The ∞-category of spectra

Definition 4.4.1. The \(\infty \)-category of spectra \(\Sp \) is defined as the stabilization of the \(\infty \)-category of animae, \[ \Sp \quad := \quad \Sp (\An ). \] Objects of \(\Sp \) are called spectra. Concretely, a spectrum is a sequence of pointed animae \(X_n\) equipped with isomorphisms \[ \sigma _n\colon X_n \iso \Omega (X_{n+1}). \]

Definition 4.4.2 (Sphere spectrum). We define the sphere spectrum as \(\S := \Sigma ^{\infty }(S^0) \simeq \Sigma ^{\infty }_+(*) \in \Sp \).

For an anima \(X\), we will frequently use the alternative notation \[ \S [X] \quad := \quad \Sigma ^{\infty }_+(X) \] for the unreduced suspension spectrum. We wish to think of the spectrum \(\S [X]\) as the ‘free spectrum generated by \(X\)’. This notation mimics the notation for the free abelian group \(\Z [S]\) generated by a set \(S\), characterized by the property that group homomorphisms \(\Z [S] \to A\) correspond to maps of sets \(S \to A\).

Remark 4.4.3. Given a pointed anima \((X,x) \in \An _*\), note that we may write \(X\) as the cofiber of the basepoint inclusion \(*_+ \to X_+\). Since \(\Sigma ^{\infty }(-)\) preserves cofibers, we may write the reduced suspension spectrum of \(X\) as \[ \Sigma ^{\infty }(X,x) \quad \simeq \quad \cofib (\S \xrightarrow {x} \S [X]). \]

Proposition 4.4.4. For a stable \(\infty \)-category \(C\), there exists a unique functor \[ \hom _C(-,-)\colon C\catop \times C \to \Sp \] that is exact in both variables and fits in a commutative triangle as follows:

Terminology 4.4.5. We refer to the spectra \(\hom _C(X,Y)\) as the hom spectra or mapping spectra of \(C\).

Corollary 4.4.6 (Stable Yoneda lemma). For a stable \(\infty \)-category \(C\), the functor \[ C \,\to \,\Fun (C\catop ,\Sp ), \qquad Y \, \mapsto \, \hom _C(-,Y) \] is fully faithful. □

Exercise 4.4.7. Show that for \(C = \Sp \), the Hom spectrum \(\hom (X,Y) := \hom _{\Sp }(X,Y)\) may be explicitly described at level \(n\) by \[ \hom (X,Y)_n \quad \simeq \quad \Hom _{\Sp }(X,Y[n]). \]

Lemma 4.4.8. The functor \(\hom (\S ,-)\colon \Sp \to \Sp \) is equivalent to the identity.

Proposition 4.4.9 (Tensoring by spectra). Let \(C\) be a stable \(\infty \)-category with small colimits. Then \(C\) is tensored over \(\Sp \): there exists a unique functor \[ - \otimes - \colon \Sp \times C \to C \] equipped with a natural equivalence of spectra \[ \hom _C(X \otimes Y, Z) \quad \cong \quad \hom _{\Sp }(X, \hom _C(Y,Z)) \] for \(X \in \Sp \) and \(Y,Z \in C\).

Corollary 4.4.10. Let \(C\) be a stable \(\infty \)-category with small colimits. For every object \(Y \in C\), the functor \(\hom _C(Y,-)\colon C \to \Sp \) admits a left adjoint \[ - \otimes Y\colon \Sp \to C. \]

Lemma 4.4.11. The functor \(- \otimes -\colon \Sp \times C \to C\) preserves colimits in both variables.

Lemma 4.4.12. The functor \(\S \otimes -\colon C \to C\) is equivalent to the identity.

Lemma 4.4.13. For a colimit-preserving functor \(F\colon \Sp \to C\), there exists a natural isomorphism \[ F(-) \, \cong \, - \otimes F(\S ). \] Moreover, this natural isomorphism can be chosen naturally in \(F \in \Fun ^{\colim }(\Sp ,C)\).

