This pretty much continues on from my last post on homological algebra by the same name. Have a look there first if you haven’t seen it. If you have, welcome to part 2! Today, I’ll apply our new intuition to a fundamental part of homological algebra: *homology*.

## Homology

OK, this all seems pretty good. Homology and cohomology let us recover presentations when applying left or right exact functors to presentations, ie. functors that damage either generators or relations (but not both). LES’s were completely necessary to include in our concept of ‘presentation’ to achieve this quite profound technique.

Homology and cohomology both seem pretty neat. What are they? Are they more generally interesting? They sure are. Let’s have a look with our new insight.

Recall the situation of derived functors. In particular, remember when we had a SES of projective resolutions? We’re gonna generalize it a bit. If we have, for three chain complexes $C_\bullet$, $B_\bullet$ and $A_\bullet$ a *right exact sequence of chain complexes*, ie. a SES $0 \rightarrow C_\bullet \rightarrow B_\bullet \rightarrow A_\bullet \rightarrow 0$ in an abelian category of chain complexes, we can induce a long exact sequence
$$
\cdots \rightarrow H_i(C_\bullet) \rightarrow H_i(B_\bullet) \rightarrow H_i(A_\bullet) \rightarrow H_{i-1}(C_\bullet) \rightarrow \cdots
$$ by a quick diagram chase, where $H_i(C_\bullet) := \textbf{ker}(d_{i-1})/\textbf{im}(d_i)$ as with $L_iF$. We call $H_i(C_\bullet)$ the $i^{th}$ *homology group* of $C_\bullet$. If the maps go $C^i \rightarrow C^{i+1}$ instead of $C_i \rightarrow C_{i-1}$, we call it *cohomology* and write $H^i(C^\bullet)$. The above LES’s existence is called the *snake lemma*.

OK… who cares? If we have a levelwise SES of chain complexes, don’t we already have a levelwise ‘presentation’ of these chain complexes? What’s so special about this LES? It might look strictly less interesting, since it describes this new artificial ‘homology’ thing. What *is* special is that we now have a single LES of homologies of *all* levels at once. Information about level $i$ now tells us about level $i-1$. This is far more powerful presentation than before. Hidden in relations between weak structures are strong relations between smaller substructures. Homology lies dormant in any chain complex, the special fragment that yields nice presentations by design.

Alright, so if we have an incomplete presentation of a chain complex, we can get a complete presentation of… this other thing. Why would we *want* to compute it though? Well, suppose we have a chain complex $C_\bullet$ and we manage to compute all its homologies $H_i(C_\bullet)$. Now we know $C_\bullet$ isn’t really a good presentation since it isn’t exact. However, what does it mean to not be exact? It means $\textbf{im}(d_i) \subsetneq \textbf{ker}(d_i)$. Something maps to $0$, but isn’t in the image above. But wait - *homology exactly contains this lost information*. So, with the chain complex and homologies combined, we should be able to recover some kind of presentation of each $C_i$! Indeed, this was exactly what we did with left and right derived functors. We used (co)homology to extend a left or right exact sequence to a LES, since in those situations homology agreed with the chain complex at level $0$. I’m currently not so sure on other methods, but I’ll get back to you on that. It’s a hunch.

Usually however, it seems people just learn to care about homology. It’s a computable and nontrivial invariant. I guess that’s fine.

## Singular Homology

A good example of learning to care about homology arises in *singular (co)homology*. In algebraic topology classes, this is usually presented in a way totally disjoint from what I’ve said thus far. Let’s try and interpret it in our new way.

The place to start is the singular chain complex. Let $X \in \textbf{Top}$. We want a good invariant of $X$ which we can understand algebraically. In particular, we’d like to *compute* it from some simple topological information about $X$.

We’ll start with the set of simplices in $X$. This is a decently accurate probe of $X$’s homotopic data - simplices capture all the ‘higher order paths’ in the space. If this doesn’t capture $X$’s homotopy data very well, it probably means $X$ is ill-suited to study by homotopy theory anyway. See my post on homotopy beyond the real line if you’re interested.

We take particular interest in the *boundary* operation on an $n$-simplex $\Delta^n \rightarrow X$, by which I mean precomposition with $\partial \Delta^n \hookrightarrow \Delta^n$. This will tell us how the various simplices fit together and into each other, giving a neat combinatorial structure that captures the vague shape of $X$.

