We now construct the ordinary homology and cohomology theories that were used axiomatically in Chapter 3. The construction is intrinsic to animae: instead of choosing a topological model and taking singular chains, we use the universal property of \(\An \) to produce chain and cochain complexes directly in the derived category \(\D (\Z )\). The bridge from chain complexes to spectra is provided by Eilenberg–MacLane spectra.
6.2.1 Eilenberg-MacLane spaces and spectra
Throughout this section we fix the abelian category \(\Aa = \Ab \) of abelian groups and work in its derived \(\infty \)-category \(\D (\Z )\). By forming mapping spectra out of the complex \(\Z [0]\), we may associate a spectrum to every complex:
Definition 6.2.1 (Eilenberg-MacLane spectrum). Given a complex \(A \in \D (\Z )\), we define its Eilenberg-MacLane spectrum as the mapping spectrum \[ HA \quad := \quad \hom _{\D (\Z )}(\Z [0],A) \qin \Sp \] Similarly, for \(n \in \Z \) we define its \(n\)-th Eilenberg MacLane anima as \[ K(A,n) \quad := \quad \Hom _{\D (\Z )}(\Z [-n],A) \qin \An . \] These define functors \(H \colon \D (\Z ) \to \Sp \) and \(K(-,n)\colon \D (\Z ) \to \An \).
In particular, every abelian group \(A \in \Ab \) defines an Eilenberg-MacLane spectrum \(HA\) and an \(n\)-th Eilenberg MacLane anima \(K(A,n)\) by regarding it as a chain complex concentrated in degree \(0\).
Eilenberg-MacLane spectra play an important role in stable homotopy theory. In the next subsection, we will use them to classify ordinary (co)homology theories with coefficients in an abelian group.
Exercise 6.2.2. Show that for an associative ring \(R\) the forgetful functor \(\RMod _R(\Ab ) \to \Ab \) induces a functor \(\D (R) \to \D (\Z )\). Show that the composite \(\D (R) \to \D (\Z ) \xrightarrow {H} \Sp \) may be identified with \(\hom _{\D (R)}(R[0],-)\).
The main property of the Eilenberg-MacLane animae and spectra is that their homotopy groups are precisely the homology groups of \(A\):
Proposition 6.2.3. For a complex \(A \in \D (\Z )\) and integers \(n \in \Z \), \(k \geq 0\), there are natural isomorphisms of abelian groups \[ \pi _k(K(A,n)) \quad \cong \quad H_{k-n}(A) \qquadtext { and } \pi _n(HA) \quad \cong \quad H_n(A). \]
Remark 6.2.4. If \(A\) is an abelian group regarded as a complex concentrated in degree \(0\), then Proposition 6.2.3 shows that \(HA\) is connective and that \(\Omega ^{\infty }HA\) is the discrete commutative group \(A\). Under the recognition equivalence of Theorem 5.4.6, it therefore corresponds to \(A\in \CGrp (\Set )\), giving a natural isomorphism \[ HA \iso \bB ^{\infty }A. \] Thus this construction agrees with the Eilenberg–MacLane spectrum of Definition 5.4.9.
This has the following useful consequence:
Proof. Since it is exact, it remains to show it preserves arbitrary coproducts: the map \(\bigoplus _{i \in I} H(A_i) \to H(\bigoplus _{i \in I} A_i)\) is an isomorphism of spectra. This may be tested at the level of homotopy groups. Combining with Proposition 6.2.3, this then follows from the fact that both \(\pi _n(-)\colon \Sp \to \Ab \) and \(H_n(-)\colon \D (\Z ) \to \Ab \) preserve coproducts. □
To prove the proposition, we introduce an auxiliary \(\infty \)-category in which the complexes \(\Z [n]\) corepresent homology directly. We will then transport this corepresentability across the further localization to \(\D (\Z )\).
Definition 6.2.6. Let \(\Aa \) be an abelian category. We define the \(\infty \)-category \(\Kk (\Aa )\) as the localization \[ \Kk (\Aa ) \quad := \quad \Ch (\Aa )[\{\text {chain homotopy equivalences}^{-1}\}]. \] There are localization functors \(\Ch (\Aa ) \to \Kk (\Aa ) \to \D (\Aa )\).
A chain map \(i\colon C_{\bullet } \to D_{\bullet }\) is called a degreewise split monomorphism if each component \(i_n\colon C_n \to D_n\) is a split monomorphism in \(\Aa \). The chosen retractions are not required to assemble into a chain map. Degreewise split epimorphisms are defined dually.
Proposition 6.2.7. For every abelian category \(\Aa \), the \(\infty \)-category \(\Kk (\Aa )\) is stable. Under the localization functor \(\Ch (\Aa ) \to \Kk (\Aa )\), the suspension of a chain complex \(C_{\bullet }\) is naturally isomorphic to the shifted complex \(C[1]_{\bullet }\).