Theorem 4.4.14 (Universality of \(\Sp \)). Let \(C\) be a stable \(\infty \)-category with small colimits. Then evaluation at the sphere spectrum induces an equivalence \[ \ev _{\S } \colon \Fun ^{\colim }(\Sp ,C) \quad \iso \quad C. \]

Corollary 4.4.15. Let \(C\) be a stable \(\infty \)-category with small colimits. Then every colimit-preserving functor \(F\colon \Sp \to C\) admits a right adjoint \(F^R\colon C \to \Sp \).

Proposition 4.4.16. There exists a unique tensor product functor \[ - \otimes -\colon \Sp \times \Sp \to \Sp \] which preserves colimits in both variables and satisfies \(\S \otimes \S \cong \S \).

Lemma 4.4.17. For spectra \(X,Y,Z \in \Sp \), there are natural isomorphisms of spectra \[ \S \otimes X \cong X \cong X \otimes \S , \qquad X \otimes Y \cong Y \otimes X \qquad (X \otimes Y) \otimes Z \cong X \otimes (Y \otimes Z) \] and \[ \hom (Y,\hom (X,Z)) \cong \hom (X \otimes Y, Z) \cong \hom (X,\hom (Y,Z)). \]

Lemma 4.4.18. There exists a natural isomorphism \[ \S [X] \otimes \S [Y] \quad \cong \quad \S [X \times Y] \] for all \(X,Y \in \An \).

Lemma 4.4.19. There exists a natural isomorphism \[ \Sigma ^{\infty }(X) \otimes \Sigma ^{\infty }(Y) \quad \cong \quad \Sigma ^{\infty }(X \wedge Y) \] for all \(X,Y \in \An _*\).

Remark 4.4.20. Let \(X\) and \(Y\) be spectra. The standard presentations of \(X\) and \(Y\) provide the following description of their tensor product: \[ X \otimes Y \quad \cong \quad \colim _n \colim _m (\Sigma ^{\infty -n} X_n \otimes \Sigma ^{\infty - m} Y_m) \quad \simeq \quad \colim _n \colim _m \Sigma ^{\infty - n - m} (X_n \wedge Y_m), \] where the last equivalence holds by the previous lemma. However, this formula should be used with care: it makes implicit use of the additivity of the shift functors \(X[-n][-m] \cong X[-(n+m)]\) from Warning 4.2.11. To avoid introducing sign errors, we will not work with this formula for the tensor product.

Definition 4.4.21. For a spectrum \(X\) and an integer \(k\), we define its \(k\)-th homotopy group \(\pi _k(X) \in \Ab \) as \[ \pi _k(X) := [\S [k],X] := \pi _0\Hom _{\Sp }(\S [k],X). \] The assignment \(X \mapsto \pi _k(X)\) defines a functor \(\pi _*\colon \Sp \to \Ab \).

Exercise 4.4.22. Show that we have \(\pi _{k}(X[n]) \simeq \pi _{k-n}(X)\) for all \(k,n \in \Z \).

Exercise 4.4.23. Show that for \(n \geq 0\) and \(k + n \geq 0\) we have \(\pi _k(X) \cong \pi _{n+k}(X_n)\). In particular we have \(\pi _k(X) = \pi _k(\Omega ^{\infty }X)\) for \(k \geq 0\).

Example 4.4.24. For a pointed anima \(X\), the negative homotopy groups of its suspension spectrum \(\Sigma ^{\infty }X\) vanish. For \(k \geq 0\), Remark 4.3.27 gives \[ \pi _k(\Sigma ^{\infty }(X)) \; \cong \; \pi _k(\Omega ^{\infty } \Sigma ^{\infty }(X)) \; \cong \; \colim _{n \geq 0} \pi _k(\Omega ^n \Sigma ^nX) = \colim _n \pi _{n+ k}(\Sigma ^n X). \] These groups are known as the stable homotopy groups of \(X\). By the classical Freudenthal suspension theorem [Freudenthal (1937)] the maps \(\pi _{n+k}(\Sigma ^nX) \to \pi _{n+k+1}(\Sigma ^{n+1}X)\) are isomorphisms for \(n \geq k+2\).

