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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)

  1. Cohomology $\sim Hom$ - Homology $\sim \otimes$

To make this precise consider the suspension $\infty-$functor sending spaces to their suspension spectra $\Sigma^{\infty}_+ :Spaces \to Sp$. 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 $E_*X:=\pi_*(\Sigma^{\infty}_+X \otimes_{\mathbb{S}} E)$. The $E$-cohomology of $X$ in this picture are the homotopy groups of the mapping spectrum $E^*X:=\pi_*(Map(\Sigma^{\infty}_+X,E))$.

  1. Homology $\sim$ Abelianization

To make this precise one can consider the tangent category to $Spaces$ which is the fiberwise stabilization of the codomain fibration $Spaces$. 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 $X \to pt$ 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.

  1. Cohomology $\sim limits$ - Homology $\sim colimits$

To make this precise start with a local system over a space $X$. Lets 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 $L:X \to Sp$ and define $L$-cohomology of X to be $LimL$ (this coincides with the sheaf cohomology definition) and $L$-homology to be $Colim L$ (giving the same answer as 1 for the case of a constant functor $L=E$).

  1. Homology $\sim$ dual to Cohomology

This is the most cheeky definition. There are many flavors of this I believe the basic archetype being the poincare duality for oriented manifolds $H^i_c(M) \cong H_{n-i}(M)$. The main idea is to define homology in such a way that one gets "poincare 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 $OR_M$ the orientation sheaf (perhaps shifted depends on ones conventions). The point is that this definition is concoted 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:

  1. 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 persisting problems when trying to translate the above inetrpretations into a relative general setting like this.
  2. 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!
  3. 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.
  4. 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 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!

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    $\begingroup$ I thought there was a definition of cosheaf homology where you just reverse the arrows in the definition of sheaves. Google finds multiple papers on this topic. $\endgroup$ – Will Sawin Jul 23 '17 at 11:58
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    $\begingroup$ +1: "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!" $\endgroup$ – Jason Starr Jul 23 '17 at 11:59
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    $\begingroup$ @WillSawin I tried reading several times about this. But I never read anything bold enough to make me think it was worthwhile to read about. Maybe I need some expert saying something like: "Cosheaf Homology is exactly the answer you're looking for, you can read about this in the following papers..." $\endgroup$ – Saal Hardali Jul 23 '17 at 12:02
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    $\begingroup$ Small comment: 1 is a special case of 3. The lim resp. colimit of the constant local system is hom resp. tensor. $\endgroup$ – Qiaochu Yuan Jul 23 '17 at 18:36
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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.

