Topology

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August 18, 2024

Sheaves and Topology

Posted by Bartosz Milewski under Category Theory, Topology | Tags: Category Theory, Topology |
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In all branches of science we sooner or later encounter the global vs. local duality. Topology is no different.

In topology we have the global definition of continuity: counter-images of all open sets are open. But we perceive a discontinuity as a local jump. How are the two pictures related, and can we express this topologically, that is without talking about sizes and distances?

All we have at our disposal are open sets, so exactly what properties of open sets are the most relevant? They do form a (thin) category with inclusions as arrows, but so does any set of subsets. As it turns out open sets can be stitched together to create coverings. Such coverings let us zoom in on finer and finer details, thus creating the bridge between the global and the local picture.

Open sets are plump–they can easily fill the bulk of space. They are also skinless, so they can’t touch each other without some overlap. That makes them perfect for constructing covers.

Covering, unlike tiling, requires overlapping. To create a leak-free roof, you need your tiles to overlap. The idea is that, if we were defining functions over a tiling, it would be possible for them to make sudden jumps at tile boundaries. Open coverings overlap, so such functions have to flow continuously.

An open cover of a set $u$ is a family of open sets $\{u_i\}$ such that $u$ is their union:

$u = \bigcup_{i \in I} u_i$

Here $I$ is a set used for indexing the family.

If we have a continuous function $f$ defined over $u$ , then all its restrictions $f|_{u_i}$ are also continuous (this follows from the condition that an intersection of open sets is open). Thus going from global to local is easy.

The converse is more interesting. Suppose that we have a family of functions $f_i$ , one per each open set $u_i$ , and we want to reconstruct the global function $f$ defined over the set $u$ covered by $u_i$ ‘s. This is only possible if the individual functions agree on overlaps.

Take two functions: $f_i$ defined over $u_i$ , and $f_j$ defined over $u_j$ . If the two sets overlap, each of the functions can be restricted to the overlap $u_i \cap u_j$ . We want these restrictions to be equal:

$f_i|_{u_i \cap u_j} = f_j|_{u_i \cap u_j}$

If all individual continuous functions agree on the overlaps then they uniquely determine a global continuous function $f$ defined over the whole set $u$ . You can stitch or collate functions that are defined locally.

In the language of category theory we talk about functions in bulk. We define a functor–a presheaf $P$ –that maps all open sets to sets of continuous functions. In this language, to an open cover $\{u_i\}$ corresponds a family of functions $\{f_i\}$ that are members of the individual sets $P u_i$ . Every such selection forms a giant $I$ -indexed tuple, that is an element of the cartesian product:

$\{f_i | i \in I\} \in \prod_{i} P u_i$

Similarly, we can gather functions that are defined over the intersections of sets into a product:

$\prod_{i j} P (u_i \cap u_j)$

(Notice that every empty intersection corresponds to a single trivial function that we call absurd in Haskell.)

Set inclusions generate function restrictions. In particular, for every intersection $u_i \cap u_j$ we have a pair of restrictions:

$f_i \mapsto f_i|_{u_i \cap u_j}$

$f_j \mapsto f_j|_{u_i \cap u_j}$

These restrictions can be seen as functions between sets:

$P u_i \to P (u_i \cap u_j)$

$P u_j \to P (u_i \cap u_j)$

If all such restrictions are pairwise equal, we call $\{f_i\}$ a matching family, and for every such matching family there is a unique element $f \in P u$ such that $f_i = f|_{u_i}$ , for all $i$ .

