Learning algebraic topology, homotopy always felt like a very intuitive and natural sort of invariant to attach to a space whereas for homology I don’t think I have anywhere as close of an intuitive handle or sense of naturality of this concept as I do for homotopy. So I tried to collect some frames / results for homology I’ve learned to see if it helps convince my intuition that this concept is indeed something natural in mathspace. I’d be very curious to know if there are any other frames or Deeper Answers to “Why homology?” I’m missing:
Measuring “holes” of a space
Singular homology: This is the first example I encountered, which will serve as intuition / motivation for the later abstract definitions.
Fixing some notations (feel free to skip this bullet point if you’re familiar with the notations):
Let’s fix some space X, and recall our goal associating to that space an algebraic object invariant under homeomorphism / homotopy equivalence.
First, a singularp-simplex is a map σ:Δp→X, intuitively representing a simplex living inside the space X. So there is a natural σ(i):Δp−1→X map which represents each of the i faces. Then, it is natural to consider the set {σ(i)}pi=0 as representing the “boundary” of the singular p-simplex σ.
To make this last idea more precise, we define singular p-chain, which is a free abelian group generated from all the singular p-simplicies of a space, denoted Δp(X). In short, its elements look like (finite) formal sums ∑σ:Δp→Xnσσ. A singular p-simplex σ is naturally an element of this group via 1σ∈Δp(X).
This construction, again, is motivated by the boundary idea earlier, since we now can define the boundary of a singular p-simplex σ as formal sum ∑pi=0σ(i)∈Δp−1(X).
In fact, the boundary of a singular p-simplex σ is actually ∑pi=0(−1)iσ(i)∈Δp−1(X).
Why? Intuition: if we draw these σ(i) of simple shapes like triangles (so σ:Δ2→X, hence σ(i):Δ1→X, which is identified with a (directed) edge), we will note that they are oriented kind of weird, contra our intuition that the “boundary” of a triangle ought to be these directed edges that are oriented consistently, clockwise or counterclockwise. The signs correct this.
So, we now generally define the boundary of a singular p-chain (not just a simplex) as ∂p:Δp(X)→Δp−1(X) where ∑σnσσ↦∑σnσ∂σ, i.e. the obvious extension of the earlier map as group homs of the free group.
Also, brute force application of the definitions show that ∂p∘∂p+1=0, which matches the intuition that a boundary of a boundary should be empty. Yet another motivation for the sign fix earlier!
Let’s collect all the objects so far: (Δ∗(X),∂∗). Then, abstractly, we associated to a topological space X, a collection of groups and maps inbetween of order 2 (i.e. ∂p∘∂p+1=0). We call this object a singular complex.
The singular chain groups Δ∗(X) are obviously invariant under homeomorphism! But it’s too large to be useful invariants.
So the natural thing to do is to take some quotients and make them smaller. Conveniently, note that ∂p∘∂p+1=0, so im∂p+1⊆ker∂p, and because they’re abelian, we can take the quotient.
This is the part where it’s like measuring holes—this video explains it nice.
So our singular complex (Δ∗(X),∂∗) induces a new collection of groups, H∗(X) where Hp(X):=ker∂p/im∂p+1. This is the p-th singular homology group ofX, also a homotopy invariant.
So singular homology has two objects: 1) singular chain complex (Δ∗(X),∂∗), and 2) singular homology groups H∗(X).
Homological algebra on topological spaces
We can abstract the structure of singular homology, and talk of “homology theory” in general as anything with the following data:
chain complex(C∗,∂∗) - a collection of some groups indexed by the integers and group homomorphisms between consecutive groups of order 2: ∂p∘∂p+1=0.
homology groupsH∗:=ker∂∗/im∂∗+1 (note that the definition of our abstract chain complex above suffices to make this well-defined)
(Note this doesn’t invoke anything about topology—though these chain complexes often arise from topological spaces, as in the singular homology example.)
