Foundations of Category Theory

Introduction to Category Theory Why Category Theory? Category theory emerged in the mid-20th century as a way to find common patterns across different mathematical disciplines. Mathematicians noticed that certain constructions—like quotients, products, and duality—appeared again and again in algebraic topology, algebra, geometry, and other fields. Rather than proving the same results over and over in different contexts, category theory provides a unifying framework that captures these patterns in abstract form. The core insight of category theory is remarkably simple: instead of studying individual mathematical objects in isolation, we study the relationships between objects and the structure-preserving maps that connect them. This shift in perspective has proven extraordinarily powerful, and today category theory provides the foundational language for much of modern mathematics and computer science, including functional programming, type theory, and algebraic geometry. What Is a Category? At its heart, a category is a very simple structure consisting of two main ingredients: A class of objects (these are abstract entities—we don't assume they have any internal structure) A class of morphisms (also called arrows or maps) that go between objects Think of a category as a collection of objects with all the allowed ways to move from one object to another. The key requirement is that these morphisms satisfy certain composition rules, which we'll discuss shortly. Hom-Classes For any two objects $a$ and $b$ in a category, the collection of all morphisms from $a$ to $b$ is called the hom-class, denoted $\operatorname{Hom}(a,b)$. (The term "hom" comes from "homomorphism.") If we write a morphism as $f\colon a \to b$, this means $f$ is an element of $\operatorname{Hom}(a,b)$—that is, $f$ "goes from" $a$ "to" $b$. The crucial point is that objects themselves have no required internal structure. An object is just whatever it is; we only care about the morphisms that connect it to other objects. Morphisms: The Real Focus While we call the constituents of a category "objects and morphisms," the emphasis in category theory is really on the morphisms. Objects are important mainly because morphisms go between them, but the real mathematical content lives in the morphisms. Morphisms can take many forms depending on the category. In the category of sets, morphisms are ordinary functions. In other categories, morphisms might be group homomorphisms, continuous functions, differentiable maps, or even abstract algebraic structures themselves. The remarkable thing is that we can study all these situations using the same categorical framework. To make this concrete: in the category Set, the objects are all sets and the morphisms are all functions between sets. But we could also create a category where the single object is a monoid, and the morphisms are the elements of that monoid (composed by the monoid operation). The abstract definition of a category encompasses both situations. Composition: Putting Morphisms Together The key structure in a category is the ability to compose morphisms. If we have a morphism $f\colon a \to b$ and another morphism $g\colon b \to c$ (notice that the target of $f$ matches the source of $g$), then we can compose them to get a new morphism $g \circ f \colon a \to c$. This composition operation must satisfy two fundamental properties: Associativity: If we have three composable morphisms $f\colon a \to b$, $g\colon b \to c$, and $h\colon c \to d$, then: $$ (h \circ g) \circ f = h \circ (g \circ f) $$ This means we can compose three morphisms without worrying about the order in which we group them—the result is the same either way. This is exactly like function composition in ordinary mathematics. Identity Morphisms: Doing Nothing Every object $x$ in a category must have an identity morphism $1x\colon x \to x$ (sometimes written as $\text{id}x$). This morphism does nothing—it "returns" you to where you started. The identity morphism must satisfy the property that composing it with any other morphism just gives back that morphism: If $f\colon x \to y$, then $f \circ 1x = f$ If $f\colon x \to y$, then $1y \circ f = f$ Think of the identity morphism as playing the role of the number 1 in multiplication—when you multiply anything by 1, you get back what you started with. Important note: While this sounds simple, a useful principle in category theory is that objects are determined by their relationships to other objects. The identity morphism is the unique morphism that "relates" an object to itself in a trivial way. Example: The Category of Sets The most intuitive example is the category Set: Objects: All sets (finite or infinite) Morphisms: All functions between sets Composition: Ordinary composition of functions Identity morphisms: The identity function on each set This example may seem trivial, but it's surprisingly useful as a testing ground for categorical concepts. When you're learning category theory, you can often understand a new concept