 0} A \to A \to \tau _{\leq -1} A\) in \(\D (\Aa )\). □

Proof. Property (1) is immediate from the behavior of homotopy groups under shifts: \(\pi _k(X[1]) \cong \pi _{k-1}(X)\).

For property (2), let \(X \in \Sp _{\geq 0}\) and \(Y \in \Sp _{\leq 0}\). By the recognition principle for connective spectra (Theorem 5.4.6), we may write \(X \simeq \bB ^{\infty }(A)\) for some \(A \in \CGrp (\An )\). By adjunction, \[ \Hom _{\Sp }(X, Y[-1]) \quad \cong \quad \Hom _{\CGrp (\An )}(A, \Omega ^{\infty }(Y[-1])). \] But \(\Omega ^{\infty }(Y[-1])\) is contractible: its homotopy groups \(\pi _k(\Omega ^{\infty }(Y[-1])) \cong \pi _k(Y[-1]) \cong \pi _{k+1}(Y)\) vanish for all \(k \geq 0\) since \(Y \in \Sp _{\leq 0}\).

For property (3), given a spectrum \(X\), we define \(\tau _{\geq 0} X := \bB ^{\infty }\Omega ^{\infty }X\). The counit of the adjunction \(\bB ^{\infty } \dashv \Omega ^{\infty }\) provides a map \(\tau _{\geq 0} X \to X\). Define \(\tau _{\leq -1} X\) as its cofiber. From the long exact sequence on homotopy groups (Proposition 4.4.30), one verifies that \(\tau _{\geq 0} X\) is connective and \(\tau _{\leq -1} X\) has homotopy groups concentrated in degrees \(\leq -1\). □

Proof. We prove (1); the proof of (2) is dual. By shifting, we may assume that \(n = 0\); in general we may then set \(\tau _{\geq n}(X) := \tau _{\geq 0}(X[-n])[n]\). Using the pointwise formula for adjunctions, it suffices to show that for every object \(X \in C\) there exists some \(\tau _{\geq 0}X\) equipped with a morphism \(\tau _{\geq 0}X \to X\) inducing equivalences \(\Hom _{C}(Y, \tau _{\geq 0}X) \iso \Hom _{C}(Y,X)\) for every \(Y \in C_{\geq 0}\). For this, we consider the exact sequence \(\tau _{\geq 0}X \to X \to \tau _{\leq -1}X\) provided by axiom (3). This gives rise to a long exact sequence \[ \dots \to \pi _{n+1}\Hom _C(Y,\tau _{\leq -1} X) \to \pi _n\Hom _C(Y,\tau _{\geq 0}X) \to \pi _n \Hom _C(Y,X) \to \pi _n\Hom _C(Y,\tau _{\leq -1} X) \to \dots \] Since \(\Hom _C(Y,\tau _{\leq -1} X) = 0\) by axiom (2), this implies that the map \(\Hom _{C}(Y, \tau _{\geq 0}X) \to \Hom _{C}(Y,X)\) induces isomorphisms on all homotopy groups, hence is an equivalence. □

Proof. We prove (1) first. Let \(X \in C_{\leq m}\). If \(n>m\), then \(\Hom _C(Z,X)=0\) for every \(Z \in C_{\geq n}\) by orthogonality. By adjunction, this gives \(\Hom _C(Z,\tau _{\geq n}X)=0\) for every \(Z \in C_{\geq n}\). Taking \(Z=\tau _{\geq n}X\), we see that the identity of \(\tau _{\geq n}X\) is null, so \(\tau _{\geq n}X\) is zero.

Assume instead that \(n \leq m\). The truncation sequence \[ \tau _{\geq n}X \to X \to \tau _{\leq n-1}X \] exhibits \(\tau _{\geq n}X\) as the fiber of a morphism between objects of \(C_{\leq m}\), since \(C_{\leq n-1} \subseteq C_{\leq m}\). By Corollary 6.3.8, it follows that \(\tau _{\geq n}X \in C_{\leq m}\).

