nLab ordinary homology

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

Contents

Idea

Ordinary homology is generalized homology with respect to an Eilenberg-MacLane spectrum $HA$, and often understood over $Hℤ$.

Equivalently this is computed by singular homology with coefficients in $A$.

Properties

Relation to homotopy groups

Ordinary homology with coefficients in $ℤ$ of a topological space $X$ serves to approximate the homotopy groups of $X$.

See for instance at Hurewicz theorem

Description in terms of higher linear algebra

We discuss (twisted) ordinary homology and ordinary cohomology in terms of sections of (∞,1)-module bundles over the Eilenberg-MacLane spectrum.

Let $k$ be a commutative ring. Write

$Hk\in {\mathrm{CRing}}_{\infty }$H k \in CRing_\infty

for the Eilenberg-MacLane spectrum of $k$, canonically regarded as an E-∞ ring. Write

$\left(Hk\right)\mathrm{Mod}\in \left(\infty ,1\right)\mathrm{Cat}$(H k) Mod \in (\infty,1)Cat

for the (∞,1)-category of (∞,1)-modules over $Hk$.

Proposition

There is an equivalence of (∞,1)-categories

$\left(Hk\right)\mathrm{Mod}\simeq {L}_{\mathrm{qi}}{\mathrm{Ch}}_{•}\left(k\mathrm{Mod}\right)$(H k)Mod \simeq L_{qi} Ch_\bullet(k Mod)

between the (∞,1)-category of (∞,1)-modules over the Eilenberg-MacLane spectrum $Hk$ and the simplicial localization of the category of unbounded chain complexes of ordinary (1-categorical) $k$-modules.

This is the statement of the stable Dold-Kan correspondence, see at (∞,1)-category of (∞,1)-modules – Properties – Stable Dold-Kan correspondence.

Definition

Let $X$ be a topological space and write $\Pi \left(X\right)\in$ ∞Grpd for its underlying homotopy type (its fundamental ∞-groupoid). Then we say that an (∞,1)-functor

$\Pi \left(X\right)\to \left(Hk\right)\mathrm{Mod}$\Pi(X) \to (H k) Mod

is (the higher parallel transport) a flat (∞,1)-module bundle over $X$, or a local system of $Hk$-(∞,1)-modules over $X$.

Proposition

(Riemann-Hilbert correspondence)

If $X$ is an oriented closed manifold, then there is an equivalence of (∞,1)-categories

$\left[\Pi \left(X\right),\left(Hk\right)\mathrm{Mod}\right]\simeq \left(TX\right)\mathrm{Mod}$[\Pi(X), (H k)Mod] \simeq (T X)Mod

between flat (∞,1)-module bundles/local systems and L-∞ algebroid representations of the tangent Lie algebroid of $X$. From right to left the equivalece is established by sending an L-∞ algebroid representation given (as discussed there) by a flat $ℤ$-graded connection on bundles of chain complexes (via prop. 1), to its higher holonomy defined in terms of iterated integrals.

This is the main theorem in (Block-Smith 09).

Definition

Write

$\Gamma \phantom{\rule{thickmathspace}{0ex}}≔\phantom{\rule{thickmathspace}{0ex}}\underset{\to }{\mathrm{lim}}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\left[\Pi \left(X\right),\left(Hk\right)\mathrm{Mod}\right]\to \left(Hk\right)\mathrm{Mod}$\Gamma \;\coloneqq\; \underset{\to}{\lim} \;\colon\; [\Pi(X), (H k)Mod] \to (H k) Mod

for the (∞,1)-colimit functor.

Remark

We may think of $\Gamma$ equivalently as

Definition

Write

${𝕀}_{X}^{Hk}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\Pi \left(X\right)\to \left(Kk\right)\mathrm{Mod}$\mathbb{I}_X^{H k} \;\colon\; \Pi(X) \to (K k)Mod

for the flat (∞,1)-module bundle which is constant on the chain complex concentrated on $k$ in degree 0, the tensor unit in $\left[\Pi \left(X\right),\left(Hk\right)\mathrm{Mod}\right]\simeq \left[\Pi \left(X\right),{L}_{\mathrm{qi}}{\mathrm{Ch}}_{•}\left(k\mathrm{Mod}\right)\right]$.

