Stabilization

Proof. The projection \(\PSp (C) \to C_*^{\N }\) is conservative because a morphism in the arrow category is an isomorphism precisely when its source and target maps are isomorphisms. The projections \(C_*^{\N } \to C_*\) are jointly conservative by definition of the product. □

Proof. This is just the definition of \(\PSp (C)\) as the lax equalizer of \(\id \) and \(\Omega \sh \). Indeed, a morphism \(X\to Y\) in \(\PSp (C)\) is a map of the underlying sequences together with a homotopy filling the square

Equivalently, it is a point of the stated equalizer. □

Proof. For limits, let \(X\colon I \to \PSp (C)\) be a diagram and write \(\overline {X}\colon I \to C_*^{\N }\) for the underlying diagram of sequences. Since \(C_*\) has \(I\)-indexed limits and \(\Omega \) preserves limits, the endofunctor \(\Omega \sh \) preserves \(I\)-indexed limits. Hence the limit \(L := \lim _i \overline {X}_i\) in \(C_*^{\N }\) inherits a structure map \[ L \simeq \lim _i \overline {X}_i \to \lim _i \Omega \sh (\overline {X}_i) \simeq \Omega \sh (L), \] which makes it the limit in \(\PSp (C)\).

For colimits, let \(D := \colim _i \overline {X}_i\) in \(C_*^{\N }\). For every \(i\), the composite \[ \overline {X}_i \xrightarrow {\sigma ^{X_i}} \Omega \sh (\overline {X}_i) \to \Omega \sh (D) \] is compatible with the diagram, and therefore induces a structure map \(D \to \Omega \sh (D)\). We claim that the resulting prespectrum is the colimit. Indeed, for any prespectrum \(Y\) the hom anima \(\Hom _{\PSp (C)}(D,Y)\) is the equalizer of the two maps \[ \Hom _{C_*^{\N }}(D,Y) \rightrightarrows \Hom _{C_*^{\N }}(D,\Omega \sh Y) \] given by postcomposition with \(\sigma ^Y\) and precomposition with the structure map of \(D\), by Lemma 4.3.7. Since \(D\) is the colimit of the \(\overline {X}_i\) in \(C_*^{\N }\), and since limits commute with limits in \(\An \), this equalizer identifies with \[ \lim _i \Hom _{\PSp (C)}(X_i,Y), \] which is the desired universal property. □

Proof. We construct an inverse. Given \(Y\in \Sp ^{(2)}(C)\) with structure maps \(\tau _k\colon Y_k \iso \Omega ^2Y_{k+1}\), define a spectrum \(QY\) by \[ (QY)_{2k}:=Y_k \qquadtext { and } \qquad (QY)_{2k+1}:=\Omega Y_{k+1}. \] The structure maps of \(QY\) are \(\tau _k\colon Y_k\to \Omega ^2Y_{k+1}=\Omega (QY)_{2k+1}\) in even degrees and the identity map \(\Omega Y_{k+1}\to \Omega Y_{k+1}=\Omega (QY)_{2k+2}\) in odd degrees.

It is immediate that \(EQ=\id \). Conversely, for \(X\in \Sp (C)\) there is a natural isomorphism \(X\to QEX\) which is the identity in even degrees and the structure map \(\sigma ^X_{2k+1}\colon X_{2k+1}\to \Omega X_{2k+2}\) in odd degrees. The compatibility with the structure maps follows directly from the definition of \(\widetilde {\sigma }^X_k\). □

Proof. This is the opposite of the universal property of the free finite cocompletion of the point, which is a special case of Proposition 22.1.2. □

Proof. Precomposition with \((-)_+\colon \An ^{\fin }\to \An _*^{\fin }\) defines a functor \[ \Fun ^{\lex }((\An _*^{\fin })\catop ,D) \longrightarrow \Fun ^{\lex }((\An ^{\fin })\catop ,D). \] An inverse sends \(F\) to the functor \[ (Y,y)\longmapsto \fib (F(Y)\to F(*)). \] This is pointed and left exact. The two inverse identities follow from \(Y_+=Y\sqcup *\) and from the pushout \(Y_+\sqcup _{S^0}*\simeq (Y,y)\). The result now follows from Proposition 4.3.11. □

