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edited Mar 8, 2021 at 16:32

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As the title suggests i'mI'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 $\sim Hom$ -$\sim \operatorname{Hom}$ — Homology $\sim \otimes$

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

To make this precise one can consider the tangent category to $Spaces$$\mathrm{Spaces}$ which is the fiberwise stabilization of the codomain fibration $Spaces$$\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 pt$$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 limits$$\sim \mathrm{limits}$ - Homology $\sim colimits$$\sim \mathrm{colimits}$

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

This is the most cheeky definition. There are many flavors of this I believe the basic archetype being the poincarePoincaré duality for oriented manifolds $H^i_c(M) \cong H_{n-i}(M)$$H^i_{\mathrm c}(M) \cong H_{n-i}(M)$. The main idea is to define homology in such a way that one gets "poincare"Poincaré duality". For example in verdierVerdier duality for locally compact (sufficiently nice) spaces one can define homology with coefficients in a sheaf $F$ as the compactly supported cohomology with coefficients in the verdierVerdier dual of $F$. For example on a manifold if $F= \mathbb{Z}$ is the constant sheaf then the verdierVerdier dual will be $OR_M$$\operatorname{OR}_M$ the orientation sheaf (perhaps shifted depends on onesone's conventions). The point is that this definition is concotedconcocted 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).

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 persistingpersistent problems when trying to translate the above inetrpretationsinterpretations 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 II'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!

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 $\sim Hom$ - 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 $E^*X:=\pi_*(Map(\Sigma^{\infty}_+X,E))$.

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.

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 $L=E$).

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).

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.
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 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!

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 $E^*X\mathrel{:=}\pi_*(\operatorname{Map}(\Sigma^{\infty}_+X,E))$.

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 $L=E$).

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).

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!