These pairs of restrictions define two mappings between our big products:

$p, q : \prod_i P u_i \rightrightarrows \prod_{i j} P (u_i \cap u_j)$

Think of each function as acting on a tuple $\{f_k\}$ and producing a matrix indexed by elements of $I$ :

$(p\; \{f_k\})_{i j} = f_i|_{u_i \cap u_j}$

$(q\; \{f_k\})_{i j} = f_j|_{u_i \cap u_j}$

Our matching condition can be expressed in the language of category theory by saying that the following diagram is an equalizer of $p$ and $q$ (the two parallel arrows):

$P u \xrightarrow{e} \prod_i P u_i \rightrightarrows \prod_{i j} P (u_i \cap u_j)$

Here $e$ is defined as mapping a function $f \in P u$ to a tuple of its restrictions $\{ f|{u_i}\}$ . These restrictions are then required to match when further restricted by $p$ and $q$ to all possible intersections.

A presheaf $P$ is called a sheaf if, for every open covering $\{u_i\}$ , a matching family $\{f_i\}$ uniquely determines the element of $P u$ of the equalizer above. This element corresponds to the function $f$ that is the result of stitching of individual functions.

Notice that, even though we tried to use the categorical language as much as possible, we still had to rely on the language of sets to define coverings. To abstract away from set theory and traditional topology, we need to talk about sites.

Next: Coverages and Sites .

August 7, 2024

Presheaves and Topology

Posted by Bartosz Milewski under Category Theory, Topology | Tags: Category Theory, Topology |
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Previously: Topology as a Dietary Choice.

Category theory lets us change the focus from individual objects to relationships between them. Since topology is defined using open sets, we’d start by concentrating on relations between sets.

One such obvious relation is inclusion. It imposes a categorical structure on the subsets of a given set $X$ . We draw arrows between two sets whenever one is a subset of the other. These arrows satisfy the axioms of a category: there is an identity arrow for every object (every set is its own subset) and arrows compose (inclusion is transitive). Not every pair of objects is connected by an arrow–some sets are disjoint, others overlap only partially. We may include the whole space as the terminal object (with arrows coming from every subset) and the empty set $\emptyset$ as the initial object (with arrows going to every set). As categories go, this is a thin category, because there is at most one arrow between any two objects.

Every topological space gives thus rise to a thin category that abstracts the structure of its open sets. But the real reason for defining a topology is to be able to talk about continuous functions. These are functions between topological spaces such that the inverse image of every open set is open. Here, again, category theory tells us not to think about the details of how these functions are defined, but rather what we can do with them. And not just one function at a time, but the whole bunch at once.

So let’s talk about sets of functions. We have our topological space $X$ , and to each open subset $u$ we will assign a set of continuous function on it. These could be functions to real or complex numbers, or whatever–we don’t care. All we care about is that they form a set.

Since open sets in $X$ form a (thin) category, we are talking about assigning to each object (open set) $u$ its own set (of continuous functions) $P u$ . Notice however that these sets of functions are not independent of each other. If one open set is a subset of another, it inherits all the functions defined over the larger set. These are the same functions, the only difference being that their arguments are restricted to a smaller subset. For instance, given two sets $v \subseteq u$ and a function $f \colon u \to \mathbb R$ , there is a function $f|_{v} \colon v \to \mathbb R$ such that $f|_{v} = f$ on $v$ .

Let’s restate these statements in the categorical language. We already have a category $X$ of open sets with inclusion. The sets of functions on these open sets are objects in the category $\mathbf{Set}$ . We have defined a mapping $P$ between these two categories that assigns sets of functions to open sets.

Notice that we are dealing with two different categories whose objects are sets. One has inclusions as arrows, the other has functions as arrows. (To confuse matters even more, the objects in the second category represent sets of functions.)

To define a functor between categories, we also need a mapping of arrows to accompany the mapping of objects. An arrow $v \to u$ means that $v \subseteq u$ . Corresponding to it, we have a function $P u \to P v$ that assigns to each $f \in P u$ its restriction $f|_{v} \in P v$ .

Together, these mappings define a functor $P \colon X^{op} \to \mathbf{Set}$ . The “op” notation means that the directions of arrows are reversed: the functor is “contravariant.”