Proof. Given \(X\in D\), Lemma 4.3.12 gives a pointed left exact functor \[ X^{(-)}\colon (\An _*^{\fin })\catop \to D \] satisfying \(X^{S^0}\simeq X\). Since \(X^{(-)}\) sends pushouts in \(\An _*^{\fin }\) to pullbacks in \(D\), we have \[ X^{S^{n+1}} \simeq X^{\Sigma S^n} \simeq \Omega X^{S^n} \] for every \(n\geq 0\). Thus the comparison automorphism of \(\Omega ^3X\) is induced by the analogous automorphism of \(S^3\) in \(\An _*^{\fin }\).

It remains to prove the claim in \(\An _*^{\fin }\) for the generator \(X = S^0\). We must show that the composite \[ \tau \colon S^3 \simeq \Sigma (\Sigma ^2S^0) \xrightarrow {\cong } \Sigma ^2(\Sigma S^0) \simeq S^3 \] is pointed homotopic to the identity. By Remark 2.4.20, it suffices to show that \(\tau \) has degree \(1\). Using the swap map \(\sigma _{X,Y}\) from Definition 2.4.10, we may identify \(\tau \) with \(\sigma _{S^1,S^2}\). The isomorphism \(S^2 \cong S^1 \wedge S^1\) from Lemma 2.4.16 then gives a decomposition \[ S^3 \simeq S^1\wedge S^1\wedge S^1 \xrightarrow {\id \wedge \sigma _{S^1,S^1}} S^1\wedge S^1\wedge S^1 \xrightarrow {\sigma _{S^1,S^1}\wedge \id } S^1\wedge S^1\wedge S^1 \simeq S^3. \] Both maps in this decomposition are suspensions of \(\sigma _{S^1,S^1}\), up to the natural identifications of smash products with suspensions. Their degrees are therefore equal to that of \(\sigma _{S^1,S^1}\) by Remark 2.4.20. Since this swap map is an isomorphism, its degree is \(\pm 1\). We conclude that \[ \deg (\tau )=\deg (\id \wedge \sigma _{S^1,S^1})\cdot \deg (\sigma _{S^1,S^1}\wedge \id )=\deg (\sigma _{S^1,S^1})^2=1, \] as desired. The construction is natural in \(X\), since the comparison is induced by the automorphism \(\tau \) of \(S^3\) under the equivalence of Lemma 4.3.12. □

Proof. By Lemma 4.3.8, finite limits in \(\PSp (C)\) are computed pointwise. Since \(\Omega \) preserves finite limits, the full subcategory \(\Sp (C) \subseteq \PSp (C)\) is closed under finite limits. It is also pointed: the constant zero prespectrum is both initial and terminal, since the zero object of \(C_*\) is both initial and terminal.

It remains, by Theorem 4.2.2, to show that the loop functor on \(\Sp (C)\) is an equivalence. Define the shift functor \[ \shift \colon \Sp (C) \to \Sp (C), \qquad \shift (X)_n:=X_{n+1}, \] with structure maps inherited from \(X\). This functor is an equivalence: an inverse sends \(X\) to the spectrum \(\shift ^{-1}X\) with \[ (\shift ^{-1}X)_0:=\Omega X_0 \qquadtext { and } \qquad (\shift ^{-1}X)_n:=X_{n-1}\quad (n\geq 1), \] whose structure map in degree \(0\) is the identity of \(\Omega X_0\), and whose structure map in degree \(n\geq 1\) is \(\sigma ^X_{n-1}\).