More formally, let’s define $C_i(X, R)$ as the free $R$-module generated by the $i$-simplices $\Delta^i \rightarrow X$. We get natural maps $d_i: C_i(X, R) \rightarrow C_{i-1}(X, R)$ by sending an $i$-simplex to an alternating sum of its boundary $(i-1)$-simplices, each component being added if in a chosen orientation and subtracted if opposing said orientation. For instance, a $1$-simplex is sent to its target $0$-simplex minus its source. This is our attempt to make the homotopy data ‘algebraic’; we’ve ascribed meaning to sums of simplices as boundaries.

These form a chain complex
$$
\cdots \overset{d_4}{\rightarrow} C_3(X, R) \overset{d_3}{\rightarrow} C_2(X, R) \overset{d_2}{\rightarrow} C_1(X, R) \overset{d_1}{\rightarrow} C_0(X, R)
$$continuing to the left forever. Indeed, for any $i$, $d_{i-1} \circ d_i = 0$, as the sums will precisely cancel out. Intuitively, a boundary has no boundary of its own. This chain complex in general won’t be exact: for instance, $X = S^1$ and $R = \mathbb{Z}$. (If you unwind the definitions, the sequence fails to be exact when $X$ has a sub-simplicial complex with no boundary that isn’t the boundary of a higher simplex, ie. a *hole*.)

At the moment, it isn’t clear how to compute this structure. What kinds of things might we be able to compute from information about $X$? Well, as we’ve learned, homology is pretty nice for computation. Note that the chain complex construction $C_\bullet(X, R)$ is functorial in $X$. This means continuous maps $X \rightarrow Y$ induce maps $C_\bullet(X, R) \rightarrow C_\bullet(Y, R)$ of chain complexes. So, if we can find structures on $X$ that induce SES’s involving $C_\bullet(X, R)$, we can start devising presentations of $H_i(C_\bullet(X, R))$ in terms of other simpler spaces! These homology structures seem like a good candidate for a ‘computable invariant’. Interestingly, they’re also invariant under homotopies, so they give homotopy data. That’s nice.

As an example, take a subspace $U \hookrightarrow X$. We can immediately find a LES
$$
\cdots \rightarrow H_i(C_\bullet(U, R)) \rightarrow H_i(C_\bullet(X, R)) \rightarrow H_i(C_\bullet(X, R)/C_\bullet(U, R)) \rightarrow H_{i-1}(C_\bullet(U, R)) \rightarrow \cdots
$$ which means we can compute $X$’s homology from the homologies of (hopefully) simpler structures. To me, this particular kind of LES is really the point of singular homology. Together with the five lemma, we get the all-important *excision theorem*, which makes topological the computability of $H_i$.

Similar structure emerges when we dualize everything - define $C^i(X, R) := \textbf{Hom}(C_i(X, R), R)$ and boundary maps $C^i(X, R) \rightarrow C^{i+1}(X, R)$ by precomposition. Taking the same quotients gives us *cohomology of $X$* and induces largely dual LES’s. This invariant seems to have even more structure however; it has a natural graded product, the preservation of which makes the invariant even more fine-grained than homology. Maybe I’ll look at it a bit more closely another time.

## Sheaf Cohomology

So, what’s up with sheaf cohomology? Yet another homological structure to think through. In truth, this one’s alot closer to our original motivation via derived functors. We really do just apply the constructions we learned for Tor and Ext.

