Category Theory in Context by Emily Riehl | eBook

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Category Theory In Context

By Emily Riehl

Dover Publications, Inc.

CHAPTER 1

Categories, Functors, Natural Transformations

Frequently in modern mathematics there occur phenomena of "naturality".

Samuel Eilenberg and Saunders Mac Lane, "Natural isomorphisms in group theory" [EM42b]

A group extension of an abelian group H by an abelian group G consists of a group E together with an inclusion of G [??] E as a normal subgroup and a surjective homomorphism E ->> H that displays H as the quotient group E]/G. This data is typically displayed in a diagram of group homomorphisms:

0 -> G -> E -> H -> 0.

A pair of group extensions E and E' of G and H are considered to be equivalent whenever there is an isomorphism E? [congruent to] E' that commutes with the inclusions of G and quotient maps to H, in a sense that is made precise in §1.6. The set of equivalence classes of abelian group extensions E of H by G defines an abelian group Ext(H, G).

In 1941, Saunders Mac Lane gave a lecture at the University of Michigan in which he computed for a prime p that Ext(z[1/p]/z, z) [congruent to] zp, the group of p-adic integers, where [ ]/ is the Prüfer p-group. When he explained this result to Samuel Eilenberg, who had missed the lecture, Eilenberg recognized the calculation as the homology of the 3-sphere complement of the p-adic solenoid, a space formed as the infinite intersection of a sequence of solid tori, each wound around p times inside the preceding torus. In teasing apart this connection, the pair of them discovered what is now known as the universal coefficient theorem in algebraic topology, which relates the homology H* and cohomology groups H* associated to a space X via a group extension [ML05]:

(1.01.1) [MATHEMATICAL EXPRESSION OMITTED]

To obtain a more general form of the universal coefficient theorem, Eilenberg and Mac Lane needed to show that certain isomorphisms of abelian groups expressed by this group extension extend to spaces constructed via direct or inverse limits. And indeed this is the case, precisely because the homomorphisms in the diagram (1.1.1) are natural with respect to continuous maps between topological spaces.

The adjective "natural" had been used colloquially by mathematicians to mean "defined without arbitrary choices." For instance, to define an isomorphism between a finite-dimensional vector space V and its dual, the vector space of linear maps from V to the ground field , requires a choice of basis. However, there is an isomorphism between V and its double dual that requires no choice of basis; the latter, but not the former, is natural.

To give a rigorous proof that their particular family of group isomorphisms extended to inverse and direct limits, Eilenberg and Mac Lane sought to give a mathematically precise definition of the informal concept of "naturality." To that end, they introduced the notion of a natural transformation, a parallel collection of homomorphisms between abelian groups in this instance. To characterize the source and target of a natural transformation, they introduced the notion of a functor. And to define the source and target of a functor in the greatest generality, they introduced the concept of a category. This work, described in "The general theory of natural equivalences" [EM45], published in 1945, marked the birth of category theory.

While categories and functors were first conceived as auxiliary notions, needed to give a precise meaning to the concept of naturality, they have grown into interesting and important concepts in their own right. Categories suggest a particular perspective to be used in the study of mathematical objects that pays greater attention to the maps between them. Functors, which translate mathematical objects of one type into objects of another, have a more immediate utility. For instance, the Brouwer fixed point theorem translates a seemingly intractable problem in topology to a trivial one (0 ≠ 1) in algebra. It is to these topics that we now turn.

Categories are introduced in §1.1 in two guises: firstly as universes categorizing mathematical objects and secondly as mathematical objects in their own right. The first perspective is used, for instance, to define a general notion of isomorphism that can be specialized to mathematical objects of every conceivable variety. The second perspective leads to the observation that the axioms defining a category are self-dual. Thus, as explored in §1.2, for any proof of a theorem about all categories from these axioms, there is a dual proof of the dual theorem obtained by a syntactic process that is interpreted as "turning around all the arrows."

Functors and natural transformations are introduced in §1.3and §1.4 with examples intended to shed light on the linguistic and practical utility of these concepts. The category-theoretic notions of isomorphism, monomorphism, and epimorphism are invariant under certain classes of functors, including in particular the equivalences of categories, introduced in §1.5. At a high level, an equivalence of categories provides a precise expression of the intuition that mathematical objects of one type are "the same as" objects of another variety: an equivalence between the category of matrices and the category of finite-dimensional vector spaces equates high school and college linear algebra.