We claim that there is a natural isomorphism \[ \alpha \colon X \xrightarrow {\cong } \Omega (\shift X). \] By Lemma 4.3.9, it suffices to construct this map after passing to the even-level model. There we set \[ \alpha _{2k}:=\sigma ^X_{2k}\colon X_{2k} \xrightarrow {\cong } \Omega X_{2k+1} = \Omega (\shift X)_{2k}. \] We must check compatibility with the two-step structure maps. For \(X\), the relevant structure map is \[ \widetilde {\sigma }^X_k = \left (X_{2k}\xrightarrow {\sigma ^X_{2k}}\Omega X_{2k+1} \xrightarrow {\Omega (\sigma ^X_{2k+1})}\Omega ^2X_{2k+2}\right ). \] For \(\Omega (\shift X)\), finite limits are computed pointwise, but its two-step structure map is not obtained by merely writing down the same composite with an extra \(\Omega \) in front. After precomposing with \(\alpha _{2k}=\sigma ^X_{2k}\), the first composite in the compatibility square is \[ X_{2k} \xrightarrow {\sigma ^X_{2k}} \Omega X_{2k+1} \xrightarrow {\Omega (\sigma ^X_{2k+1})} \Omega ^2X_{2k+2} \xrightarrow {\Omega (\Omega (\sigma ^X_{2k+2}))} \Omega (\Omega ^2X_{2k+3}) \xrightarrow {\cong } \Omega ^2(\Omega X_{2k+3}), \] where the final arrow is the canonical comparison coming from the fact that \(\Omega ^2\) preserves loop objects. The other composite in the compatibility square is \[ X_{2k} \xrightarrow {\sigma ^X_{2k}} \Omega X_{2k+1} \xrightarrow {\Omega (\sigma ^X_{2k+1})} \Omega ^2X_{2k+2} \xrightarrow {\Omega ^2(\sigma ^X_{2k+2})} \Omega ^2(\Omega X_{2k+3}). \] Thus the two composites whose equality is required differ only by the resulting automorphism of \(\Omega ^3X_{2k+3}\). This automorphism is the identity by Proposition 4.3.13. Hence \(\alpha \) is an isomorphism of spectra.

Consequently \(\id _{\Sp (C)}\simeq \Omega \circ \shift \). Since \(\shift \) is an equivalence, \(\Omega \) is an equivalence as well. □

Proof. The proof of Theorem 4.3.14 gives a natural isomorphism \(\id _{\Sp (C)} \cong \Omega \circ \shift \). Since \(\Sigma \) is inverse to \(\Omega \), this identifies \(\Sigma \) with \(\shift \). □

Proof. Let \(D\) be a stable \(\infty \)-category. We need to show that postcomposition with \(\Omega ^{\infty }\) induces an equivalence \[ \Fun ^{\lex }(D,\Sp (C)) \iso \Fun ^{\lex }(D,C). \]

We construct an inverse. Let \(F\colon D \to C\) be left exact. Since \(D\) is pointed, the functor \(F\) has a canonical lift to \(C_*\): the object \(F(d)\) is pointed by the map \(* \simeq F(0) \to F(d)\) induced by \(0 \to d\). For every \(n \geq 0\), define a functor \[ \widetilde {F}_n\colon D \to C_*, \qquad \widetilde {F}_n(d) := F(d[n]). \] The unit equivalences \(d[n] \to \Omega _Dd[n+1]\), together with left exactness of \(F\), provide natural isomorphisms \[ \widetilde {F}_n \iso \Omega \widetilde {F}_{n+1}. \] Explicitly, these are obtained by applying \(F\) to the unit equivalences and then using the natural isomorphisms \(F(\Omega _Dd[n+1]) \iso \Omega F(d[n+1])\) provided by left exactness. Thus the functors \(\widetilde {F}_n\) and these structure isomorphisms assemble into a functor \[ \widetilde {F}\colon D \to \Sp (C). \] This construction is natural in \(F\), and hence defines a functor \(\Fun ^{\lex }(D,C)\to \Fun ^{\lex }(D,\Sp (C))\). Since limits in \(\Sp (C)\) are computed levelwise, and since each shift \(d\mapsto d[n]\) is an equivalence of \(D\), the functor \(\widetilde {F}\) is left exact.