Example 4.4.25. By Remark 2.4.20, there is an isomorphism \(\deg \colon \pi _n(S^n) \xrightarrow {\cong } \Z \) for every \(n\geq 1\), and the suspension maps correspond to the identity of \(\Z \). It follows that there is an isomorphism \(\pi _0(\S ) \cong \colim _n \pi _n(S^n) \cong \Z \).

Example 4.4.26 (The stable Hopf map). The first positive stable stem already contains information that ordinary homology does not detect. Let \[ h\colon S^3 \longrightarrow S^2 \] be the Hopf fibration. Applying the suspension-spectrum functor and shifting by \(-2\) gives the stable Hopf map \[ \eta \colon \S [1] \longrightarrow \S . \] The classical calculation \(\pi _4(S^3)\cong \Z /2\) identifies the suspension of \(h\) with its nonzero element. The Freudenthal suspension theorem then shows that all subsequent suspensions are isomorphisms, and hence \[ \pi _1(\S )\cong \Z /2, \] generated by \(\eta \); see, for example, [Mosher and Tangora (1968); Switzer (1975)].

Regard \(\CP ^2\) as pointed at its \(0\)-cell. Its usual cell structure \(\CP ^2\cong S^2\cup _h e^4\) gives an exact sequence of spectra \[ \S [3]\xrightarrow {\ \Sigma ^{\infty }h\ }\S [2]\longrightarrow \Sigma ^{\infty }\Piinfty {\CP ^2}. \] Thus the attaching map of the top cell remains nonzero after stabilization, even though it induces the zero map on reduced integral homology. This is the simplest example in which the stable category retains information not visible to ordinary homology groups.

Corollary 4.4.27. The functors \(\pi _k\colon \Sp \to \Ab \) for \(k \in \Z \) are jointly conservative: a morphism of spectra \(f\colon X \to Y\) is an isomorphism if and only if the induced map \(\pi _k(f) \colon \pi _k(X) \to \pi _k(Y)\) is an isomorphism.

Lemma 4.4.28. For every \(k \in \Z \), the functor \(\pi _k\colon \Sp \to \Ab \) preserves filtered colimits.

Lemma 4.4.29. For every \(k \in \Z \), the functor \(\pi _k\colon \Sp \to \Ab \) preserves arbitrary coproducts.

Proposition 4.4.30 (Long exact sequence of homotopy groups). Let \(X \xrightarrow {f} Y \xrightarrow {g} Z\) be an exact sequence of spectra. Then there is a long exact sequence of homotopy groups of the form \[ \dots \xrightarrow {} \pi _{1}(Y) \xrightarrow {g_*} \pi _{1}(Z) \to \pi _0(X) \xrightarrow {f_*} \pi _0(Y) \xrightarrow {g_*} \pi _0(Z) \to \pi _{-1}(X) \xrightarrow {f_*} \pi _{-1}(Y) \xrightarrow {} \dots . \]

Definition 4.4.31 (\(E\)-homology). Let \(E\) be a spectrum and let \(X \in \An _*\) be a pointed anima. For \(k\in \Z \), we define the reduced \(E\)-homology of \(X\) by \[ \widetilde E_k(X):=\pi _k(\Sigma ^{\infty }X\otimes E). \]

Proposition 4.4.32. For every spectrum \(E\), the functors \(\widetilde E_*\) admit natural suspension isomorphisms that make them into a reduced homology theory in the sense of Definition 3.1.2.

Remark 4.4.33 (Comparison with the earlier definition). The cohomology theory associated to \(E\) in Definition 3.2.3 can now be expressed in terms of the mapping spectrum. Indeed, the adjunction \(\Sigma ^{\infty }\dashv \Omega ^{\infty }\) gives \begin {align*} \pi _{-k}\hom (\Sigma ^{\infty }X,E) &\cong [\Sigma ^{\infty }X[-k],E] \\ &\cong [\Sigma ^{\infty }X,E[k]] \\ &\cong [X,\Omega ^{\infty }(E[k])]_* \\ &\cong [X,E_k]_* = \widetilde E^k(X). \end {align*}

Here the last isomorphism uses the spectrum structure maps for \(k\geq 0\) and the convention \(E_k=\Omega ^{-k}E_0\) for \(k<0\). All displayed isomorphisms are isomorphisms of abelian groups: under \(\Omega ^{\infty }(E[k])\simeq E_k\), the canonical infinite-loop group structure agrees with the double-loop group structure used in Definition 3.2.3.

