The Yoneda Lemma
Alasdair M
1 Background The Yoneda Lemma is a fundamental result in category theory. I provide a proof, explaining all relevant categorical concepts. My main aim is to supply the reader with all the tools to understand this proof, along with some relevant motivation. Though only a reader with an exceptional memory will benefit from an entirely linear reading of any mathematical text, the sections of this document are intended to be read in order, with many parts relying on previously explained concepts. 1.1 Motivation The structure of a category is highly general and can be applied to almost any situation in mathematics. As such, learning facts about categories in general allows one to know at least a little bit about very nearly every mathematical structure. The Yoneda Lemma is the foremost such result, meaning it has a vast range of applications. For example, for those familiar with group theory, it generalizes Cayley’s Theorem. 1.2 A note on sources Most of the definitions I provide here are fairly standard; if not, this is noted. I do not claim that my proof is not replicated elsewhere, but it is my own work, though loosely based on the partial sketch of a proof found at [7]. Most of my knowledge of category theory is due to [3], which does not cover the Yoneda Lemma, so my comments beyond the preliminary definitions are original. 1.3 What is a category? Categories are the fundamental objects of study in category theory, but their definition can be somewhat opaque to an unfamiliar reader. Intuitively, a category is a collection of objects with arrows between them, along with rules for how to follow paths through multiple arrows, but it is often difficult to see how that notion can represent many of the structures that categories generalize. Small categories can represent objects such as an ordered set, or, if the reader is familiar with the notion, a group, while large categories can represent an entire algebraic system: for example, one could have the category of all ordered sets, or the category of all groups. Definition 1 A category 1 C consists of a collection Ob( C ) , called its objects , together with, for every pair ( A,B ) of objects, a set Hom( A,B ) , called the morphisms from A to B. If f is a morphism from A to B we write f : A → B, and call A the domain and B the codomain. Further to this, for each pair of morphisms f : A → B and g : B → C a category also specifies a morphism f ◦ g : A → C, called the composition of f with g. 2 This composition is 1 I use the term ‘category’ throughout to refer to what is technically called a locally small category ; in fact, there is a more general notion where the hom-sets can be proper classes, but this should not concern an unfamiliar reader. For a detailed axiomatic treatment of why such a distinction might be necessary, I recommend [6]. 2 Some authors use the opposite convention for composition of morphisms, requiring f ◦ g to exist when the domain of f is the codomain of g . I find this notation much less intuitive.
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