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Notes on Category Theory

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Notes on Category Theory NOTES ON CATEGORY THEORY RENRENTHEHAMSTER This set of notes presents an introduction to category theory mainly in the context of undergraduate abstract algebra. The reader is assumed to have had prior exposure to the theory of groups, rings and modules, but no prior exposure to category theory is assumed. This set of notes will cover basic concepts such as functors, natural transformations, and universal properties. I would like to acknowledge Aluffi’s "Algebra: Chapter 0" for many of the dia- grams and examples found in this set of notes. Date: April 2017. 1 1. The Basics of Category Theory One can imagine that the ultimate mathematician is one who can see analogies between analogies. { Stefan Banach Often in the study of mathematics, we come across a class of sets with additional mathematical structure imposed on them; for example groups are sets which obey certain axioms, and topological spaces are sets of points obeying certain axioms. We also care alot about how these objects \talk" to each other via maps that respect the structure of these objects (e.g. group homomorphisms for the case of groups, and continuous functions for the case of topological spaces). In a superficial sense, Category Theory is a common language that further abstracts our study of special mathematical objects. Just as cyclic groups and symmetry groups are special examples of groups, the collection of groups and the collection of rings are two examples of categories1. By de-emphasizing the specific stuctures that the objects possess, we can study, for example, what is generally meant by \isomorphism" between two objects, or the \product" of two objects, without caring whether they are groups, or rings, or something else. The reader will come to appreciate that many propositions that he/she has learnt or will learn are actually variations on the same theme, which could have been done in one shot using the framework of categry theory. In addition, many constructions could have been more cleanly described, and registered in the reader's mind as \the same" if understood in category-theoretic terms. All in all, category theory can be seen as moving further up one level of abstrac- tion, where we study classes of mathematical objects (i.e. the categories, e.g. the category of groups, the category of rings, the category of sets), and relationships between these classes (i.e. functors between categories, e.g. how the category of groups relate to the category of sets). 1.1. Key Definition and Many Examples. We begin with the most important definition of this text. Definition 1.1.1. A category C consists of: • A class ob(C) whose elements are called objects • A class Hom(C) of morphisms between objects. For every two objects A; B 2 ob(C),we write HomC(A; B) to denote the hom-class of all mor- phisms from A to B • A binary operation called composition of morphisms, such that for any three objects A; B; C we have a map HomC(A; B) × HomC(B; C) ! HomC(A; C). Composition is governed by two axioms: 1Technically the category also includes the respective homomorphisms, but this is just a mo- tivational exposition. NOTES ON CATEGORY THEORY 3 { Associativity: If f : A ! B; g : B ! C; h : C ! D, then h(gf) = (hg)f { Identity: For every object A, there exists a morphism 1A 2 HomC(A; A) called the identity morphism, such that for every morphism f : A ! B, we have 1Bf = f1A = f A common way of thinking about categories is to draw a schematic where arrows are used to represent morphisms of interest, such as in figure 1.1. Figure 1.1. When an arrow points from an object to itself, it has a special name: Definition 1.1.2. A morphism from an object to itself is called an endomor- phism. For an object A, we write EndC(A) for HomC(A; A). Here comes a series of examples. While the examples may appear increasingly abstract and random, rest assured that these examples will show up often in the remainder of the text. As such, study them carefully. Example 1.1.3. Here are some examples which should be familiar from previous study of abstract algebra. • Sets, as objects, and set-functions, as morphisms, form a category, called `Set'. • The category `Grp' has groups as objects and group homomorphisms as morphisms. The category `Ab' has abelian groups as objects and group homomorphisms as