Sheaves and Cohomology for the Uninitiated (Part 1)
Or: Explaining what I do to my non-math friends.
A Short Anecdote and a Long Ramble
I was recently discussing my summer plans with my friend S., a non-mathematician, and mentioned that I would be attending the PCMI Graduate Summer School in Park City, Utah. At the mention of the topic of the summer school, which I mistakenly said was "motivic cohomology", I was met with a look of incredulousness - as if I had muttered a sequence of meaningless syllables.1 Such exchanges have become somewhat of an inside joke between S. and I, where my mention of an esoterically named mathematical concept is met with an incredulous blank stare, and soon followed by a knowing laugh. Acyclic Kan fibrations, anyone?
Browsing my (limited) mathematical dictionary, I can already think of numerous examples of such names. Take the word "cohomology", for example. To someone who is unacquainted with algebraic topology, algebraic geometry, homological algebra, algebraic number theory, and the like, this would be remarkably hard to connect to one's existing knowledge. Compare this to individuals describing their work in other fields. A biologist mentions they work on sea star locomotion and the listener likely knows what a sea star is, and even if not would likely be able to deduce that it is a sea-dwelling animal — given that the motion of this organism is the topic of study. A theologian mentions that they are writing their thesis on Mircea Eliade and one quickly realizes that they spend a large proportion of their time reading and interpreting, a realization seemingly independent of what Eliade wrote about.2 “Cohomology” might as well have been a collection of random syllables.
Looking at the world etymologically is not of more help here. The prefix "co" is often used to express a notion of togetherness (eg. collaboration, coworker), yet this does not convey the right notion of "co" which is used in category theory to express duality or "oppositeness" (eg. colimit, co-cone vs. limit, cone3). "Homo" from the ancient Greek homos (ὁμός) meaning same, common, or joint4 does capture something pertinent to cohomology insofar as the cohomology of a homotopic topological spaces are the same - but without a priori knowledge, it is not clear what the sameness is supposed to be of. "Logos", once again from the ancient Greek (λόγος), meaning reason, word, study,5 is pretty vacuous here - a mathematical concept inherently involves study and reason. One who still has a strong recollection of their biology might even associate the concept “homologous” from evolutionary biology, the study of shared genetic ancestry, which does not elucidate what “cohomology” is in the slightest. Putting the piecemeal etymological analysis together, we roughly get "the study of (things) with same" or, being more verbose, "the study of things in light of sameness". While this captures some of its essence, I don't think it in any way conveys a meaningful intuition for what cohomology is or how it works. To borrow the Aristotelian term, the function (ἔργον) of cohomology remains obscure.6
This high level of abstraction is ubiquitous in research-level mathematics, and likely contributes to the difficulty of communicating mathematical research broadly. My point here is not that the difficulty of good public communication is an issue isolated to mathematics, but to highlight a barrier mathematics must further overcome given how detached mathematical definitions and concepts can be. Historian Alma Steingart in sees this theme of abstraction as one of the defining features of post-WWII mathematics, developing in concert with the “High Modernism” of the time. She writes in Axiomatics: Mathematical Thought and High Modernism:
In announcing a break with the past, in turning against the long history of thinking about abstraction as an operational mode of thought (“to draw away”), abstractions came to be identified with modernism. In philosophy, mathematics, physics, and the arts, abstraction became by midcentury emblematic of a new paradigm of intellectual and artistic creativity, announcing a rupture (whether real or imagined) with the work of the past. (Steingart, p.10)
At least in the case of mathematics, I think Steingart is correct in identifying post-war mathematics with a “rupture”. As a student and practitioner of mathematics, I have come to appreciate the power of these abstractions: complicated thoughts can be expressed in fewer words, the generality such thinking affords, the cleanliness of thought itself within these new frameworks. Yet one would be incorrect to pursue boundless abstraction, a prospect that would only serve to alienate mathematics and mathematicians. I’ll put this another way: the arithmetic geometer Barry Mazur calls mathematics humanity’s longest conversation, conversation is an exchange, and exchange can only happen with some understanding. Abstraction has a dark side too.