Homology from this perspective is then basically a functor that assigns, to some object of interest, a “chain” of groups that are connected by maps ∂ such that they vanish in order 2 ∂p∘∂p+1=0, which implies im∂p+1⊆ker∂p. But not necessarily exactness, i.e. im∂p+1=ker∂p. Homology, then, measures the failure of exactness via taking quotients.
Studying properties of chain complexes and their homology groups on their own as algebraic objects (without caring about where they came from) is called homological algebra, and apparently it shows up in various places in mathematics.
So given this assumption of homological algebra’s utility, one could expect that it would be useful to do homological algebra on topological spaces by finding a way to construct chain complexes from spaces, and singular complexes for example does that.
(though from a historical perspective this reason for framing the utility of homology in topology feels like double-counting evidence; though might be a frame that convinces an expert who is already convinced of the utility of homology in other fields but doesn’t know algebraic topology?)
Elienberg-Steenrod axioms
Going back to “homology” for topological spaces, we can abstract them from singular homology alternatively via an axiomatic approach using the Eilenberg-Steenrod axioms, which are axioms that a Top to Grp should satisfy. Showing that singular homology satisfies these axioms is somewhat difficult (specifically, the Excision axiom), but once this is done, it’s easy to prove various things about singular homologies directly from the axioms.
Turns out, for nice topological spaces (CW complex), Eilenberg-Steenrod axioms fully characterize the homology of that space up to isomorphism!!! Singular homology is an example, but if you hand me some other chain complex and homology induced from that space satisfying the axioms, then their homology should match that of singular homology.
This is quite impressive! It really seems like “homology,” as a concept, isn’t “ad hoc” (as one might feel when first learning about singular homology), but rather something deep & universal about spaces, as homotopy is?
(going back to 1. Measuring holes, we can then add other examples of chain complexes and the resulting homology groups, of topological spaces, aside from singular homology—the standard ones are: cellular homology, simplicial homology. Why care about multiple homology theories of topological spaces that give you isomorphic homology groups (at least for nice spaces)? Because they have comparative advantages, eg singular: easy to prove things in, cellular: easy for humans to compute with, simplicial: easy for computers to compute with, etc)
Hurewicz theorem gives a homomorphism between the nth homotopy group πn(X) and the nth homology group Hn(X). In particular, it says that the 1st homology group is an abelianization of the fundamental group (!!!!)
But homotopy groups are always abelian for n≥2, so no hopes of this abelianization connection beyond n=1 (or not?)
Dold-Thom theorem: Hn(X)≅πnSP(X) for CW complex X
SP(X) is the infinite symmetric product space of X (in short, take finite products of X and mod by permutation—and given such collection, take a direct limit).
All the frames you are mentioning are good for intuition. I would say the deepest one is 4. and that everything falls into place cleanly once you formulate things in the language of infinity-category theory (at the price of a lot of technicalities to establish the “right” language). For example,
singular homology with coefficients in A can be characterised as the unique colimit-preserving infinity-functor from the infinity-category of spaces/homotopy types/infinity-grpoids/anima to the derived infinity-category of abelian groups which sends a one-point space (equivalently any contractible space) to A[0].
The derived infinity-category of abelian groups is itself in some sense the “(stable presentable) Z-linearization” of the infinity-category of spaces, although this is more tricky to state precisely and I won’t try to do this here.
Here are two more closely related results in the same circle of ideas. The first one gives a description (a kind of fusion of Dold-Thom and Eilenberg-Steenrod) of homology purely internal to homotopy theory, and the second explains how homological algebra falls out of infinity-category theory:
Consider functors E:S_* --> S_* from the infinity-category of spaces to itself which commutes with filtered colimits, carries pushout squares to pullback squares, sends the one-point space to itself, and the 0-sphere (aka two points!) to a discrete space. Then A=E(S^0) has a natural structure of abelian group, and E is an (infinity-categorical version of) the Dold-Thom functor and satisfies pi_n E(X)=H_n(X,A) (reduced homology). In particular, E(S^n) is an Eilenberg-Maclane space K(A,n).