A functor must preserve the structure of a category, that is identity and composition. In our case this follows from the fact that an identity $u \subseteq u$ maps to a trivial do-nothing restriction, and that restrictions compose: $(f|_v)|_w = f|_w$ for $w \subseteq v \subseteq u$ .

There is a special name for contravariant functors from any category $\mathcal C$ to $\mathbf{Set}$ . They are called presheaves, exactly because they were first introduced in the context of topology as precursors of “shaves.” Consequently, the simpler functors $\mathcal C \to \mathbf{Set}$ had to be confusingly called co-presheaves.

Presheaves on $\mathcal C$ form their own category, often denoted by $\hat{\mathcal C}$ , with natural transformations as arrows.

Next: Sheaves and Topology.

July 12, 2024

Topology as a Dietary Choice

Posted by Bartosz Milewski under Topology | Tags: Topology |
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What is Topology?

When talking about topology, people draw cups with handles turning into donuts. When I think of topology, I see nutritious food.

In mathematics, topology is defined as a family of subsets of some space $X$ . We call these subsets open. Open sets are like meaty, skinless fruits.

Watermelon

For instance, in standard topology, the inside of a ball in 3-d is considered meaty. Contrast this with an empty sphere, a curve, or a point–these are skinny when embedded in 3-d–they have no nutritional value.

In one dimension (on a line), the inside of a segment is meaty, but a segment with endpoints is not open, because it has a rind (the endpoints).

These four conditions define a topology.

The intersection of any two open sets is again an open set. This is what I mean by skinlessness. If you included skins, the intersection could end up skinny.
A union of open sets is again open. It’s even more juicy, and no skin can be produced by a union. There is a subtlety there: You can take a union of an arbitrary number of open sets and it’s still open. But you have to be careful with intersections–only finite intersections are allowed. That’s because by intersecting an infinite number of open sets you could end up with something very skinny–like a single point.
The whole space $X$ is open. In a sense, it defines what it means to be juicy and it doesn’t have a skin because it has no contact with the outside–it is its own Universe.
As usual, an empty set is an odd item. Even though it’s empty, it’s considered open. You may think of it as a union of zero open sets.

There are some extreme topologies, like the discrete topology in which all subsets are open (even individual points), and a trivial (indiscrete) topology where only the whole space and the empty set are open. But most topologies are reasonable and adhere to our intuitions. So let’s not worry about pathologies.

Continuity

Consider a function from one topological space $X$ to another topological space $Y$ . Intuitively, a function is continuous if it doesn’t make sudden jumps. So naively you might think that a continuous function would map open sets to open sets. But that’s not true. For instance a constant function maps any open set to a point which, in most topologies, is not open.

In fact any time a function stalls, or makes a turnaround (like the function $y = x^2$ at $x = 0$ ) you get a skinny point in its image.

The correct definition goes in the opposite direction: a function is continuous if and only if the pre-image of every open set is open.

First of all, a function cannot stall or turn around in the $x$ direction, since that would mean mapping one point to many.

Secondly, if a function makes a jump at some point $x$ , it’s possible to surround $f(x)$ with a small open set whose counter-image contains $x$ as its boundary.

It’s also possible to define a continuous function as a pair of functions. One function $f$ is the usual mapping of points from $X$ to $Y$ . The other function $g$ maps open sets in $Y$ to open sets in $X$ . The pair $(f, g)$ defines a continuous function if for all points $x \in X$ and open sets $O$ in $Y$ we have the following equivalence:

$f(x) \in O \iff x \in g(O)$

The left-hand side tells us that $x$ is in the pre-image of $O$ under $f$ . The right-hand side tells us that $g$ maps $O$ to this preimage. This formula looks a bit like an adjunction between $f$ and $g$ . It’s an example of a more general notion of Chu constructions.

Finally, the cups and donuts magic trick uses invertible continuous functions called homeomorphisms to deform shapes without tearing or gluing them.

Next: Presheaves and Topology.

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