Proof. The proof is the split analogue of the proofs of Proposition 6.1.27, Theorem 6.1.30. Equip \(\Ch (\Aa )\) with chain homotopy equivalences as weak equivalences and degreewise split monomorphisms as cofibrations. The mapping-cylinder factorization shows that every map factors as a cofibration followed by a weak equivalence. The pushout axiom follows from the criterion that a degreewise split monomorphism is a chain homotopy equivalence precisely when its cokernel is contractible. The dual argument uses degreewise split epimorphisms, so the localization \(\Kk (\Aa )\) admits finite limits and colimits.
For a chain complex \(C_{\bullet }\), the degreewise split short exact sequence \[ 0 \to C_{\bullet } \to \Cyl (C_{\bullet }\to 0) \to C[1]_{\bullet } \to 0 \] is sent to a cofiber sequence in \(\Kk (\Aa )\). Its middle term is contractible, so it identifies suspension with the shift \([1]\). Thus suspension is an equivalence, with inverse induced by \([-1]\), and Theorem 4.2.2 implies that \(\Kk (\Aa )\) is stable. □
Lemma 6.2.8. The homotopy category of \(\Kk (\Aa )\) is the category whose objects are chain complexes and whose morphisms are chain maps modulo chain homotopy.
Proof. As in the proof of Corollary 2.3.7, this follows from the observation that a functor \(\Ch (\Aa ) \to C\) into a 1-category \(C\) inverts chain homotopy equivalences if and only if it identifies chain homotopic chain maps. □
Corollary 6.2.9. For a chain complex \(C_{\bullet } \in \Ch (\Ab )\) and \(n \in \Z \) there is a natural isomorphism \[ \pi _0 \Hom _{\Kk (\Ab )}(\Z [n],C_{\bullet }) \cong H_n(C). \] In particular, the functor \(\Hom _{\Kk (\Ab )}(\Z [n],-)\colon \Kk (\Ab ) \to \An \) corepresented by \(\Z [n]\) inverts quasi-isomorphisms.
Proof. By Lemma 6.2.8, the left-hand side is the set of chain maps \(\Z [n] \to C_{\bullet }\) up to chain homotopy. The first claim follows, since a chain map \(\Z [n] \to C_{\bullet }\) is nothing but an element of \(\ker (d_n\colon C_n \to C_{n-1})\) and two chain maps are chain homotopic if and only if their difference lies in the image of \(d_{n+1}\colon C_{n+1} \to C_n\).
The second claim now follows from the stability of \(\Kk (\Ab )\) established in Proposition 6.2.7: for \(k \geq 0\) we have \[ \pi _k\Hom _{\Kk (\Ab )}(\Z [n],A) \cong \pi _0\Hom _{\Kk (\Ab )}(\Z [n+k], A) \cong H_{n+k}(A). \qedhere \] □
Lemma 6.2.10. Let \(\gamma \colon C \to C[W^{-1}]\) be the localization functor inverting a collection of morphisms \(W\) in some \(\infty \)-category \(C\). Let \(X \in C\) be an object with the property that \(\Hom _C(X,-)\colon C \to \An \) inverts all morphisms in \(W\). Then for every \(Y \in C\) the map \[ \Hom _C(X,Y) \to \Hom _{C[W^{-1}]}(\gamma X, \gamma Y) \] is an isomorphism of animae.
Proof. The assumption on \(X\) guarantees that \(\Hom _C(X,-)\) descends to a functor \(C[W^{-1}] \to \An \). We need to show that this functor is corepresented by \(\gamma X \in C[W^{-1}]\). But this follows immediately from the full faithfulness of \(\gamma ^*\colon \Fun (C[W^{-1}],\An ) \hookrightarrow \Fun (C,\An )\): for any other functor \(F\colon C[W^{-1}] \to \An \) we have \[ \Nat _{C[W^{-1}] \to \An }(\Hom _C(X,-),F) \iso \Nat _{C \to \An }(\Hom _C(X,-),F\circ \gamma ) \iso F(\gamma (X)). \qedhere \] □
We are now ready for the proof of Proposition 6.2.3:
Proof of Proposition 6.2.3. Combining Corollary 6.2.9 and Lemma 6.2.10, we see that for every complex \(A \in \Kk (\Ab )\) and \(n \in \Z \), the canonical map \[ \Hom _{\Kk (\Ab )}(\Z [-n],A) \to \Hom _{\D (\Z )}(\Z [-n], A) = K(A,n) \] is an isomorphism of animae. In particular, we have \[ \pi _k(K(A,n)) \cong \pi _k\Hom _{\Kk (\Ab )}(\Z [-n],A) \cong \pi _0\Hom _{\Kk (\Ab )}(\Z [k-n], A) \cong H_{k-n}(A) \] and thus \[ \pi _n(HA) \cong \pi _0 H(A[-n]) \cong \pi _0 K(A[-n],0) \cong H_{0}(A[-n]) \cong H_n(A). \qedhere \] □
6.2.2 Ordinary homology and cohomology
Recall from Section 3.3 the axiomatization of ordinary (co)homology with coefficients in an abelian group \(A\). We can now describe these as the (co)homology theory associated to the Eilenberg-MacLane spectrum \(HA\) of \(A\). More generally, we obtain versions of ordinary (co)homology with coefficients in an arbitrary complex of abelian groups.