by first understanding it in the category of sets. Special Types of Morphisms Not all morphisms are created equal. Different morphisms can have different properties, and we have special names for morphisms with particular characteristics. These special types generalize familiar concepts from other areas of mathematics. Monomorphisms (Injective Morphisms) A morphism $f\colon a \to b$ is a monomorphism (often abbreviated as monic) if it is "left-cancellative"—that is, whenever we have two morphisms $g1, g2\colon x \to a$ with $f \circ g1 = f \circ g2$, we can conclude that $g1 = g2$. In other words, if two different paths through $a$ lead to the same point after passing through $f$, they must have been the same path all along. In the category of sets, this is exactly the notion of an injective function (one-to-one)—if $f$ maps two different elements to the same output, we could find two different functions that would be identified by $f$. The definition captures the idea of being "one-to-one" without requiring the concept of elements, so it works in any category. Epimorphisms (Surjective Morphisms) A morphism $f\colon a \to b$ is an epimorphism (often abbreviated as epic) if it is "right-cancellative"—that is, whenever we have two morphisms $h1, h2\colon b \to y$ with $h1 \circ f = h2 \circ f$, we can conclude that $h1 = h2$. In other words, if two different paths out of $b$ give the same result after being reached from $f$, they must be the same path. In the category of sets, this is exactly the notion of a surjective function (onto)—if $f$ doesn't map to everything in $b$, we could distinguish two functions on $b$ by where they send the missing elements. The definition captures the idea of being "onto" in a way that works in any category. Isomorphisms (Bijective Morphisms) A morphism $f\colon a \to b$ is an isomorphism if there exists a morphism $g\colon b \to a$ (called the inverse of $f$) such that: $f \circ g = 1b$ (composing $f$ with $g$ gives the identity on $b$) $g \circ f = 1a$ (composing $g$ with $f$ gives the identity on $a$) An isomorphism is a morphism that can be completely "undone"—we can go from $a$ to $b$ via $f$ and come back exactly to where we started via $g$. In categorical terms, isomorphic objects are essentially "the same" for all purposes within that category. In the category of sets, isomorphisms are precisely the bijective functions (one-to-one and onto). If $f$ is bijective, its inverse function $g$ is the unique function that reverses $f$'s action. Endomorphisms and Automorphisms An endomorphism of an object $a$ is simply a morphism $f\colon a \to a$ that starts and ends at the same object. The collection of all endomorphisms of $a$ is denoted $\operatorname{End}(a)$. An automorphism is an endomorphism that is also an isomorphism—that is, a morphism $f\colon a \to a$ for which an inverse $g\colon a \to a$ exists with $f \circ g = 1a = g \circ f$. The collection of all automorphisms of $a$ is denoted $\operatorname{Aut}(a)$. Automorphisms are the "symmetries" of an object—they are the ways an object can transform while staying fundamentally the same. Automorphism groups are central objects of study throughout mathematics. Retractions and Sections Two more special types of morphisms are worth learning because they relate to isomorphisms in important ways. A retraction is a morphism $f\colon a \to b$ that has a right inverse. This means there exists a morphism $g\colon b \to a$ such that $f \circ g = 1b$. In other words, if you go from $b$ to $a$ via $g$ and then back to $b$ via $f$, you end up where you started. A section is a morphism $f\colon a \to b$ that has a left inverse. This means there exists a morphism $g\colon b \to a$ such that $g \circ f = 1a$. In other words, if you go from $a$ to $b$ via $f$ and then back to $a$ via $g$, you end up where you started. Think of sections and retractions as "partial inverses"—they reverse the action of a morphism on one side, but not necessarily both sides. How These Special Morphisms Relate There are important relationships between all these special types of morphisms: Every retraction is an epimorphism, and every section is a monomorphism. This makes intuitive sense: if you have a way to reverse a morphism on the right (retraction), then the original morphism must be surjective. Similarly, if you can reverse a morphism on the left (section), it must be injective. Most importantly, here is the key relationship that ties everything together: A morphism $f$ is an isomorphism if and only if it is both a monomorphism and a retraction. Equivalently, a morphism is an isomorphism if and only if it is both an epimorphism and a section. This tells us that being "one-to-one" together with having a right inverse is enough to guarantee that $f$ is completely reversible. Similarly, being "onto" together with having a left inverse guarantees complete reversibility. And of course, an isomorphism is certainly both injective (monomorphism) and surjective (epimorphism), so these relationships all fit together coherently.