Lemma 4.4.34 (Mapping telescope). Let \(C\) be a stable \(\infty \)-category with countable coproducts, and let \(Y_0 \xrightarrow {f_0} Y_1 \xrightarrow {f_1} Y_2 \to \dots \) be a sequential diagram in \(C\). Write \(\iota _k\colon Y_k \to \bigoplus _{j \geq 0} Y_j\) for the inclusion of the \(k\)-th summand. Then there is an exact sequence \[ \bigoplus _{k \geq 0} Y_k \xrightarrow {\ \id - f\ } \bigoplus _{k \geq 0} Y_k \longrightarrow \colim _k Y_k, \] where \(\id - f\) restricts on the \(k\)-th summand to the difference \(\iota _k - \iota _{k+1} \circ f_k\).

Corollary 4.4.35 (Dual mapping telescope). Let \(C\) be a stable \(\infty \)-category with countable products. For every tower \[ \dots \xrightarrow {f_2} Y_2 \xrightarrow {f_1} Y_1 \xrightarrow {f_0} Y_0 \] in \(C\), there is an exact sequence \[ \lim _k Y_k \longrightarrow \prod _{k \geq 0}Y_k \xrightarrow {\ \id -f\ } \prod _{k \geq 0}Y_k, \] where the \(k\)-th component of \((\id -f)(y)\) is \(y_k-f_k(y_{k+1})\).

Definition 4.4.36 (First derived limit). Let \(\dots \xrightarrow {f_2} A_2 \xrightarrow {f_1} A_1 \xrightarrow {f_0} A_0\) be a tower of abelian groups. Define a homomorphism \[ d\colon \prod _{k \geq 0}A_k \longrightarrow \prod _{k \geq 0}A_k, \qquad d(a)_k := a_k-f_k(a_{k+1}). \] The first derived limit of the tower is the abelian group \[ \lim _k^1 A_k := \coker (d). \] Notice that the ordinary inverse limit \(\lim _k A_k\) is the kernel of \(d\).

Corollary 4.4.37 (Milnor sequence). For every tower \(\dots \to Y_2 \to Y_1 \to Y_0\) of spectra and every \(n \in \Z \), there is a natural short exact sequence \[ 0 \longrightarrow \lim _k^1 \pi _{n+1}(Y_k) \longrightarrow \pi _n\big (\lim _k Y_k\big ) \longrightarrow \lim _k \pi _n(Y_k) \longrightarrow 0. \]

Exercise 4.1. Let \(C\) and \(D\) be \(\infty \)-categories with finite limits. Show that there is an equivalence \[ \Sp (C \times D) \iso \Sp (C) \times \Sp (D). \]

Exercise 4.2. Let \(C\) be a semiadditive \(\infty \)-category.

(1)

Show that for all objects \(X\) and \(Y\) in \(C\) the set \([X,Y] := \pi _0\Hom _C(X,Y)\) of homotopy classes of morphisms from \(X\) to \(Y\) admits a canonical structure of an abelian monoid, where addition of \(f,g \in [X,Y]\) is given by the composite \[ f + g \colon \quad X \xrightarrow {\Delta } X \oplus X \xrightarrow {f \oplus g} Y \oplus Y \xrightarrow {\nabla } Y. \]

(2)

Show that for all morphisms \(f\colon X' \to X\) and \(g\colon Y \to Y'\) in \(C\) the composition maps \[ - \circ f\colon [X,Y] \to [X',Y] \qquadtext {and} g \circ - \colon [X,Y] \to [X,Y'] \] are homomorphisms of abelian monoids.

(3)

Let \(M\) be an abelian monoid. Show that \(M\) is an abelian group if and only if the map \[ \begin {pmatrix} 1 & 1 \\ 0 & 1 \end {pmatrix}\colon M \oplus M \to M \oplus M, \qquad (x,y) \mapsto (x,x+ y) \] is a bijection of sets.