morphisms. • The category `Ring' has rings as objects and ring homomorphisms as mor- phisms. The category `CRing' has instead commutative rings as objects. • For a given ring R, the category `R-Mod' consists of left R-modules and the category `Mod-R' consists of right R-modules. The morphisms are module homomorphisms. Of course, if R is commutative, then the two categories are the same. • Topological spaces, as objects, and continuous functions, as morphisms, form a category, often called `Top'. Example 1.1.4. Given a set S, a preordering ≤ is a relation on S that is reflexive and transitive. Any preordered set (S; ≤) forms a category, where the objects are the members of S, the morphisms are arrows pointing from x to y whenever x ≤ y. This results in a thin category, i.e. a category with at most one morphism from an object to another. This makes it obvious how to define composition of morphisms. 4 RENRENTHEHAMSTER As specific examples of example 1.1.4, one may consider Z with the relation ≤, or a power set with the inclusion relation ⊆, or Z+ with the divisibility relation. Here are two ways of building new categories from old ones. Example 1.1.5. Given a category C, the dual category is the category Cop formed by reversing all the arrows. Example 1.1.6. Given two categories C and D, the product category C × D is defined as follows: • The objects are pairs (A; B) where A is an object of C and B is an object of D. • The morphisms from (A; B) to (A0;B0) are pairs (f; g) where f : A ! A0 is a morphism in C and g : B ! B0 is a morphism in D. • Composition of morphisms is done component-wise: (f; g) ◦ (f 0; g0) = (f ◦ f 0; g ◦ g0) Here are two slightly abstract examples, but these are actually recurring exam- ples throughout this set of notes. Example 1.1.7. Given a category C and an object A of C, define the category (C # A) called the slice category w.r.t. A or category of objects over A as follows: • The objects are pairs (Z; f) where Z 2 Ob(C) and f 2 HomC(Z; A). Pic- torially, an object of (C # A) is an arrow in figure 1.2. Figure 1.2. • If (Z1; f1) and (Z2; f2) are objects in (C # A), then a morphism σ : (Z1; f1) ! (Z2; f2) is defined to be a morphism σ : Z1 ! Z2 (slight abuse of notation here) which makes the following diagram (figure 1.3) commute in C: Figure 1.3. NOTES ON CATEGORY THEORY 5 • The composition of σ :(Z1; f1) ! (Z2; f2) and τ :(Z2; f2) ! (Z3; f3) is de- fined as follows: The commutativity of figure 1.4 implies that τσ (obtained from composition in C) makes the following diagram (figure 1.5) commute. This means τσ :(Z1; f1) ! (Z3; f3) can be taken as the composition. Figure 1.4. Figure 1.5. Example 1.1.8. Given a category C and an object A 2 C, define the category (A # C) called the coslice category w.r.t. A or category of objects under A where the objects are pairs (f; Z) where f 2 HomC(A; Z) and a morphism from (f1;Z1) to (f2;Z2) is a map σ : Z1 ! Z2 that makes the following diagram (figure 1.6) commute: Figure 1.6. Example 1.1.9. (Pointed Sets) In the context of example 1.1.8, take C to be Set and A to be a set with one element, say A = {∗}. Then an object is equivalent to picking (S; s) where S is a set and s 2 S, and a morphism (S; s) ! (T; t) corresponds to a set function σ : S ! T such that σ(s) = t. Hopefully the reader has gotten quite comfortable with the notion of a category. This final example is the most abstract in this section, but its importance will be unveiled shortly when we study products and coproducts. Example 1.1.10. Given a category C and two objects A; B 2 C, define the category 2 (C ⇒ A; B) which we shall call the double slice category w.r.t. A; B as follows: • Objects are diagrams of the form shown in figure 1.7 • Morphisms between the objects shown in figure 1.8 are commutative dia- grams of the form shown in figure 1.9 Similarly, we can define the category (A; B ⇒ C) which we shall call the double coslice category w.r.t. A; B by flipping all the arrows. 2unlike all other names in this section, the author is unaware of how this category is often referred to in the literature. As such, the author made up this name. 6 RENRENTHEHAMSTER Figure 1.7. Figure 1.8. Figure 1.9. 1.2. More on Morphisms. In the context of set theory, certain set-functions are special, e.g. they might be injective, surjective, or bijective. These properties are typically defined with reference to elements of the sets involved.
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