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    $\begingroup$ @Saal: In response to the question "Why isn't there a six-functor formalism for homology?", I think what I'm saying is "There is, and it is identical to the usual six-functor formalism": the latter is no more about "cohomology" than it is about "homology". If your last question is interpreted as "Can one define some version of homology, in terms of sheaf theory, for spaces that are not locally compact?", then I don't know. But that's venturing pretty far away from the usual context in which one talks about six functors, constructibility, Verdier duality, etcetera. $\endgroup$ – Jacob Lurie Jul 23 '17 at 14:32
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    $\begingroup$ Aha! Thanks a lot! I like this answer. This is a certain reassurance that anything I should really care about (relating to co/homology) in the near future can be understood in terms of $f^!,f_!,f^*,f_*,$. $\endgroup$ – Saal Hardali Jul 23 '17 at 14:46
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    $\begingroup$ Concerning definition of homology via cosheaves one may consult Lawvere distributions - his idea was to view covariant functors on the site as analogs of distributions on in the same context as contravariant ones are analogs of functions. The analogy can be pushed surprisingly far: for example, there is a nice analog of the Radon-Nikodym derivative, and it relates to things like enrichment of comodules in modules for a Hopf algebra or of right $\mathscr D$-modules in left ones. There also is a nice duality between sheaf/cosheaf conditions: $\endgroup$ – მამუკა ჯიბლაძე Jul 24 '17 at 3:31
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    $\begingroup$ While the sheaf condition on a contravariant $\mathscr F$ gives an étale space $f:E_{\mathscr F}\to X$ that is a local homeomorphism, i. e. the canonical map$$\varinjlim_{U\ni x}\Gamma_U(f)\to f^{-1}(x)$$is an isomorphism, the cosheaf condition on a covariant $\mathscr G$ gives an étale space $g:E^{\mathscr G}\to X$ that is a $\textit{complete spread}$, i. e. the canonical map$$g^{-1}(x)\to\varprojlim_{U\ni x}\pi_0(g^{-1}U)$$is an isomorphism. Note that while $\Gamma$ is right adjoint to the inverse image, $\pi_0$ is the left adjoint, but one needs local connectedness for that. $\endgroup$ – მამუკა ჯიბლაძე Jul 24 '17 at 3:41
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    $\begingroup$ does anyone know where all the facts mentioned in this answer are spelled out/proved in the literature? $\endgroup$ – Yosemite Sam Jul 24 '17 at 17:46
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I generally think about the relationship differently than Jacob, probably because I'm coming from an algebraic topology background rather than an algebraic geometry one. I would say that if $\mathcal{E}$ is any $(\infty,1)$-topos, with $f:\mathcal{E}\to \mathcal{S}$ its unique geometric morphism to $\infty$-groupoids (homotopy spaces), then $f_*$ is cohomology of $\mathcal{E}$ with coefficients in some ($\infty$-)sheaf (of spectra, say), and so $f_* f^*$ is cohomology with coefficients in a constant sheaf. The homology of $\mathcal{E}$ with coefficients in some sheaf would use instead $f_!$, the left adjoint to $f^*$: the difference is that such a left adjoint doesn't always exist, only when $\mathcal{E}$ is locally contractible.

If $X$ is a topological space, we can make it into an $(\infty,1)$-topos in multiple ways. One is the slice $\mathcal{S}/X$, where $X$ is regarded as its homotopy type: this is always locally contractible, and in this way we get ordinary algebraic-topological homology and cohomology, as well as homology and cohomology with local coefficients in the classical sense (i.e. locally constant coefficients). Another is $\mathrm{Sh}(X)$, consisting of $\infty$-sheaves on the site of opens in $X$; this is not locally contractible unless $X$ itself is. Thus we can define cohomology with coefficients in an arbitrary sheaf on an arbitrary space, whereas for homology with such coefficients we need the space to be locally contractible --- or else to define the homology as only a "pro-object".

I am not an expert on six-functor formalism, but my understanding is that it includes both algebro-topological homology and algebro-geometric compactly supported-cohomology: the former arises in the Wirthmuller context $f^* = f^!$, while the latter is a generalization of the Grothendieck context $f_! = f_*$ (which is the case when $f$ is proper, i.e. $\mathcal{E}$ is compact). In topos theory there is a fundamental duality between local-connectedness conditions and compactness conditions (see for instance chapter C3 of Sketches of an Elephant), and the two perspectives on homology come from focusing on one or the other of these worlds.

Edit in response to comment: the duality between local connectedness and properness is indeed not obvious from the usual definitions. One way to see it fairly clearly is in terms of their characterizations using Beck-Chevalley conditions. A geometric morphism is "locally $n$-connected" iff every pullback of it (in the $(\infty,2)$-category of $(\infty,1)$-toposes) satisfies the left Beck-Chevalley condition for $n$-truncated objects, and it is "$n$-proper" (or "proper of height $n$") iff every pullback of it satisfies the right Beck-Chevalley condition for $n$-truncated objects. I don't know whether this is in the literature for general $n$; the locally (-1)-connected (a.k.a. open) and locally 0-connected (a.k.a. locally connected) cases and the (-1)-proper (a.k.a. proper) and 0-proper (a.k.a. tidy) cases, for 1-toposes, are in chapter C3 of Sketches of an Elephant, and the $\infty$-proper case is in section 7.3 of Higher Topos Theory.

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