I find myself returning to Aristotle here. In Metaphysics K, he notes how it is the mathematician that strips away (περιελὼν) the accidental sense-perceptible qualities to get to quantity (ποσὸν) and continuity (συνεχές).7 This opens up a superior way of investigation. Continuing this discussion of abstraction, he writes in Metaphysics M:
Each would be best (ἄριστα) understood in this way, if someone would take that which has not been separated and separate it, as both the arithmetician and the geometer do. (Met. 1078a21-24, translation my own)8
Yet, Aristotle is not ignorant of the difficulties of abstraction. At the outset of the Metaphysics Aristotle outlines the process of inquiring into universals:
For the things known to each person, what is first known is often barely known, and possesses little to no being (ὄντος). Yet, from those things poorly known yet known to oneself, one must strive to understand what is wholly (ὅλως) known, making the transition, as has been said, through those very things [one knows]. (Met. 1029b8-11)9
Inquiry of abstractions and universals begins with what one knows (ἑκάστοις γνώριμα) which is then generalized and abstracted by the process of stripping away (περιελὼν), a process grounded in what one knows. The issue in communicating research mathematics more widely, then, starts by expanding the “what one knows”. I certainly can’t change science communication for mathematics in one blog post. But hopefully by explaining sheaves and cohomology there’ll be one less word to puzzle over at dinner.
Prefatory Note for the Initiated: I have, perhaps counterintuitively, decided to introduce sheaves using the categorical definition via sites. Though I am somewhat of an abstract-nonsense afficionado myself, I'm not actually sure if this is the right way for a practicing mathematician to learn about sheaves, but I think it will be a nice way to introduce some related ideas, especially category theory.
Categories
Categories were introduced by Eilenberg and MacLane in the 1940s to generalize and unify structures in various branches of mathematics.10 Category theory and its analogues is at times (lovingly?, disparagingly?)11 referred to “abstract nonsense” by mathematicians, and the statements of definitions and theorems can at times seem unmotivated and difficult to grasp.
Before talking about categories, we need to define what a category is. Yet even the definition of a category is not immune to the abstraction and difficulty of the field. The reader encountering categories for the first time might consider skimming the definition and revisiting it after the examples discussed below, especially the case of Sets.
Definition (Category). A category C consists of the following data:
A collection Obj(C), the objects of C.12
For any two objects a, b in Obj(C) a set C(a, b), the set of maps between a and b.
For any three objects a, b, c in Obj(C), a set function
\(C(a,b)\times C(b,c)\longrightarrow C(a,c); (f,g)\mapsto g\circ f\)taking a map f from a to b and a map g from b to c to the composite map (which goes from a to c).
Furthermore, the data above satisfies the following rules.
For each object a in Obj(C), an identity map id(a) in C(a,a) such that
\(\mathrm{id}(a)\circ f=f, g\circ\mathrm{id}(a)\circ g=g.\)whenever such compositions exist.
For any four objects a, b, c, d in Obj(C) and maps f, g, h in C(a, b), C(b, c), C(c, d), respectively, there is an equality of maps a to d
\(h\circ(g\circ f)=(h\circ g)\circ f.\)
That (long) definition seems pretty unmotivated a priori, so here’s a (hopefully) concrete example to think about. Consider the case C = Sets. So we have a collection Obj(Sets) of all possible sets — in other words, S is in the collection Obj(Sets) if S is a set. More concretely, take S = {a, b} and T = {2, 4}. Both of these are sets and hence objects of the category Sets, which we might equivalently say S and T are in Obj(Sets). There are many maps between these two sets. For example, f : S —> T takes a to 2 and b to 4, but g : S —> T takes a to 4 and b to 2. So f and g are both in the set of maps from S to T which we denoted by Sets(S, T). The set Sets(S, T) might even be empty if there are no maps from S to T. Revisiting the definition of a category above, we have the first two pieces of data.
If I had another set Q = {$, %} and a map h : Q —> S say taking $ to a and % to b, we can form maps from P to T by composing the map h with either f or g which we denote
the composition of f with h and the composition of g with h, respectively. We can think of as first doing the function h to get from P to S then doing either the function f or g. In the composition of f with h, we first do h taking $ to a and % to b and then do f taking a to 2 and b to 4 so overall the composition takes the set Q to the set T with $ mapping to 2 and % mapping to 4. Similarly in the case of the composition of g with h, we get $ mapping to 4 and % mapping to 2. So this takes a function h from P to S (ie. an element of Sets(P, S)) and a function say f from S to T (ie. an element of Sets(S, T)) and produces a new function — the composition — which lives in Sets(Q, T). This gives a composition law, which was the third piece of data we needed.