The proof of (2) is dual. □

Proof. We prove the first equivalence; the second is dual. Since \(F\) is t-exact, applying \(F\) to the exact sequence \(\tau _{\geq n+1} X \to X \to \tau _{\leq n} X\) gives another exact sequence \[ F(\tau _{\geq n+1} X) \to F(X) \to F(\tau _{\leq n} X) \] where the first object lies in \(D_{\geq n+1}\) and the last one in \(D_{\leq n}\). It follows that the second map induces an isomorphism \(\tau _{\leq n} F(X) \iso F(\tau _{\leq n} X)\). □

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 \(\fib (g) \in C_{\geq 0}\). □

Proof. This is an immediate consequence of part (4) of Proposition 6.3.15. □

Proof. Let \(0 \to X \hookrightarrow Y \twoheadrightarrow Z \to 0\) be a short exact sequence in \(C^{\heartsuit }\). By the previous corollary, the sequence \(X \to Y \to Z\) is exact in \(C\), hence induces an exact sequence \(F(X) \to F(Y) \to F(Z)\) in \(D\). Using the various parts of Proposition 6.3.15, we see:

(i)

By (2), the morphism \(F(Y) \to F(Z)\) is an epimorphism in \(D^{\heartsuit }\), since its fiber \(F(X)\) lies in \(D^{\heartsuit }\);

(ii)

By (3), the morphism \(F(X) \to F(Y)\) is a monomorphism in \(D^{\heartsuit }\), since its cofiber \(F(Z)\) lies in \(D^{\heartsuit }\);

(iii)

By (4), the exact sequence \(F(X) \to F(Y) \to F(Z)\) in \(D\) is also exact in \(D^{\heartsuit }\).

We conclude that the sequence \(0 \to F(X) \hookrightarrow F(Y) \twoheadrightarrow F(Z) \to 0\) is exact in \(D^{\heartsuit }\), as desired. □

Proof. First assume that \(X \in C^{\heartsuit }\). Then for \(Y \in C_{\geq 0}\) and \(k > 0\) we have \(\pi _k\Hom _C(Y,X) \simeq \pi _0\Hom _C(Y,X[-k]) = 0\) as \(X[-k] \in C_{\leq -k} \subseteq C_{\leq -1}\). Conversely, if \(X\) is 0-truncated in \(C_{\geq 0}\), then we have \(\pi _k\Hom _C(Y,X[-1]) \cong \pi _{k+1}\Hom _C(Y,X) \cong 0\) for all \(Y \in C_{\geq 0}\) and \(k \geq 0\). It follows that \(X[-1] \in C_{\leq -1}\), and thus \(X \in C^{\heartsuit }\). □

Proof. By the recognition principle for connective spectra (Theorem 5.4.6), the functor \(\bB ^{\infty }\) induces an equivalence \(\bB ^{\infty }\colon \CGrp (\An ) \iso \Sp _{\geq 0}\). By Corollary 6.3.19, we may identify \(\Sp ^{\heartsuit }\) with the subcategory of \(\CGrp (\An )\) spanned by the 0-truncated objects. Note that object in \(\CGrp (\An )\) is \(0\)-truncated if and only if its underlying anima is 0-truncated, which in turn happens if and only if it is contained in the full subcategory \(\Set \subseteq \An \). All in all, we get \(\Ab = \CGrp (\Set ) \iso \CGrp (\An )_{\leq 0} \iso \Sp ^{\heartsuit }\). □

Proof. By Lemma 6.3.10, both composites take values in \(C_{\leq m} \cap C_{\geq n}\). If \(n>m\), this intersection is zero by orthogonality, so both composites are zero. We may therefore assume that \(n \leq m\).

Fixing \(Y \in C\), the map \(\tau _{\geq n} Y \to Y\) induces \(\tau _{\leq m} \tau _{\geq n} Y \to \tau _{\leq m} Y\). Since the source is \(n\)-connective, this factors uniquely through a map \[ \tau _{\leq m} \tau _{\geq n} Y \to \tau _{\geq n} \tau _{\leq m} Y, \] which is the ‘canonical map’ from the statement. To show it is an isomorphism, we may equivalently show that the cofiber of \(\tau _{\leq m} \tau _{\geq n} Y \to \tau _{\leq m} Y\) is \((n-1)\)-coconnective. To this end, we may compare the truncation sequences for \(\tau _{\geq n} Y\) and \(Y\) to obtain a commutative diagram

Since \(m \geq n\), the left vertical map is an isomorphism. By the pasting law for exact squares, it follows that the cofiber of the right vertical map is the cofiber of the middle vertical map. This cofiber is \(\tau _{\leq n-1}Y\), which is \((n-1)\)-coconnective, finishing the proof. □