In addition to providing a new language to describe emerging mathematical phenomena, category theory also introduced a new proof technique: that of the diagram chase. The introduction to the influential book [ES52] presents commutative diagrams as one of the "new techniques of proof" appropriate for their axiomatic treatment of homology theory. The technique of diagram chasing is introduced in §1.6 and applied in §1.7 to construct new natural transformations as horizontal or vertical composites of given ones.

1.1. Abstract and concrete categories

It frames a possible template for any mathematical theory: the theory should have nouns and verbs, i.e., objects, and morphisms, and there should be an explicit notion of composition related to the morphisms; the theory should, in brief, be packaged by a category.

Barry Mazur, "When is one thing equal to some other thing?" [Maz08]

Definition 1.1.1. A category consists of

• a collection of objectsX, Y, Z, ...

• a collection of morphismsf, g, h, ...

so that:

• Each morphism has specified domain and codomain objects; the notation f: X ->Y signifies that f is a morphism with domain X and codomain Y.

• Each object has a designated identity morphism 1X: X ->X.

• For any pair of morphisms f, g with the codomain of fequal to the domain of g, there exists a specifiedcomposite morphismgf whose domain is equal to the domain of f and whose codomain is equal to the codomain of g, i.e.,:

[MATHEMATICAL EXPRESSION OMITTED]

This data is subject to the following two axioms:

• For any f: X ->Y, the composites 1Y f and f 1X are both equal to f.

• For any composable triple of morphisms f, g, h, the composites h(gf) and (hg) f are equal and henceforth denoted by hg f.

[MATHEMATICAL EXPRESSION OMITTED]

That is, the composition law is associative and unital with the identity morphisms serving as two-sided identities.

Remark 1.1.2. The objects of a category are in bijective correspondence with the identity morphisms, which are uniquely determined by the property that they serve as two-sided identities for composition. Thus, one can define a category to be a collection of morphisms with a partially-defined composition operation that has certain special morphisms, which are used to recognize composable pairs and which serve as two-sided identities; see [Ehr65, §I.1] or [FS90, §1.1]. But in practice it is not so hard to specify both the objects and the morphisms and this is what we shall do.

It is traditional to name a category after its objects; typically, the preferred choice of accompanying structure-preserving morphisms is clear. However, this practice is somewhat contrary to the basic philosophy of category theory: that mathematical objects should always be considered in tandem with the morphisms between them. By Remark 1.1.2, the algebra of morphisms determines the category, so of the two, the objects and morphisms, the morphisms take primacy.

Example 1.1.3. Many familiar varieties of mathematical objects assemble into a category.

(i) Set has sets as its objects and functions, with specified domain and codomain, as its morphisms.

(ii) Top has topological spaces as its objects and continuous functions as its morphisms.

(iii) Set* and Top* have sets or spaces with a specified basepoint as objects and basepoint-preserving (continuous) functions as morphisms.

(iv) Group has groups as objects and group homomorphisms as morphisms. This example lent the general term "morphisms" to the data of an abstract category. The categories Ring of associative and unital rings and ring homomorphisms and Field of fields and field homomorphisms are defined similarly.

(v) For a fixed unital but not necessarily commutative ring R, ModR is the category of left R-modules and R-module homomorphisms. This category is denoted by Vectk when the ring happens to be a fieldk and abbreviated as Ab in the case of Modz, as az-module is precisely an abelian group.

(vi) Graph has graphs as objects and graph morphisms (functions carrying vertices to vertices and edges to edges, preserving incidence relations) as morphisms. In the variant DirGraph, objects are directed graphs, whose edges are now depicted as arrows, and morphisms are directed graph morphisms, which must preserve sources and targets.

(vii) Man has smooth (i.e., infinitely differentiable) manifolds as objects and smooth maps as morphisms.

(viii) Meas has measurable spaces as objects and measurable functions as morphisms.

(ix) Poset has partially-ordered sets as objects and order-preserving functions as morphisms.

(x) ChR has chain complexes of R-modules as objects and chain homomorphisms as morphisms.

(xi) For any signature σ, specifying constant, function, and relation symbols, and for any collection of well formed sentences T in the first-order language associated to σ, there is a category Model whose objects are σ-structures that modelT, i.e., sets equipped with appropriate constants, relations, and functions satisfying the axioms T. Morphisms are functions that preserve the specified constants, relations, and functions, in the usual sense. Special cases include (iv), (v), (vi), (ix), and (x).