Definition 6.2.11 (Reduced ordinary (co)homology). Let \(A \in \D (\Z )\) be a complex of abelian groups and let \(X \in \An _*\) be a pointed anima. We define the reduced ordinary homology of \(X\) with coefficients in \(A\) and the reduced ordinary cohomology of \(X\) with coefficients in \(A\) as the (co)homology theories represented by the Eilenberg-MacLane spectrum \(HA\): \[ \widetilde {H}_n(X;A) \, := \, HA_n(X) \, = \, \pi _n(\Sigma ^{\infty }X \otimes HA) \] and \[ \widetilde {H}^n(X;A) \, := \, HA^n(X) \, = \, \pi _{-n}\hom (\Sigma ^{\infty }X,HA). \] We will mostly be interested in the case where \(A \in \Ab \) is an abelian group, regarded as a complex concentrated in degree \(0\).
The groups \(\widetilde {H}_*(X;A)\) and \(\widetilde {H}^*(X;A)\) may alternatively be computed as the homology groups of certain chain complexes associated to \(X\) and \(A\).
Construction 6.2.12 (Chain and cochain complexes). Let \(A \in \D (\Z )\) be a complex. Since \(\D (\Z )\) admits all small limits and colimits by Corollary 6.1.29, the universal property of \(\An \) from Corollary 1.8.10 implies that evaluation at the point induces equivalences \[ \ev _{\pt }\colon \Fun ^{\colim }(\An ,\D (\Z )) \iso \D (\Z ) \qquadtext {and} \ev _{\pt }\colon \Fun ^{\lim }(\An \catop ,\D (\Z )) \iso \D (\Z ). \] In particular, the object \(A \in \D (\Z )\) determines two functors \[ C_*(-;A)\colon \An \to \D (\Z ) \qquadtext {and} C^*(-;A) \colon \An \catop \to \D (\Z ), \] characterized uniquely by the properties that \(C_*(-;A)\) preserves colimits, \(C^*(-;A)\) preserves limits, and \(C_*(\pt ;A) \simeq A \simeq C^*(\pt ;A)\).
For a commutative ring \(R\), the same construction in \(\D (R)\) defines chains and cochains with coefficients in any \(A \in \D (R)\). Restriction of scalars along \(\Z \to R\) identifies their underlying complexes in \(\D (\Z )\) with the constructions above.
For a pointed anima \((X,x) \in \An _*\), we define the reduced chain complex and reduced cochain complex as \[ \widetilde {C}_*(X;A) := \cofib (A \simeq C_*(\pt ;A) \xrightarrow {x_*} C_*(X;A)) \] and \[ \widetilde {C}^*(X;A) := \fib (C^*(X;A) \xrightarrow {x^*} C^*(\pt ;A) \simeq A). \]
Definition 6.2.13 (Ordinary (co)homology). Given an anima \(X \in \An \), a chain complex \(A \in \D (\Z )\), and an integer \(n \in \Z \), we define the ordinary homology and ordinary cohomology of \(X\) with coefficients in \(A\) as \[ H_n(X;A) := H_n(C_{*}(X;A)) \qquadtext {and} H^n(X;A) := H_{-n}(C^{*}(X;A)). \] When \(A = \Z \) (regarded as a complex concentrated in degree \(0\)), we simply write \(H_n(X) := H_n(X;\Z )\) and \(H^n(X) := H^n(X;\Z )\).
The next result shows that the spectrum-theoretic and chain complex definitions of ordinary (co)homology agree.