The identity map on sets just takes each element of a set to itself. For example, id(S) is a map S to S taking a and b to themselves. In other words, we have verified the first rule. Checking that the second rule holds is just saying that “the order we compose functions in doesn’t matter”: for P = {γ, δ} and j : P —> Q by γ mapping to $ and γ mapping to % we have an equality of functions P to T
On the left, we first compose h with j taking γ to a and δ to b (through $ and %). Composing this with f gives a map P to T taking γ to 2 and δ to 4. Unpacking what happens on the right, we can check that first composing f with h and feeding it the input of j also takes γ to 2 and δ to 4 agreeing with the left side. In other words, it doesn’t matter which pair we composed first. This also checks the second rule the data must satisfy. More generally, such a property is called associativity, and a more concise definition of a category would merely appeal to this property without spelling things out.13
Here’s another running example that we’ll use throughout. Recall from calculus that an interval of the real numbers (ie. can be expressed as a decimal) is open if it doesn’t contain its endpoints, and closed if it does. Introducing some notation, we’ll write (a, b) to denote the interval of real numbers between a and b excluding the endpoints a and b themselves, (a, b] to denote the interval of real numbers between a and b excluding a and including b, and [a, b] the interval of real numbers between a and b including the endpoints. Let X denote the closed unit interval over the real numbers (ie. X = [0, 1]). It is a fact that properties of X can be deduced by the local properties of a special family of subsets of X known as open sets.
Call a subset of X open if it is the intersection of some union of open intervals with X itself. Any open set of the reals is a union of open intervals. So for the open intervals (0.2, 0.4) and (0.5, 0.6), their union
is open in the reals and fully contained in X = [0, 1] so is open in X = [0, 1]. If you are looking at opens of the reals which contains the endpoints then things get slightly more complex. Say we have a set
This is open in the reals and its intersection with X is
which is open (because we defined the open sets of X to be the intersection of X with an open set of the reals). There are a some other possible cases, but you should be able to convince yourself that the open subsets of X are sets of the following type:
where w, x, y, z satisfy the following conditions
These conditions on w, x, y, z are just to ensure that all these open sets actually are subsets of X. Pictorially, these look like the following blue subsets of the space X
where solid and open dots indicate including or excluding the endpoints, respectively. Given two open sets U, V of X, we declare there be a map from U to V only when U is a subset of V. Pictorially,
where in the first case there is no map since the union of (0.4, 0.7) and (0.6, 0.8) is not a subset of (0.3, 0.7).
It is pretty routine to verify that the data of the open sets of X together with the maps including a smaller subset into a bigger subset satisfy the conditions for being a category which we call Opens(X).
To check this more explicitly, we have the objects Obj(Opens(X)) to be the open sets as described above and the maps Opens(X)(U, V) as follows:
in other words, if U is a subset of V there is exactly one map from U to V and no maps if U is not a subset of V. This gives us the first two pieces of data. For the third piece of data, note that if U is a subset of V and V a subset of W, U is a subset of W giving a map U to W. The additional rules that the data must satisfy is pretty easy to check too. Any open set U has a map to itself - just take each point to itself, and the second rule is satisfied because taking U as a subset of V as a subset of W is the same as taking U as a subset of W in the first place.
Okay. Now that we have first met our category, we start to talk about the math that happens within a category. We now define the fibered product, which is a way of producing a new object of a category from old ones.
Definition (Fibered Product). Let C be a category, a,b,c in Obj(C), and f,g in C(a, c), C(b, c), respectively (ie. f is a function a to c and g a function b to c).
The fibered product of a and b over c, if it exists, is an object
with maps to both a and b satisfying the following rules
The maps
\(a\times_{c}b\to a\to c, a\times_{c}b\to b\to c\)are the same.
If d in Obj(C) is such that the maps
\(d\to a\to c, d\to b\to c\)are the same, there is a unique map from d to the fibered product.
Mathematical Remark. The second condition is the universal property of the fibered product. In particular, the fibered product can be expressed as the final object of some slice category over a diagram. Yoneda’s lemma thus implies that it is unique up to unique isomorphism.14
Let us grant that fibered products exist in Opens(X), a setting in which fibered products have a particularly nice interpretation. When we discussed the maps in Opens(X), we noted that maps only exist when the source set is a subset of the target set. Suppose we have maps
in Opens(X), so U and V are both subsets of W. Since we are assuming fibered products exist we have that
agree as maps from the fibered product of U and V over W to W. In particular the maps from the fibered product to U and V tell us that the fibered product is a subset of U and V. This gives us a natural candidate for what the fibered product is: the intersection of U and V. Looking at the following example, we can quickly convince ourselves that the intersection satisfies the first property.