The preceding are all examples of concrete categories, those whose objects have underlying sets and whose morphisms are functions between these underlying sets, typically the "structure-preserving" morphisms. A more precise definition of a concrete category is given in 1.6.17. However, "abstract" categories are also prevalent:

Example 1.1.4.

(i)For a unital ring R, MatR is the category whose objects are positive integers and in which the set of morphisms from n to m is the set of m × n matrices with values in R. Composition is by matrix multiplication

[MATHEMATICAL EXPRESSION OMITTED]

with identity matrices serving as the identity morphisms.

(ii) A group G (or, more generally, a monoid) defines a category BG with a single object. The group elements are its morphisms, each group element representing a distinct endomorphism of the single object, with composition given by multiplication. The identity element e [member of] G acts as the identity morphism for the unique object in this category.

(iii) A poset (P, ≤) (or, more generally, a preorder) may be regarded as a category. The elements of P are the objects of the category and there exists a unique morphism x ->y if and only if x ≤ y. Transitivity of the relation "≤" implies that the required composite morphisms exist. Reflexivity implies that identity morphisms exist.

(iv) In particular, any ordinal a = {ß | ß< a} defines a category whose objects are the smaller ordinals. For example, 0 is the category with no objects and no morphisms. 1 is the category with a single object and only its identity morphism. 2 is the category with two objects and a single non-identity morphism, conventionally depicted as 0 -> 1. ω is the categoryfreely generated by the graph

[MATHEMATICAL EXPRESSION OMITTED]

in the sense that every non-identity morphism can be uniquely factored as a composite of morphisms in the displayed graph; a precise definition of the notion of free generation is given in Example 4.1.13.

(v) A set may be regarded as a category in which the elements of the set define the objects and the only morphisms are the required identities. A category isdiscrete if every morphism is an identity.

(vi) Htpy, like Top, has spaces as its objects but morphisms are homotopy classes of continuous maps. Htpy* has based spaces as its objects and basepoint-preserving homotopy classes of based continuous maps as its morphisms.

(vii) Measure has measure spaces as objects. One reasonable choice for the morphisms is to take equivalence classes of measurable functions, where a parallel pair of functions are equivalent if their domain of difference is contained within a set of measure zero.

Thus, the philosophy of category theory is extended. The categories listed in Example 1.1.3 suggest that mathematical objects ought to be considered together with the appropriate notion of morphism between them. The categories listed inExample 1.1.4 illustrate that these morphisms are not always functions. The morphisms in a category are also called arrows or maps, particularly in the contexts of Examples 1.1.4 and 1.1.3, respectively.

Remark 1.1.5. Russell's paradox implies that there is no set whose elements are "all sets." This is the reason why we have used the vague word "collection" in Definition 1.1.1. Indeed, in each of the examples listed in 1.1.3, the collection of objects is not a set. Eilenberg and Mac Lane address this potential area of concern as follows:

... the whole concept of a category is essentially an auxiliary one; our basic concepts are essentially those of a functor and of a natural transformation. ... The idea of a category is required only by the precept that every function should have a definite class as domain and a definite class as range, for the categories are provided as the domains and ranges of functors. Thus one could drop the category concept altogether and adopt an even more intuitive standpoint, in which a functor such as "Hom" is not defined over the category of "all" groups, but for each particular pair of groups which may be given. [EM45]

The set-theoretical issues that confront us while defining the notion of a category will compound as we develop category theory further. For that reason, common practice among category theorists is to work in an extension of the usual Zermelo–Fraenkel axioms of set theory, with new axioms allowing one to distinguish between "small" and "large" sets, or between sets and classes. The search for the most useful set-theoretical foundations for category theory is a fascinating topic that unfortunately would require too long of a digression to explore. Instead, we sweep these foundational issues under the rug, not because these issues are not serious or interesting, but because they distract from the task at hand.

For the reasons just discussed, it is important to introduce adjectives that explicitly address the size of a category.

(Continues...)

Excerpted from Category Theory In Context by Emily Riehl. Copyright © 2016 Emily Riehl. Excerpted by permission of Dover Publications, Inc..
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Category Theory in Context

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