Proof. By Lemma 6.3.13 we have natural isomorphisms \[ F(\pi _n X) = F(\tau _{\geq 0} \tau _{\leq 0}(X[-n])) \iso \tau _{\geq 0} \tau _{\leq 0} (F(X)[-n]) = \pi _n(F(X)). \qedhere \] □

Proof. Since every such exact sequence in \(C\) induces exact sequences \(Z[-1] \to X \to Y\) and \(Y \to Z \to X[1]\) and we have \(\pi _n(Z[-1]) = \pi _{n+1}(Z)\) and \(\pi _n(X[1]) = \pi _{n-1}(X)\), it suffices by induction to show that the sequence \[ \pi _0(X) \xrightarrow {\pi _0(f)} \pi _0(Y) \xrightarrow {\pi _0(g)} \pi _0(Z) \] is exact in \(C^{\heartsuit }\). We will prove this in progressively more general cases.

Case 1: Assume that \(X,Y,Z \in C_{\geq 0}\). We claim that in this case, the sequence \[ \pi _0(X) \to \pi _0(Y) \to \pi _0(Z) \to 0 \] is exact. To see this, it suffices to show that for every \(W \in C^{\heartsuit }\), the induced sequence of abelian groups \[ 0 \to \Hom _{C^{\heartsuit }}(\pi _0(Z),W) \to \Hom _{C^{\heartsuit }}(\pi _0(Y),W) \to \Hom _{C^{\heartsuit }}(\pi _0(X),W) \] is exact. Since \(Z \in C_{\geq 0}\), we have \(\pi _0(Z) = \tau _{\leq 0}Z\), so by adjunction we obtain isomorphisms \[ \Hom _{C^{\heartsuit }}(\pi _0(Z),W) \cong \pi _0\Hom _C(Z,W) \cong \pi _0 \hom _C(Z,W), \] and similarly for \(X\) and \(Y\). Furthermore, we have \(\pi _1\hom _C(X,W) \cong \pi _0\hom _C(X[1],W)\cong 0\) as \(\tau _{\leq 0}(X[1]) \cong 0\). The claim now follows from the long exact sequence on homotopy groups associated to the exact sequence of mapping spectra \[ \hom _C(Z,W) \to \hom _C(Y,W) \to \hom _C(X,W). \]

Case 2: Assume that \(X \in C_{\geq 0}\), but \(Y\) and \(Z\) are arbitrary. For every \(W\in C_{\leq -1}\), orthogonality and the exact sequence \(X\to Y\to Z\) identify \(\Hom _C(Z,W)\) with \(\Hom _C(Y,W)\). By adjunction and the Yoneda lemma in \(C_{\leq -1}\), the induced map \(\tau _{\leq -1}Y\to \tau _{\leq -1}Z\) is therefore an isomorphism, so we get a cofiber sequence \[ 0 \cong \tau _{\leq -1}X \to \tau _{\leq -1}Y \to \tau _{\leq -1}Z, \] showing that \(\tau _{\leq -1}Y \cong \tau _{\leq -1}Z\). Consider now the following diagram:

Since all three columns are exact and the bottom two rows are exact, it follows that the top row is also exact. Since \(\pi _0(X) \cong \pi _0(\tau _{\geq 0}X)\), and similarly for \(Y\) and \(Z\), the exactness of \(\pi _0(X) \to \pi _0(Y) \to \pi _0(Z) \to 0\) has thus been reduced to Case 1.

Case 3: Assume that \(Z \in C_{\leq 0}\), but \(X\) and \(Y\) are arbitrary. Using an argument dual to that of Cases 1 and 2, we obtain an exact sequence \[ 0 \to \pi _0(X) \to \pi _0(Y) \to \pi _0(Z). \]

Case 4: Finally, we consider the case for general \(X,Y,Z\). Let \(Q\) denote the cofiber of the composite \(\tau _{\geq 0}X \to X \to Y\). We then have the following commutative diagram:

The three rows are exact, as are the left two columns, and hence so is the third column. Applying Cases 2 and 3, we get exact sequences \[ \pi _0(X) \to \pi _0(Y) \twoheadrightarrow \pi _0(Q) \to 0 \qquadtext { and } 0 \to \pi _0(Q) \hookrightarrow \pi _0(Z) \to \pi _{-1}(X). \] Gluing them together then gives the desired exact sequence \(\pi _0(X) \to \pi _0(Y) \to \pi _0(Z)\). □

Proof. This is [Lurie (2017), Proposition 1.2.1.19]. □