The composite \(\Omega ^{\infty }\widetilde {F}\) is \(F\), since evaluation at level zero gives \(\widetilde {F}(d)_0=F(d)\). Conversely, let \(G\colon D \to \Sp (C)\) be left exact. Since \(D\) and \(\Sp (C)\) are stable, \(G\) is exact by Corollary 4.2.27. Hence \[ G(d[n]) \simeq G(d)[n]. \] By Corollary 4.3.15, evaluation at level zero gives natural isomorphisms \[ (\Omega ^{\infty }G)(d[n]) = G(d[n])_0 \simeq G(d)_n. \] These isomorphisms are compatible with the structure maps, by naturality of the unit \(d[n]\to \Omega _Dd[n+1]\) and of the equivalence between suspension and shift in \(\Sp (C)\). Thus \(\widetilde {\Omega ^{\infty }G}\simeq G\), naturally in \(G\). □

Proof. By the pointwise criterion for adjunctions from Lemma 21.1.4, it suffices to show that for every stable \(\infty \)-category \(D\), composition with \(\Omega ^{\infty }\colon \Sp (C) \to C\) induces an equivalence \[ \Omega ^{\infty } \circ -\colon \Hom _{\Cat ^{\st }_{\infty }}(D,\Sp (C)) \iso \Hom _{\Cat ^{\lex }_{\infty }}(D,C). \] But since a functor \(D \to \Sp (C)\) is exact if and only if it is left exact, this map is obtained from the defining equivalence \(\Fun ^{\lex }(D,\Sp (C)) \iso \Fun ^{\lex }(D,C)\) by passing to groupoid cores. □

Proof. By Lemma 4.3.8, the limit of a diagram of spectra can first be computed in \(\PSp (C)\). Since \(\Omega \) preserves limits, the structure maps of this pointwise limit are again equivalences. Thus the limit prespectrum is a spectrum. □

Proof. The endofunctor \(\Omega \sh \) of \(C_*^{\N }\) induces an endofunctor, still denoted \(\Omega \sh \), of \(\PSp (C)\) by \[ (\Omega \sh X)_n := \Omega X_{n+1}, \] with structure maps \(\Omega (\sigma ^X_{n+1})\colon \Omega X_{n+1} \to \Omega ^2X_{n+2}\). The structure maps of \(X\) define a natural map \(\eta _X\colon X \to \Omega \sh X\). Since \(\sh \) preserves colimits and \(\Omega \) preserves sequential colimits, the endofunctor \(\Omega \sh \) preserves sequential colimits of prespectra.

For a prespectrum \(X\), define \[ X^{\mathrm {sp}} := \colim \left ( X \xrightarrow {\eta _X} \Omega \sh X \xrightarrow {\Omega \sh (\eta _X)} (\Omega \sh )^2X \to \cdots \right ) \] in \(\PSp (C)\). Then \[ \Omega \sh (X^{\mathrm {sp}}) \simeq \colim _{k \geq 0} (\Omega \sh )^{k+1}X \simeq \colim _{k \geq 0} (\Omega \sh )^kX = X^{\mathrm {sp}}, \] where the second isomorphism uses that the successor map \(k \mapsto k+1\) is cofinal in \(\N \). Under this isomorphism, the structure map of \(X^{\mathrm {sp}}\) is an isomorphism, so \(X^{\mathrm {sp}}\) is a spectrum.

It remains to prove the universal property. Let \(Y\) be a spectrum. Since \(\eta _Y\colon Y \to \Omega \sh Y\) is an isomorphism, composition with \(\eta _X\) induces an equivalence \[ \Hom _{\PSp (C)}(\Omega \sh X,Y) \xrightarrow {\sim } \Hom _{\PSp (C)}(X,Y); \] an inverse sends a map \(f\colon X \to Y\) to the composite \(\Omega \sh X \xrightarrow {\Omega \sh (f)} \Omega \sh Y \xrightarrow {\eta _Y^{-1}} Y\). Therefore \[ \Hom _{\Sp (C)}(X^{\mathrm {sp}},Y) \simeq \lim _k \Hom _{\PSp (C)}((\Omega \sh )^kX,Y) \simeq \Hom _{\PSp (C)}(X,Y), \] which is the desired adjunction. □