The category of functors E:S_* --> S_* satisfying all the properties above except the one about E(S^0) being discrete is a model for the infinity-category Sp of spectra, i.e. the “stabilization” (in a precise categorical sense) of the infinity-category of spaces. From this perspective, the functors from the previous points are called the Eilenberg-Maclane spectra HA. Moreover, the infinity-category of spectra has a symmetric monoidal structure (the “smash product”), HR is naturally an algebra object for this structure whenever R is a ring, and it makes sense to talk about the infinity-category LMod(HR) of left HR-modules in Sp. Then LMod(HR) is equivalent (essentially by a stable version of the Dold-Kan correspondence) to the infinity derived category of left R-modules D(R). In other words, for homotopy theorists, (chain complexes,quasi-isomorphisms) are just a funny point-set model for HR-module spectra!
All of this is discussed in the first chapter of Lurie’s Higher Algebra, except the last point which is not completely spelled out because monoidal structures and modules are only introduced later on.
I should point out that this perspective is largely a reformulation of the results you already mentioned, and in themselves certainly do not bring new computational techniques for singular homology. However they show that 1) homological algebra comes out “structurally” from homotopy theory, which itself comes out “structurally” from infinity-category theory and 2) homological algebra (including in more sophisticated contexts than just abelian groups, e.g. dg-categories), homotopy theory, sheaf theory… can be combined inside of a common flexible categorical framework, which elegantly subsumes previous point-set level techniques like model categories.
In 3, there’s also the Brown representability theorem: Cohomology groups are just homotopy groups, with the sphere spectrum replaced with the Eilenberg-MacLane spectrum.
Have you looked at the historical development of homology? IME that’s generally a good way to build intuitions on a topic, as it tends to make concepts appear natural.
If you do read up on the history of homology, I’d be grateful if you’d tell me whether it helps. I’m curious if my advice was actually a good idea, or if it just sounds good.
Learning algebraic topology, homotopy always felt like a very intuitive and natural sort of invariant to attach to a space whereas for homology I don’t think I have anywhere as close of an intuitive handle or sense of naturality of this concept as I do for homotopy. So I tried to collect some frames / results for homology I’ve learned to see if it helps convince my intuition that this concept is indeed something natural in mathspace. I’d be very curious to know if there are any other frames or Deeper Answers to “Why homology?” I’m missing:
Measuring “holes” of a space
Singular homology: This is the first example I encountered, which will serve as intuition / motivation for the later abstract definitions.
Fixing some notations (feel free to skip this bullet point if you’re familiar with the notations):
Let’s fix some space X, and recall our goal associating to that space an algebraic object invariant under homeomorphism / homotopy equivalence.
First, a singular p-simplex is a map σ:Δp→X, intuitively representing a simplex living inside the space X. So there is a natural σ(i):Δp−1→X map which represents each of the i faces. Then, it is natural to consider the set {σ(i)}pi=0 as representing the “boundary” of the singular p-simplex σ.
To make this last idea more precise, we define singular p-chain, which is a free abelian group generated from all the singular p-simplicies of a space, denoted Δp(X). In short, its elements look like (finite) formal sums ∑σ:Δp→Xnσσ. A singular p-simplex σ is naturally an element of this group via 1σ∈Δp(X).
This construction, again, is motivated by the boundary idea earlier, since we now can define the boundary of a singular p-simplex σ as formal sum ∑pi=0σ(i)∈Δp−1(X).
In fact, the boundary of a singular p-simplex σ is actually ∑pi=0(−1)iσ(i)∈Δp−1(X).
Why? Intuition: if we draw these σ(i) of simple shapes like triangles (so σ:Δ2→X, hence σ(i):Δ1→X, which is identified with a (directed) edge), we will note that they are oriented kind of weird, contra our intuition that the “boundary” of a triangle ought to be these directed edges that are oriented consistently, clockwise or counterclockwise. The signs correct this.