Sheaves are all about ’local-to-global’ problems. Given a collection of structures defined in local regions, can we glue them together into one single structure? As it turns out, thinking about epimorphisms of sheaves is a good way to phrase this problem. Zhen Lin gave a pretty good example here that I’ll use. Let $X = \mathbb{R}^2 \setminus \{(0, 0)\}$ and write $\Omega^\bullet_X$ for the de Rham complex, as a chain complex of sheaves. Since $X$ is 2-dimensional, we get an exact sequence
$$
\Omega^0_X \rightarrow \Omega^1_X \rightarrow \Omega^2_X \rightarrow 0
$$ in $\textbf{Sh}(X, \textbf{Vect}_\mathbb{R})$ as it’s always possible to, in a local contractible region, integrate a closed $(n+1)$-form to get an $n$-form by the Poincare lemma. So, if we define $B^1(\Omega_X)$ to be the image of the map $\Omega^0_X \rightarrow \Omega^1_X$, or equivalently the subsheaf of closed $1$-forms by exactness, we have the usual epimorphism of sheaves $\Omega^0_X \rightarrow B^1(\Omega_X)$. However, if we look at global sections, the induced map $\Gamma(X, \Omega^0_X) \rightarrow \Gamma(X, B^1(\Omega_X))$ is anything but surjective! For instance, the 1-form
$$
\text{d}\theta = \frac{-y\text{d}x + x\text{d}y}{x^2 + y^2}
$$ is indeed closed but cannot be exact - after all,
$$
\int_{S^1} \text{d}\theta = 2\pi
$$ where $S^1$ is the unit circle in $\mathbb{R}^2$. So, while every closed form is *locally* exact, it isn’t necessarily *globally* exact. This is the general issue I’m getting at.

Epimorphisms of sheaves are ’local surjections’. They surject in a small enough region. Any section in the codomain sheaf will locally everywhere be in the image of the epimorphism. However, this says nothing about whether the section itself is in the image. Lots of ’local-to-global’ problems can be phrased as a sheaf epimorphism $f : S \rightarrow T$, then restricting to global sections $\Gamma(X, f) :\Gamma(X, S) \rightarrow \Gamma(X, T)$. We know that every section of $T$ *locally* has some property represented by being in the image of $S$. The question is when that is *globally* the case. More formally, **how surjective is $\Gamma(X, f)$**?

The answer is given by sheaf cohomology. Any epimorphism $f$ immediately induces a SES of sheaves $$ 0 \rightarrow K \rightarrow S \overset{f}{\rightarrow T} \rightarrow 0 $$ where $K := \textbf{ker}(f)$. The functor $\Gamma(X, -)$ is left exact - it preserves monomorphisms. Saying it isn’t left exact is precisely saying it destroys epimorphisms, as we’ve seen. So, we can take right derived functors of $\Gamma(X, -)$ to recover a new LES $$ 0 \rightarrow H^0(X, K) \rightarrow H^0(X, S) \rightarrow H^0(X, T) \rightarrow H^1(X, K) \rightarrow H^1(X, S) \rightarrow H^1(X, T) \rightarrow H^2(X, K) \rightarrow \cdots $$ where $H^0(X, -) \cong \Gamma(X, -)$. This LES answers precisely how much a local epimorphism fails to be a global epimorphism - it ‘completes the presentation of the global sections’. So, sheaf cohomology addresses the problem of globalizing local properties. Awesome!

Are singular and sheaf cohomology connected in any way? It turns out that they agree in ’nice’ situations, ie. $X$ is locally contractible and we have a locally constant sheaf of abelian groups. Conceptually though? I’m unclear. We usually think of sheaf cohomology as when we vary the *sheaves* $S$ in $H^\bullet(X, S)$. Somehow, singular cohomology should happen when we vary $X$ instead. Again, this only works in nice spaces and I haven’t really read the proofs enough to understand. Look forward to a future post on that, I suppose.

## What Now?

So those are my thoughts on homological algebra. It’s a category theorist’s answer to presentations. By applying the philosophies of category theory, we have realized how presentations extend naturally to the more powerful notion of *exact sequences*, where we may ‘present’ one object in terms of others and vice versa. We have removed the bias on what is being presented and have been rewarded with useful new constructions for our insight, like cohomology, to better understand other things we care about (operations.)

Now what? Well, we wanna compute presentations. We wanna compute (co)homology, derived functors and so forth to understand operations in terms of presentations. We wanna infer more things from exact sequences. We wanna use (co)homology to understand properties of topological spaces and sheaves, so we can learn about geometry and ’local-to-global’ problems.

I’m no expert in this area, so I’ll be learning as I write. I might write about derived categories next time; they’re a major step forwards in derived functor theory and lead us into the domain of *model categories* and *$\infty$-categories*. Maybe I’ll have a look at triangulated categories, topoi, some applications of homological algebra in physics… hard to say. But I’ll look forward to sharing what I find. And I hope you look forward to reading.