Proposition 6.2.14. Let \(A \in \D (\Z )\) be a chain complex. There are natural equivalences of functors \[ H(C_*(-;A)) \,\simeq \, \Sigma ^{\infty }_+(-) \otimes HA \colon \An \to \Sp \] and \[ H(C^*(-;A)) \,\simeq \, \hom (\Sigma ^{\infty }_+(-), HA) \colon \An \catop \to \Sp . \] In particular, for any anima \(X\) there are natural isomorphisms \[ H_n(X;A) \cong \pi _n(\Sigma ^{\infty }_+ X \otimes HA) \qquadtext {and} H^n(X;A) \cong \pi _{-n}\hom (\Sigma ^{\infty }_+ X, HA). \]
Proof. Since the Eilenberg-MacLane functor \(H\colon \D (\Z ) \to \Sp \) preserves colimits by Lemma 6.2.5, it follows that \(H \circ C_*(-;A)\colon \An \to \Sp \) preserves colimits. On the other hand, the functor \(\Sigma ^{\infty }_+(-) \otimes HA\) also preserves colimits, since both \(\Sigma ^{\infty }_+\) and \(- \otimes HA\) do. Since both evaluate to \(HA\) on the point, it follows they are naturally equivalent. The equivalence \(H(C^*(-;A)) \simeq \hom (\Sigma ^{\infty }_+(-), HA)\) follows similarly, as both sides preserve limits.
The final statement follows by taking homotopy groups and using Proposition 6.2.3. □
Corollary 6.2.15. For a pointed anima \(X \in \An _*\) and a chain complex \(A \in \D (\Z )\), there are natural isomorphisms \[ \widetilde {H}_n(X;A) \cong H_n(\widetilde {C}_*(X;A)) \qquadtext {and} \widetilde {H}^n(X;A) \cong H_{-n}(\widetilde {C}^*(X;A)). \]
Proof. We have \[ \widetilde {H}_n(X;A) = \pi _n(\Sigma ^{\infty } X \otimes HA) \cong \pi _n(\cofib (\S \otimes HA \to \Sigma ^{\infty }_+ X \otimes HA)) \] since \(\Sigma ^{\infty } X = \cofib (\S \to \Sigma ^{\infty }_+ X)\) and \(- \otimes HA\) is exact. By Proposition 6.2.14, this is isomorphic to \[ \pi _n(H(\cofib (A \to C_*(X;A)))) = \pi _n(H(\widetilde {C}_*(X;A))) = H_n(\widetilde {C}_*(X;A)). \] The cohomological case is similar. □
Lemma 6.2.16. For any anima \(X\) there is an isomorphism \(H_0(X) \cong \bigoplus _{\pi _0(X)} \Z \).
Proof. Since both \(\Sigma ^{\infty }_+X\) and \(H\Z \) are connective spectra, so is \(\Sigma ^{\infty }_+X \otimes H\Z \). The functor \(H_0(-)\colon \An \to \Ab \) may thus be written as the composite \[ \An \xrightarrow {\Sigma ^{\infty }_+} \Sp _{\geq 0} \xrightarrow {- \otimes H\Z } \Sp _{\geq 0} \xrightarrow {\pi _0} \Ab , \] each of which preserves colimits. Here \(\pi _0\colon \Sp _{\geq 0} \to \Ab \) preserves colimits because it is left adjoint to the inclusion of abelian groups as Eilenberg–MacLane spectra. On the other hand, the assignment \(X \mapsto \bigoplus _{\pi _0(X)} \Z \) may be written as the composite of \(\pi _0\colon \An \to \Set \) with the free abelian group functor \(\Set \to \Ab \), both of which preserve colimits as well. By the universal property of \(\An \), it thus suffices to check the claim for \(X = \pt \), where both sides are \(\Z \). □
For an abelian group \(A\), one can show that the assignments \(X \mapsto C_*^{\mathrm {sing}}(X;A)\) and \(X \mapsto C^*_{\mathrm {sing}}(X;A)\) from Remark 3.3.2 are homotopy invariant, send disjoint unions to (co)products, satisfy the Mayer–Vietoris property, and carry skeletal colimits of CW-complexes to colimits and limits, respectively; see for example Hatcher (2002). By restricting these assignments to CW-complexes, it follows that they descend to functors \[ C_*^{\mathrm {sing}}(-;A)\colon \An \to \D (\Z ) \qquadtext {and} C^*_{\mathrm {sing}}(-;A)\colon \An \catop \to \D (\Z ) \] that preserve colimits and limits, respectively. Since \(C_*^{\mathrm {sing}}(\pt ;A) \simeq A[0] \simeq C^*_{\mathrm {sing}}(\pt ;A)\), the universal property of \(\An \) implies that these functors are naturally equivalent to \(C_*(-;A)\) and \(C^*(-;A)\) from Construction 6.2.12. Thus the intrinsic ordinary (co)homology constructed here agrees with singular (co)homology for animae coming from topological spaces.
Finally, for an abelian group \(A\) and the pointed anima \(S^0\), we have \[ \widetilde H_n(S^0;A) \cong \pi _n(HA) \cong H_n(A) \qquadtext {and} \widetilde H^n(S^0;A) \cong \pi _{-n}(HA) \cong H_{-n}(A). \] Both groups are therefore \(A\) when \(n=0\) and vanish otherwise. This verifies the dimension axiom and completes the construction of the ordinary theories postulated in Theorem 3.3.1.
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