Category Theory or: How to live without elements, using arrows instead (Part I)

Jun. 4, 2023

After my recent attempt to learn the core of category theory and exploration of Saunders Mac Lane’s insightful books, I have tried to organize my notes, thoughts, and highlights in this post. The post is divided into two parts: the origins of category theory are covered in the first one, and next comes the core machinery for working with categories.

Content Hierarchy :


Mathematics is said to have its origins in “human cultural activities,” as Mac Lane put it. These activities lead to some ideas, like the activity of observing, which leads to a notion of the “symmetry” of objects, and this general idea of “symmetry” is then formalized as a symmetry group of figures or formulae (which when defined on an infinite set becomes a transformation group). This perspective comes from the intuitionists, who argue that mathematics is a human construct, not an objective truth, and that is to say that there would be no math without the brain. The ideas that have their origins in these activities are said to have some intuitive content, which can then be formalized. By formal, we mean a list of rules, axioms, or methods of proof that can be applied without attention to the “meaning” but which give results that do have the correct interpretation.

ArgueProofLogical Connectives
CountingNextSuccessor; Ordinal Number
ObservingSymmetryTransformation Group

If we consider mathematics to be the science of number and space, then we can construct some formal notions. It is the morphosis of fact into form.

What is the idea of the formal?

“With Bourbaki, we hold that Mathematics deals with such “mother structures”. Against the historical order, we hold that they arise directly from the basic stuff of mathematics.”

Where does category theory come from?

Category Theory as a Foundation for Mathematics

We can use set theory as a foundation, primarily because essentially all mathematics is reducible to sets. Another way to look at this is that instead of looking at mathematics as a whole, it’s better viewed as its objects and mappings between them. This approach works for numerous branches of mathematics.

Set TheorySetsFunctions
TopologyTopological SpacesContinuous Maps
Euclidean GeometryInner Product SpacesOrthogonal Transformations

Machinery of Category Theory

What is a Category?


Functor Composition

Important Functors

Natural Transformations

Composition of Natural Transformations

Functor Category

Natural Isomorphisms

A natural isomorphism is an isomorphism in a functor category. If \(F : \mathbf{C} \to \mathbf{D}\) and \(G : \mathbf{C} \to \mathbf{D}\) are two functors, a natural isomorphism between them is a natural transformation \(\eta : F \Rightarrow G\) whose components are isomorphisms. In this case, the inverse natural transformation \(\eta^{−1} : G \Rightarrow F\) is given by \((\eta^{-1})_A = (\eta_A)^{-1}\). We write \(F \cong G\) when F and G are naturally isomorphic.

Equivalence of Categories

An equivalence (\(\simeq\)) of categories is a pair of functors, \(F\) and \(G\) such that,


Category of Presheaves


A presheaf is a functor, \(F: \mathbf{C}^{op} \to \textbf{Set}\) such that for any \(x \in \mathbf{C}\), \(Fx\) in \(\textbf{Set}\) is the set that represents the ways \(x\) can occur in \(F\) and any mapping \(f: x \to y\) where \(f,y \in \mathbf{C}\) the corresponding \(Ff : Fy \to Fx\) maps each of the \(y\)’s of \(Fy\) to each of the \(x\)’s in \(Fx\).

Representable Presheaf

The above presheaf \(F\) becomes a representable when it is naturally isomorphic to a hom-functor \(\textbf{hom}_{\mathbf{C}} (\_ , X) : \mathbf{C}^{op} \to \textbf{Set}\) which maps any object \(c \in \mathbf{C}\) to the hom-set \(\textbf{hom}_{\mathbf{C}} (c , X)\) and each \(f : c’ \to c\) where \(f, c’ \in \mathbf{C}\) to the function which maps each morphism \(c \to X\) to the composite \((c’ \to c) \to X\). Here the object \(X\) is determined uniquely upto an isomorphism in \(\mathbf{C}\) and is called the representing object.

Yoneda Lemma

“The set of morphisms from a representable presheaf \(y( c)\) into an arbitrary presheaf \(X\) is in natural bijection with the set \(X( c)\) assigned by \(X\) to the representing object \(c\).”

In simple words if we have a functor \(F\) that goes from \(\mathbf{C} \to \textbf{Set}\) then the natural transformation between \(F\) and the hom-functor \(\textbf{hom}_{\mathbf{C}} (\_ , c)\) corresponds by a natural isomorphism (set theoretic bijection) to the set \(Fc\).

Proof Sketch