Turning to the second condition, we first observe that if some T of Opens(X) had the property that the maps
are the same, we know that T is a subset of both U and W, that is, T is a subset of the intersection, our previous candidate for the fibered product. But since T is a subset of the intersection of U and V, there is a unique map from T to the intersection. This shows that the intersection satisfies both the properties of the fibered product. We could have also gotten to this another way: the fibered product must be a subset of the intersection of U and V and it must be the intersection itself since we know the intersection satisfies the property, but if the fibered product were smaller than the intersection, there would not be map from the intersection to the fibered product.
Sites
Sites were developed by the mathematician Alexander Grothendieck to generalize the concept of topological spaces (whatever those may be). It is for this reason that sites are at times called Grothendieck sites, or a Grothendieck topology on a category.15 Instead of stating the definition directly, we first explore Opens(X). First note that an open set in Opens(X) can be written as the union of smaller open sets, motivating the definition of a covering.
Definition (Covering). Let U be an object of Opens(X). We say a collection of maps with target U16
is a covering if
Pictorially,
since the union of (0.1, 0.4) and (0.3, 0.7) is U = (0.1, 0.7). Furthermore, observe that:
For U an object of Opens(X),
\(U \to U\)is a covering.
For
\(\{U_{i}\to U\}_{i=1}^{n}\)a covering of U and V another object of Opens(X) with a map to U,
\(\{U_{i}\times_{U}V\to V\}_{i=1}^{n}\)is a covering for V. Take, as above, (0.1, 0.4) and (0.3, 0.7) covering U = (0.1, 0.7) and V = (0.2, 0.5). We have already seen that fibered products are intersections which are (0.2, 0.4) and (0.3, 0.5). Evidently these form a cover for V.
For
\(\{U_{i}\to U\}_{i=1}^{n}\)a covering of U and
\(\{U_{ij}\to U_{i}\}_{j=1}^{m_{i}}\)a covering of Ui for each i,
\(\{U_{ij}\to U_{i}\to U\}_{1\leq i\leq n, 1\leq j\leq m_{i}}\)is a covering of U. Pictorially:
where the sets in purple cover U1 and the sets in orange cover U2, but the purple and orange sets together cover all of U.
This in fact shows that Opens(X) is a site. More generally, we define a site as follows.
Definition (Grothendieck Topology). Let C be a category with fibered products. A Grothendieck Topology on C is the data of a set Cov(X) consisting of maps
for each object X of Obj(C) satisfying the following conditions:
If Y is isomorphic to X,
\(\{Y\to X\}\)is a covering of X.
If
\(\{X_{i}\to X\}_{i=1}^{n}\)is a covering of X and W in Obj(C) with a map to X then
\(\{X_{i}\times_{X}W\to W\}_{i=1}^{n}\)is a covering for W.
If
\(\{X_{i}\to X\}_{i=1}^{n}\)is a covering for X and
\(\{X_{ij}\to X_{i}\}_{j=1}^{m_{i}}\)a covering of Xi for each i,
\(\{X_{ij}\to X_{i}\to X\}_{1\leq i\leq n, 1\leq j\leq m_{i}}\)is a covering for X.
Definition (Site). A site S is the data of a category C with a Grothendieck topology.
So the coverings in Opens(X) give a Grothendieck topology on Opens(X). In other words, we have proven the following theorem.
Theorem. Opens(X) is a site.
In this way, sites generalize the idea of a (topological) space.
Recap and Next Time…
We have already seen what a category is, saw how fibered products work, met Opens(X), and defined sites. Next time:
We will meet the functor, which is a map of categories. Just as we can map sets to each other, we can map categories to each other too.
We will construct the opposite category. The opposite category of Opens(X) will allow us to define presheaves.
We will look at the category-theoretic equalizer. This will allow us to define sheaves.
We will define cohomology and look at some examples.
The title of the summer school is in fact "motivic homotopy", a title that I am certain would have elicited the same reaction from S.
Mircea Eliade (1907-1986) was a highly influential Romanian-born theorist of religion who taught at the University of Chicago Divinity School.
I was once told by Michael Hopkins of the inaccuracy of the term cone. The category of cones is defined to be the slice category over a fixed diagram. One then defines the limit to be the final object in the category of cones. Yet the word "cones" has a "co" in it, which suggests the property of an initial object of a slice category under a fixed diagram, such as in the case of the colimit or coproduct. Cones ought instead be named nes, omitting the "co".
https://logeion.uchicago.edu/%CE%BB%CF%8C%CE%B3%CE%BF%CF%82. Certainly one of the most nuanced words in classical philology and theology, though I believe the above is sufficient as a working definition.