(4)

Conclude that the semiadditive \(\infty \)-category \(C\) is in fact additive if and only if the abelian monoid \([X,Y]\) is an abelian group for all \(X\) and \(Y\).

Exercise 4.3. Let \(C\) be a pointed \(\infty \)-category with finite limits.

(1)

Show that for every object \(X\), the loop space \(\Omega X\) admits the structure of a group object in the homotopy category \(\Ho (C)\): there are maps \[ e\colon * \to \Omega X, \qquad m\colon \Omega X \times \Omega X \to \Omega X, \qquadtext { and } i\colon \Omega X \to \Omega X \] that satisfy the usual group relations up to homotopy.

(2)

Show that for objects \(X\) and \(Y\) in \(C\), the set \([X,\Omega Y]\) is canonically a group. Show that \([X,\Omega ^2Y]\) is an abelian group.

(3)

Deduce that when \(C\) is stable, the Hom-set \([X,Y]\) in \(\Ho (C)\) is an abelian group for all \(X\) and \(Y\), and that composition in \(C\) defines group homomorphisms.

(4)

Show that the resulting abelian group structure on \([X,Y]\) agrees with the one defined in Chapterexercise 4.2.

Remark 4.4.38. Applying the previous exercise to \(C\catop \), we deduce that for every \(X\) the suspension \(\Sigma X\) is a cogroup object in \(\Ho (C)\), and hence that \([\Sigma X, Y]\) admits an abelian group structure for all \(Y\).

Remark 4.4.39. In Chapter 5, we will upgrade the group structure of \(\Omega X\) in \(\Ho (C)\) to a group structure in \(C\).

Exercise 4.4. Let \(X\) be a pointed anima. Consider the prespectrum \(\Sigma ^{\infty , \mathrm {pre}}(X) := (\Sigma ^nX, \sigma _n)\), where the structure maps are given by the unit maps \[ \Sigma ^nX \to \Omega \Sigma (\Sigma ^n X) = \Omega \Sigma ^{n+1}X \] of the adjunction \(\Sigma \dashv \Omega \). Show that the spectrum associated to the prespectrum \(\Sigma ^{\infty , \mathrm {pre}}(X)\) is \(\Sigma ^{\infty }(X)\).

Hint: show that all the maps \[ \Sigma ^{\infty - n}(\Sigma ^nX) \to \Sigma ^{\infty - (n+1)}(\Sigma ^{n+1}X) \] appearing in the definition of the associated spectrum are isomorphisms.

Exercise 4.5. Let \(X\) be a finite cell complex (Definition 2.3.1). Show that \(\Pi _{\infty }(X)\) is a finite anima.

Exercise 4.6. Show that for every spectrum \(X\), the functor \(\hom (-,X)\colon \Sp \catop \to \Sp \) admits a left adjoint, which is given by \(\hom (-,X)\catop \colon \Sp \to \Sp \catop \).

Exercise 4.7 (Shifts of hom spectra). Let \(C\) be a stable \(\infty \)-category and let \(X,Y\in C\). Prove that \[ \hom _C(X[k],Y)\cong \hom _C(X,Y)[-k] \qquad \text {and}\qquad \hom _C(X,Y[k])\cong \hom _C(X,Y)[k] \] Conclude that \(\hom (\S [m],\S [n])\cong \S [n-m]\) for all \(m,n\in \Z \).

Exercise 4.8 (Represented theories of spheres and Moore spectra). For \(m\geq 2\), let \(\S /m\) be the cofiber of multiplication by \(m\) on the sphere spectrum. Let \(E\) be a spectrum.

(1)

Prove that \(\widetilde E_k(S^n)\cong \pi _{k-n}(E)\) for every \(n\geq 0\).

(2)

Deduce natural short exact sequences \[ 0\to \pi _k(E)/m\to \pi _k(E\otimes \S /m)\to \pi _{k-1}(E)[m]\to 0. \]

(3)

Deduce natural short exact sequences \[ 0\to \pi _{k+1}(E)/m\to \pi _k\hom (\S /m,E)\to \pi _k(E)[m]\to 0. \]

(4)

Express the reduced \(E\)-homology of the Moore anima from Chapterexercise 3.3 in terms of the groups in part (2).