So, we now generally define the boundary of a singular p-chain (not just a simplex) as ∂p:Δp(X)→Δp−1(X) where ∑σnσσ↦∑σnσ∂σ, i.e. the obvious extension of the earlier map as group homs of the free group.
Also, brute force application of the definitions show that ∂p∘∂p+1=0, which matches the intuition that a boundary of a boundary should be empty. Yet another motivation for the sign fix earlier!
Let’s collect all the objects so far: (Δ∗(X),∂∗). Then, abstractly, we associated to a topological space X, a collection of groups and maps inbetween of order 2 (i.e. ∂p∘∂p+1=0). We call this object a singular complex.
The singular chain groups Δ∗(X) are obviously invariant under homeomorphism! But it’s too large to be useful invariants.
So the natural thing to do is to take some quotients and make them smaller. Conveniently, note that ∂p∘∂p+1=0, so im∂p+1⊆ker∂p, and because they’re abelian, we can take the quotient.
This is the part where it’s like measuring holes—this video explains it nice.
So our singular complex (Δ∗(X),∂∗) induces a new collection of groups, H∗(X) where Hp(X):=ker∂p/im∂p+1. This is the p-th singular homology group of X, also a homotopy invariant.
So singular homology has two objects: 1) singular chain complex (Δ∗(X),∂∗), and 2) singular homology groups H∗(X).
Homological algebra on topological spaces
We can abstract the structure of singular homology, and talk of “homology theory” in general as anything with the following data:
chain complex (C∗,∂∗) - a collection of some groups indexed by the integers and group homomorphisms between consecutive groups of order 2: ∂p∘∂p+1=0.
homology groups H∗:=ker∂∗/im∂∗+1 (note that the definition of our abstract chain complex above suffices to make this well-defined)
(Note this doesn’t invoke anything about topology—though these chain complexes often arise from topological spaces, as in the singular homology example.)
Homology from this perspective is then basically a functor that assigns, to some object of interest, a “chain” of groups that are connected by maps ∂ such that they vanish in order 2 ∂p∘∂p+1=0, which implies im∂p+1⊆ker∂p. But not necessarily exactness, i.e. im∂p+1=ker∂p. Homology, then, measures the failure of exactness via taking quotients.
Studying properties of chain complexes and their homology groups on their own as algebraic objects (without caring about where they came from) is called homological algebra, and apparently it shows up in various places in mathematics.
So given this assumption of homological algebra’s utility, one could expect that it would be useful to do homological algebra on topological spaces by finding a way to construct chain complexes from spaces, and singular complexes for example does that.
(though from a historical perspective this reason for framing the utility of homology in topology feels like double-counting evidence; though might be a frame that convinces an expert who is already convinced of the utility of homology in other fields but doesn’t know algebraic topology?)
Elienberg-Steenrod axioms
Going back to “homology” for topological spaces, we can abstract them from singular homology alternatively via an axiomatic approach using the Eilenberg-Steenrod axioms, which are axioms that a Top to Grp should satisfy. Showing that singular homology satisfies these axioms is somewhat difficult (specifically, the Excision axiom), but once this is done, it’s easy to prove various things about singular homologies directly from the axioms.
Turns out, for nice topological spaces (CW complex), Eilenberg-Steenrod axioms fully characterize the homology of that space up to isomorphism!!! Singular homology is an example, but if you hand me some other chain complex and homology induced from that space satisfying the axioms, then their homology should match that of singular homology.
This is quite impressive! It really seems like “homology,” as a concept, isn’t “ad hoc” (as one might feel when first learning about singular homology), but rather something deep & universal about spaces, as homotopy is?
(going back to 1. Measuring holes, we can then add other examples of chain complexes and the resulting homology groups, of topological spaces, aside from singular homology—the standard ones are: cellular homology, simplicial homology. Why care about multiple homology theories of topological spaces that give you isomorphic homology groups (at least for nice spaces)? Because they have comparative advantages, eg singular: easy to prove things in, cellular: easy for humans to compute with, simplicial: easy for computers to compute with, etc)
Homology = (Abelianized? Symmetrized? Linearized?) Homotopy
Hurewicz theorem gives a homomorphism between the nth homotopy group πn(X) and the nth homology group Hn(X). In particular, it says that the 1st homology group is an abelianization of the fundamental group (!!!!)