Proof. Let \(X\colon I \to \Sp (C)\) be a diagram. Compute its colimit in \(\PSp (C)\), where colimits are pointwise by Lemma 4.3.8, and then apply spectrification. Since spectrification is left adjoint to the inclusion \(\Sp (C) \hookrightarrow \PSp (C)\), the resulting spectrum has the universal property of the colimit in \(\Sp (C)\). □

Proof. Filtered colimits in \(\An _*\) are created in \(\An \), and the loop functor \(\Omega \colon \An _* \to \An _*\) preserves them. Hence the levelwise colimit in \(\PSp (\An )\) of a filtered diagram of spectra again has isomorphisms as its structure maps, so it is already a spectrum. The final claim follows by evaluating at level zero. □

Proof. First define the free prespectrum functor \[ \Sigma ^{\infty ,\pre }\colon C_* \to \PSp (C) \] by \[ \Sigma ^{\infty ,\pre }(X)_n := \Sigma ^n X, \] with structure maps \(\Sigma ^nX \to \Omega \Sigma ^{n+1}X\) given by the unit of the adjunction \(\Sigma \dashv \Omega \) in \(C_*\). We claim that \(\Sigma ^{\infty ,\pre }\) is left adjoint to evaluation at level zero \(\ev _0\colon \PSp (C) \to C_*\). Indeed, a map \(\Sigma ^{\infty ,\pre }(X) \to Y\) is determined by its level-zero component \(X \to Y_0\): once \(f_n\colon \Sigma ^nX \to Y_n\) is known, compatibility with the structure maps forces \(f_{n+1}\colon \Sigma ^{n+1}X \to Y_{n+1}\) to be adjoint to the composite \[ \Sigma ^nX \xrightarrow {f_n} Y_n \xrightarrow {\sigma ^Y_n} \Omega Y_{n+1}. \] Conversely, this recursive construction produces a compatible map of prespectra. The construction is natural in \(X\) and \(Y\) and applies equally to parametrized families of maps, so it gives an isomorphism of hom animae \[ \Hom _{\PSp (C)}(\Sigma ^{\infty ,\pre }X,Y) \simeq \Hom _{C_*}(X,Y_0). \]

Now set \[ \Sigma ^{\infty }(X) := (\Sigma ^{\infty ,\pre }X)^{\mathrm {sp}}. \] For \(Y \in \Sp (C)\), spectrification gives natural equivalences \[ \Hom _{\Sp (C)}(\Sigma ^{\infty }X,Y) \simeq \Hom _{\PSp (C)}(\Sigma ^{\infty ,\pre }X,Y) \simeq \Hom _{C_*}(X,\Omega ^{\infty }Y), \] which proves the adjunction. □

Proof. Since adjunctions compose, this is immediate from the previous lemma and the fact that a left adjoint to the forgetful functor \(C_* \to C\) is given by \((-)_+\colon C \to C_*, \, X \mapsto X_+ := X \sqcup *\). □

Proof. This follows from stability of \(\Sp (C)\) and the fact that \(\Omega ^{\infty }\colon \Sp (C)\to C_*\) preserves finite limits. □

Proof. Since adjunctions compose, the functor \((\Sigma ^{\infty }(-))[-n]\) is left adjoint to \(\Omega ^{\infty }((-)[n])=\Omega ^{\infty -n}\). The stated relations follow by passing to left adjoints in the previous lemma. □

Proof. Let \(Y\) be a spectrum in \(\Sp (C)\). Mapping out of the displayed colimit gives \[ \lim _n \Hom _{\Sp (C)}(\Sigma ^{\infty -n}X_n,Y) \simeq \lim _n \Hom _{C_*}(X_n,\Omega ^{\infty -n}Y). \] The transition maps in this limit are precisely the compatibility conditions for a map of prespectra \(X \to Y\). Hence the right-hand side identifies with \(\Hom _{\PSp (C)}(X,Y)\). By the spectrification adjunction from Proposition 4.3.19, this is naturally equivalent to \(\Hom _{\Sp (C)}(X^{\mathrm {sp}},Y)\). The claim follows from Yoneda. □

Proof. Apply the previous proposition to the underlying prespectrum of \(X\). Since \(X\) is already a spectrum, spectrification fixes it. □