See, for example, 1097b24 of the Nicomachean Ethics. As is customary, Aristotle is cited by the Bekker (marginal) pages and short name (eg. NE 1097b24).
καθάπερ δ᾽ ὁ μαθηματικὸς περὶ τὰ ἐξ ἀφαιρέσεως τὴν θεωρίαν ποιεῖται(περιελὼν γὰρ πάντα τὰ αἰσθητὰ θεωρεῖ, οἷον βάρος καὶ κουφότητα καὶ σκληρότητα καὶ τοὐναντίον, ἔτι δὲ καὶ θερμότητα καὶ ψυχρότητα καὶ τὰς ἄλλας αἰσθητὰς ἐναντιώσεις, μόνον δὲ καταλείπει τὸ ποσὸν καὶ συνεχές, τῶν μὲν ἐφ᾽ ἓν τῶν δ᾽ ἐπὶ δύο τῶν δ᾽ ἐπὶ τρία, καὶ τὰ πάθη τὰ τούτων ᾗ ποσά ἐστι καὶ συνεχῆ, καὶ οὐ καθ᾽ ἕτερόν τι θεωρεῖ, καὶ τῶν μὲν τὰς πρὸς ἄλληλα θέσεις σκοπεῖ καὶ τὰ ταύταις ὑπάρχοντα, τῶν δὲ τὰς συμμετρίας καὶ ἀσυμμετρίας, τῶν δὲ τοὺς λόγους, ἀλλ᾽ ὅμως μίαν πάντων καὶ τὴν αὐτὴν τίθεμεν ἐπιστήμην τὴν γεωμετρικήν)(Met. 1061a28-b3, Retrieved from Perseus 3/26/24)
Just as the mathematician creates theory through abstraction (for, having removed all perceivables, he considers, for instance, weight and lightness, hardness and its opposite, as well as warmth and coldness and the other perceptible oppositions, leaving only quantity and continuity, in some cases on one, in others on two, and in others on three dimensions, and the qualities of these insofar as they are quantifiable and continuous, and he considers nothing else), and in the case of some, he examines their relative positions and the properties that result from these and of these, he examines the symmetries and asymmetries, and of others, the ratios, yet still we assign one and the same science, geometry, to all of them. (Translation my own)
ἄριστα δ᾽ ἂν οὕτω θεωρηθείη ἕκαστον, εἴ τις τὸ μὴ κεχωρισμένον θείη χωρίσας, ὅπερ ὁ ἀριθμητικὸς ποιεῖ καὶ ὁ γεωμέτρης. (Retrieved from Perseus 3/26/24)
τὰ δ᾽ ἑκάστοις γνώριμα καὶ πρῶτα πολλάκις ἠρέμα ἐστὶ γνώριμα, καὶ μικρὸν ἢ οὐθὲν ἔχει τοῦ ὄντος: ἀλλ᾽ ὅμως ἐκ τῶν φαύλως μὲν γνωστῶν αὐτῷ δὲ γνωστῶν τὰ ὅλως γνωστὰ γνῶναι πειρατέον, μεταβαίνοντας, ὥσπερ εἴρηται, διὰ τούτων αὐτῶν. (Retrieved from Perseus 3/26/24)
I refer the reader interested in the socio-historical development of category theory to Steingart’s discussion on p.46 ff. of Axiomatics.
en.wikipedia.org/wiki/Abstract_nonsense. My use of it here and previously is affectionate self-mockery. One would do well to heed aphorism 107 of Nietzsche’s Gay Science: "“We must now and then be joyful in our folly, that we may continue to be joyful in our wisdom.”
I am being deliberately vague here. Formally, the objects of a category can be a class — in order to avoid Russel’s paradox and the like. Here and throughout I will suppress set-theoretic difficulties.
See, for example, Stacks Project Tag 0014 .
Those particularly intent on obscurantism will say instead “can be classified up to membership in a codiscrete groupoid”.
See the notes of M. Artin’s 1962 seminar at Harvard (hdl.handle.net/2027/mdp.39015056607966), though Giraud defines sites differently in the 1962/63 Séminaire Bourbaki.
More precisely, the covering maps should be taken to be jointly surjective over an arbitrary indexing set, not a finite one. I write finite indexing sets everywhere for ease of understanding.