But homotopy groups are always abelian for n≥2, so no hopes of this abelianization connection beyond n=1 (or not?)
Dold-Thom theorem: Hn(X)≅πnSP(X) for CW complex X
SP(X) is the infinite symmetric product space of X (in short, take finite products of X and mod by permutation—and given such collection, take a direct limit).
This is crazy!
Dold-Kan correspondence (?)
All the frames you are mentioning are good for intuition. I would say the deepest one is 4. and that everything falls into place cleanly once you formulate things in the language of infinity-category theory (at the price of a lot of technicalities to establish the “right” language). For example,
singular homology with coefficients in A can be characterised as the unique colimit-preserving infinity-functor from the infinity-category of spaces/homotopy types/infinity-grpoids/anima to the derived infinity-category of abelian groups which sends a one-point space (equivalently any contractible space) to A[0].
The derived infinity-category of abelian groups is itself in some sense the “(stable presentable) Z-linearization” of the infinity-category of spaces, although this is more tricky to state precisely and I won’t try to do this here.
Here are two more closely related results in the same circle of ideas. The first one gives a description (a kind of fusion of Dold-Thom and Eilenberg-Steenrod) of homology purely internal to homotopy theory, and the second explains how homological algebra falls out of infinity-category theory:
Consider functors E:S_* --> S_* from the infinity-category of spaces to itself which commutes with filtered colimits, carries pushout squares to pullback squares, sends the one-point space to itself, and the 0-sphere (aka two points!) to a discrete space. Then A=E(S^0) has a natural structure of abelian group, and E is an (infinity-categorical version of) the Dold-Thom functor and satisfies pi_n E(X)=H_n(X,A) (reduced homology). In particular, E(S^n) is an Eilenberg-Maclane space K(A,n).
The category of functors E:S_* --> S_* satisfying all the properties above except the one about E(S^0) being discrete is a model for the infinity-category Sp of spectra, i.e. the “stabilization” (in a precise categorical sense) of the infinity-category of spaces. From this perspective, the functors from the previous points are called the Eilenberg-Maclane spectra HA. Moreover, the infinity-category of spectra has a symmetric monoidal structure (the “smash product”), HR is naturally an algebra object for this structure whenever R is a ring, and it makes sense to talk about the infinity-category LMod(HR) of left HR-modules in Sp. Then LMod(HR) is equivalent (essentially by a stable version of the Dold-Kan correspondence) to the infinity derived category of left R-modules D(R). In other words, for homotopy theorists, (chain complexes,quasi-isomorphisms) are just a funny point-set model for HR-module spectra!
All of this is discussed in the first chapter of Lurie’s Higher Algebra, except the last point which is not completely spelled out because monoidal structures and modules are only introduced later on.
I should point out that this perspective is largely a reformulation of the results you already mentioned, and in themselves certainly do not bring new computational techniques for singular homology. However they show that 1) homological algebra comes out “structurally” from homotopy theory, which itself comes out “structurally” from infinity-category theory and 2) homological algebra (including in more sophisticated contexts than just abelian groups, e.g. dg-categories), homotopy theory, sheaf theory… can be combined inside of a common flexible categorical framework, which elegantly subsumes previous point-set level techniques like model categories.
In 3, there’s also the Brown representability theorem: Cohomology groups are just homotopy groups, with the sphere spectrum replaced with the Eilenberg-MacLane spectrum.
Have you looked at the historical development of homology? IME that’s generally a good way to build intuitions on a topic, as it tends to make concepts appear natural.
Thank you for the suggestion! That sounds like a good idea, this thread seems to have some good recommendations, will check them out.
If you do read up on the history of homology, I’d be grateful if you’d tell me whether it helps. I’m curious if my advice was actually a good idea, or if it just sounds good.
It was a good idea!