Disclaimer: I don't feel qualified to ask this question and yet it's been troubling me for some time now and I lost my patience and decided to ask to get some kind of answer. If there are any stupid mistakes please treat them as such and try to focus on the main issue raised if at all possible.
As the title suggests I'm struggling with the meaning of "Homology". In particular how are "Homology" and "Cohomology" related. By the end of my question I hope it will be clear what I mean. Let me start with some of the possible interpretations I'm (somewhat) familiar with, and after that let me say what troubles me. (All categories and functors are $\infty$ unless stated otherwise)
- Cohomology $\sim \operatorname{Hom}$ — Homology $\sim \otimes$
To make this precise consider the suspension $\infty$-functor sending spaces to their suspension spectra $\Sigma^{\infty}_+ :\mathrm{Spaces} \to \mathrm{Sp}$. The category of spectra is a symmteric monoidal $\infty$-category so for every space $X$ and spectrum $E$ one can
define the $E$-homology of $X$ as the homotopy groups of the smash
product $E_*X\mathrel{:=}\pi_*(\Sigma^{\infty}_+X \otimes_{\mathbb{S}} E)$.
The $E$-cohomology of $X$ in this picture is the homotopy groups of
the mapping spectrum $E^*X\mathrel{:=}\pi_*(\operatorname{Map}(\Sigma^{\infty}_+X,E))$.
- Homology $\sim$ Abelianization
To make this precise one can consider the tangent category to $\mathrm{Spaces}$ which is the fiberwise stabilization of the codomain fibration $\mathrm{Spaces}$. The fiber over a space $X$ will be the category spectra parametrized by $X$. Then one can define the Homology of $X$ as the image of the identity map $X \to X$ under the stabilization procedure. This is the "absolute cotangent complex" $L_X$. One has a kind of shriek pushforward for these parametrized spectra which for the case $X \to \mathrm{pt}$ sends $L_X$ to $\Sigma^{\infty}_+X$ and one recovers some of the above from this viewpoint (I'm not so sure about this statement suddenly, is this true?). In a sense this is the relative setting for the above.
- Cohomology $\sim \mathrm{limits}$ – Homology $\sim \mathrm{colimits}$
To make this precise start with a local system over a space $X$. Let's take as a definition for a local system a functor from $X$ considered as an infinity groupoid to some category of coefficients (say spectra). Take this local system $L:X \to \mathrm{Sp}$ and define $L$-cohomology of X to be $\operatorname{Lim} L$ (this coincides with the sheaf cohomology definition) and $L$-homology to be $\operatorname{Colim} L$ (giving the same answer as 1 for the case of a constant functor $L=E$).
- Homology $\sim$ dual to Cohomology
This is the most cheeky definition. There are many flavors of this I believe the basic archetype being the Poincaré duality for oriented manifolds $H^i_{\mathrm c}(M) \cong H_{n-i}(M)$. The main idea is to define homology in such a way that one gets "Poincaré duality". For example in Verdier duality for locally compact (sufficiently nice) spaces one can define homology with coefficients in a sheaf $F$ as the compactly supported cohomology with coefficients in the Verdier dual of $F$. For example on a manifold if $F= \mathbb{Z}$ is the constant sheaf then the Verdier dual will be $\operatorname{OR}_M$ the orientation sheaf (perhaps shifted depends on one's conventions). The point is that this definition is concocted so that one always has a duality between homology and cohomology. This can be done in any cohomology theory which has good duality properties (i.e. six functors).
Why am I not satisfied?
Here are my concerns. Some of the interpretations above answer some of the concerns but none of them answer all of the concerns in a satisfactory way:
- Lack of convenient relative framework: For sheaf cohomology one has a very convenient framework for working in a relative situation (push/pull) in any context no matter how general. All one needs is a site and one immediately can ask questions about how cohomology behaves in this site, what kind of properties does it satisfy? Does it have 6 functor formalism? If not maybe at least 5 or 4? Does it have any interesting dualities? etc.… For Homology one seems to run into several persistent problems when trying to translate the above interpretations into a relative general setting like this.
- Using duality as a crutch: As much as I like dualities sometimes I feel like we're being a bit unfair to "Homology" treating it like a deformed creature which only has a right to exist as a dual to cohmology when in fact homology is the older brother of the two!
- Asymmetry between co/homology: In cohomology one has sheaves, sections, resolutions etc.… What do we have in homology? I'm kind of wishing that all the homology business is part of a bigger story Cosheaf Homology — Sheaf Cohomology. Unfortunately I have no idea what the words in the left hand side mean or even what they should mean. I just wish there was some way to put homology and cohomology on an equal footing.
- Only locally constant data: This is related to the above point. Why is there no "Constructible Homology" or "Coherent Homology"? Why doesn't Homology deserve these variants?
I hope by now I've made it clear what's my "problem" with my current understanding of Homology. As I said I don't feel like I'm qualified to ask this question so if anyone has any suggestion for an edit or a revision please don't even ask permission just edit away!
Best Answer
Let's take coefficients in a field $k$, for simplicity.
On 2): the singular cohomology of a topological space $X$ is the dual of its singular homology, almost by definition. But if $X$ is a space for which singular cohomology is not the same as sheaf cohomology, then the sheaf cohomology of $X$ need not have a predual. For example, if $X$ is the Cantor set, then the sheaf cohomology of $X$ with coefficients in the constant sheaf $\underline{k}$ is the vector space of locally constant functions from $X$ into $k$. This is a vector space of countable dimension over $k$, so it cannot arise as the dual of anything.
On 1) and 4): part of the point of the six-functor formalism is that it incorporates things like homology automatically. For nice spaces $X$, singular cohomology = sheaf cohomology with coefficients in the constant sheaf, and singular homology = compactly supported sheaf cohomology with coefficients in the dualizing sheaf. Or, in six-functor notation,
Cohomology of $X$ = $f_* f^* k$ and homology of $X$ = $f_! f^! k$ (here $f$ is the projection map from $X$ to a point, and all functors are derived). These constructions are related as follows:
a) If the topological space $X$ is locally nice (so that the constant sheaf satisfies Verdier biduality), then cohomology $f_* f^* k$ is the dual of homology $f_! f^! k$. This is satisfied for many spaces of interest (for example, finite simplicial complexes, underlying topological spaces of complex algebraic varieties, ...)
b) If the topological space $X$ is compact, then homology $f_! f^! k$ is the dual of cohomology $f_* f^* k$. This applies even when $X$ is locally very badly behaved, like the Cantor set.
If $X$ is both compact and locally nice, then both of these arguments apply, and the homology and cohomology of $X$ are forced to be finite-dimensional.