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.

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.

Bartosz Milewski's Programming Cafe

Presheaves and Topology

Topology as a Dietary Choice

What is Topology?

Continuity

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