Sections and Retractions

About two weeks ago, I went over the concepts of Sections and Retractions while I was studying Category Theory. Even though they are relatively simple within that field of study, it has been a journey to properly understand exactly what they are, and how to use them in real life examples. I had to look for several sources of definitions, and to read them many times, but now I feel confident I’ve managed to get the correct mindset.

I will be focusing this post on Sections and Retractions, and the terminology around them. I’ll start by giving you an intuitive approach, then I’ll go through their strict definitions, and at the end I will be providing real life examples. Let’s get into it…

Intuitively

Let’s say we have a group of animal species and a larger set of animals composed of many individuals of each specie. Within the large group, if you could relate each animal to a specific common ancestor, the relation from the group of species to each individual ancestor is a section.

A retraction based on the common ancestors co-exists in the definition of section, and reveals how every animal is related to their common ancestor as much as they are related to their corresponding specie too. We could see it as a way to sort the larger group into subgroups (however, a retraction could exist for a smaller group).

Both the section and retraction ultimately reveal the set of species, no matter how many times you try to follow the section with the retraction (but first the section then the retraction).

If these relationships happen to exist, you could be able to deduce the full picture even if you’re presented with some elements of it. For example, if you only know about all the animals and their common ancestor, you can immediately deduce the corresponding species.

In Topology, a section s of a bundle p:E→B allows the base space B to be identified with a subspace s(B) of E.

The concepts of sections and retractions eventually lead us to a couple of other definitions:

• The relationship between every individual and its ancestor is called an idempotency, which stands for an operation that gives the same result after its first execution (Wikipedia).

Just like clicking the elevator button repeatedly 👉🔘

• How the set of species is related to itself is called identity, which stands for an operation that results in exactly what it received (Wikipedia).

What you see is what you get.

Note: Does it sound confusing? Please reply with more intuitive ways of defining these concepts.

Visual proof of the Pythagorean identity. For any angle θ, The point (cos(θ),sin(θ)) lies on the unit circle, which satisfies the equation x2+y2=1. Thus, cos2(θ)+sin2(θ)=1. Also, cos(θ) and sin(θ) are both sections of the unit circle. Image source.

Strict definitions

We have been talking about a group of relationships between two objects of possibly many elements. Any relationship itself is properly named morphism, and the set of all the relationships between two objects is called the “hom” set. The object that originates any given relation is named domain, and the destination is called codomain. A morphism f between X and Y can be represented using the syntax f:X→Y, and two morphisms f and g can be composed as functions, where f(g(x))=z means the z result of f after g, for any element x of the domain of g.

An important rule to remember is that all of the elements of the domain must have one and only one relationship going out to any element in the codomain. However, more than one element of the domain can be mapped to a single element in the codomain.

Back to Sections and Retractions: For a morphism f from any domain A to any co-domain B, denoted as f:A→B …

• f is a section of another morphism g only if after f:A→B, the morphism g:B→A results in the identity of A, 1A.
• f is a retraction of another morphism g only if g:B→A followed by f:A→B results in the identity of B, 1B.
• If f is both a section and a retraction between A and B, both the domain and the codomain are isomorphic.

In the case where two morphisms, f:A→B and g:B→A result in the identity of A when composed as f(g(x))→1A, then:

• f is the section of g, because g after f is the identity of A, 1A.
• g is the retraction of f, because of the same reason.

f is a section of g, g is a retraction of f.

We can also deduce that if f is a section for g, and g is a retraction for f, then r is exactly f. So that, if f necessarily has a retraction r, the only possible section for f is r itself.

More resources:

As described by Wikipedia (on sections):

Wikipedia also states that “every section is a monomorphism, and every retraction is an epimorphism”, that “a monomorphism is an injective homomorphism” and that “a homomorphism is a structure-preserving map between two algebraic structures of the same type”. Also that “Epimorphisms are categorical analogues of surjective functions”. Therefore, sections are injective functions and so they require a left inverse, and retractions are surjective functions and so they require a right inverse.

Image found in: Understanding Visualisation: A Formal Foundation using Category Theory and Semiotics.

Now, specifically on retractions, Wikipedia says that: “In topology, a retraction is a continuous mapping from a topological space into a subspace which preserves the position of all points in that subspace. A deformation retraction is a mapping which captures the idea of continuously shrinking a space into a subspace.” this definition encompasses the book of Conceptual Mathematics: A First Introduction to Categories, where it says that retractions are easy to find if an idempotent map is found between all the elements of a larger object into itself, effectively “sorting” it and thus maintaining the order.

Real Life examples

Having sorted out the definitions, we can try to find these relationships in the world surrounding us.

Cross Sections

The name “sections” comes from the concept of “Cross Sections”. A Cross Section is composed of all the points within an N dimensional object that cross plane in a lower dimension.

Source. I’m sure my mother will get hungry if she sees this picture 😁

If you cut a coconut in half, it exhibits two circular shapes, just as a shadow behind any object facing a source of light exposes the shape of the object itself.

We can think on the concept of retraction to be how the more complex structure is reduced into a simpler one, and the section on how the simple shape resembles the greater one.

The House of Representatives

The United States House of Representatives is the lower chamber of the United States Congress. It is composed of representatives who sit in congressional districts which are allocated to each of the 50 states on a basis of population as measured by the U.S. Census, with each district entitled one representative.

There’s an isomorphic relationship between each representative and each district, however we can also build a greater category between the set of all people living in the United States, and the set of all chairs in the House of Representatives. In that case, we have:

• Two objects, one composed of the set of all people living in the United States, called A, and another composed of all the chairs in the House of representatives, called B.
• There’s a morphism f:A→B which indicates how, ideally, every person is being represented in B.
• The identity we’re looking for can only happen in the smaller set, since once we start retractions from it, we won’t be able to recover all of it’s elements.
• The existence of each representative indicates a section g:B→A. the seats will be occupied by a selection of people in A, so that each selected person has a specific seat in the House of Representatives (1B).
• The retraction of that section is f itself, for the same reason, retraction f after section g are the identity in B.
• There’s an idempotence in A, since every person is related to a representative.

Source. Charts are beautiful, aren’t they?

Idempotency in Fractions

There’s a great example in Conceptual Mathematics: A First Introduction to Categories that I’m going to share. Back when we went to school, we learned to simplify fractions through a reduction process where we had to:

• Cancel the greatest common factor in the numerator and in the denominator. So that 2/-10 would become 1/-5.
• If the denominator is negative, change the signs of both the numerator and the denominator, so that 1/-5 would become -1/5, and -3/-4 would become 3/4.

We can define this process to be a morphism “f” from the object of all fractions to itself. The nature of “f” is going to be idempotent, because no matter how many times we run this reduction process, we will always get the same result.

Categories, morphisms, sections, retractions, idempotences and isomorphisms may be unfortunately named, and perhaps even confusing for a while, but with a little patience and imagination, you might also see them everywhere around you. It took me some time to get the whole picture, but they don’t need to be hard. I hope that the intuitively section at the beginning of this post could have shown how general and simple these relationships are. In any case, if you find yourself in doubt, keep in mind that you can actually use their proper definitions to prove them. If you happen to have in mind more questions about their terrible naming though, I’m sure that a bit of history will illustrate the reasoning behind them — but we’ll leave that for another post.

If by any chance I got you interested, use this energy to continue learning, and perhaps share back what you discover ❤️. On the other hand, in the case that I’ve made a mistake, or you’ve spotted a fault in my approach, please let me know, I will appreciate it.

I will leave it here, people. Have a wonderful time!

Cheers! 🍷