Proposition

For $X$ a topological space, we have a natural equivalence (with the identification of prop. 1 understood) of the form

$\Gamma \left({𝕀}_{X}^{Hk}\right)\simeq {C}_{•}\left(X,k\right)\phantom{\rule{thinmathspace}{0ex}},$\Gamma(\mathbb{I}_X^{H k}) \simeq C_\bullet(X,k) \,,

between the $\left(Hk\right)$-(∞,1)-module of sections of the trivial $\left(Hk\right)$-(∞,1)-module bundle ${𝕀}_{X}^{Hk}$ and the singular chain complex of $X$ for ordinary homology with coefficients in $k$.

Proof

This is a classical basic (maybe folklore) statement. Here is one way to see it in full detail.

First notice that the (∞,1)-colimit of functors out of ∞-groupoids and constant on the tensor unit in $\left(Hk\right)\mathrm{Mod}$ is by definition the (∞,1)-tensoring operation of $\left(Hk\right)\mathrm{Mod}$ over ∞Grpd. Now if we find a presentation of $\left(Hk\right)\mathrm{Mod}$ by a simplicial model category the by the dicussion at (∞,1)-colomit – Tensoring and cotensoring – Models this (∞,1)-tensoring is given by the left derived functor of the sSet-tensoring in that simplicial model category.

To obtain this, use prop. 1 and then the discussion at model structure on chain complexes in the section Projective model structure on unbounded chain complexes which says that there is a simplicial model category structure on the category of simplicial objects in the category of unbounded chain complexes which models ${L}_{\mathrm{qi}}{\mathrm{Ch}}_{•}\left(k\mathrm{Mod}\right)$, and whose weak equivalences are those morphisms that produce quasi-isomorphism under the total chain complex functor.

In summary it follows that with any simplicial set $\left(\Pi \left(X\right){\right)}_{•}\mathrm{in}{L}_{\mathrm{whe}}\mathrm{sSet}$ representing $\Pi \left(X\right)$ (under the homotopy hypothesis-theorem) we have

$\underset{\to }{\mathrm{lim}}\left({𝕀}_{X}^{Hk}\right)\simeq {\int }_{\left[n\right]\in \Delta }\left(\Pi \left(X\right){\right)}_{n}\cdot 𝕀\phantom{\rule{thinmathspace}{0ex}},$\underset{\to}{\lim} (\mathbb{I}_X^{H k}) \simeq \int_{[n] \in \Delta} (\Pi(X))_n \cdot \mathbb{I} \,,

where on the right we have the coend over the simplex category of the tensoring (of simplicial sets with simplicial objects in the category of unbounded chain complexes) of the standard cosimplicial simplex with the simplicial diagram constant on the tensor unit chain complex.

The result on the right is manifestly, by the very definition of singular homology, under the ordinary Dold-Kan correspondence the chain complex of singular simplices:

${\int }_{\left[n\right]\in \Delta }\left(\Pi \left(X\right){\right)}_{n}\cdot 𝕀\simeq \Pi \left(X\right)\otimes k\simeq {C}_{•}\left(X,k\right)\phantom{\rule{thinmathspace}{0ex}}.$\int_{[n] \in \Delta} (\Pi(X))_n \cdot \mathbb{I} \simeq \Pi(X) \otimes k \simeq C_\bullet(X,k) \,.
Proposition

If $X\in {L}_{\mathrm{whe}}\mathrm{Top}$ is a Poincaré duality space of dimension $n$, then $\Gamma \left({𝕀}_{X}^{HA}\right)\in \left(HA\right)\mathrm{Mod}$ is a dualizable object which is almost self-dual except for a degree twist of (itself) degree 0:

${\left(\Gamma \left({𝕀}_{X}^{HA}\right)\right)}^{\vee }\simeq {\Sigma }^{-n}\left(\Gamma \left({𝕀}_{X}^{HA}\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\left( \Gamma\left(\mathbb{I}_X^{H A}\right) \right)^\vee \simeq \Sigma^{-n} \left( \Gamma\left(\mathbb{I}_X^{H A}\right) \right) \,.
Proof

Under the identifications of prop. 1 and prop. 3 this is the theorem discussed at Poincaré duality in the section Poincaré duality – Refinement to homotopy theory.

The Riemann-Hilbert correspondence/de Rham theorem for $HA$-modules is established in