Galois, Abel, Cauchy, Cayley and other stalwarts of their era developed group theory. Kummer, Dedekind, Noether were among those who enriched our understanding of rings and ideals, especially over \mathbb Z and over \mathbb Q. This marked the beginning of what is coined as “Abstract Algebra” in many undergraduate texts and curricula. But as Herstein puts it, abstractness, being a relative term, this algebra of rings, fields, modules and vector-spaces might not be so abstract, as far as today’s development in Algebra is concerned. Perhaps I might call today’s research in Algebraic Geometry, the pinnacle of abstraction. Indeed, Grothendieck  took the subject to dizzying heights of abstraction. There was another abstractness introduced by founders of homological algebra including McLane and Eilenberg. “Category Theory” is sometimes notoriously humoured as “Abstract Nonsense”.

What is a Category?

A category, roughly speaking, consists of objects and relations between objects. To be more formal, I would say a category is a collection of objects (Obj) and for any two objects A, B \in Obj, a (possibly empty) set (Mor(A, B)) called as morphisms from A to B (which can be thought of as the set of all maps from A to B); and for any three objects A, B, C \in Obj, a law of composition (i.e. a map)

Mor(A, B) \times Mor(B, C) \to Mor(A, C)

satisfying a few properties that we may want them to satisfy. Before I start-off with some examples, let me tell a fundamental category that can stand as a guiding example in better understanding properties of categories. Set is a category whose objects are sets (not classes of sets; for I don’t want to start off paradoxes again :P) and morphisms are functions (or plain set-maps).

The properties a category \mathfrak C(Obj, Mor) is expected to satisfy are obvious —

  1. Two sets Mor(A, B) and Mor(A', B') are disjoint unless A=A' and B=B', in which case they are equal.
  2. For each object A of Obj, there is a morphism \text{id}_A \in Mor(A, A) which acts as left and right identity for the elements of Mor(A, B) and Mor(B, A) respectively, for all objects B\in Obj(\mathfrak C).
  3. The law of composition is associative whenever defined (whatever that means!)

(In (2) above, note that Mor(A,A) may contain more than one object; there are so many bijections from a set onto itself.)

Notice that these properties are satisfied by the aforementioned category Set quite trivially. Categories are abundant in Mathematics. Here are a few to begin with:

(PS: Fat rigorous books in Maths make comments like: Henceforth, by abuse of notation, we will denote a category by its objects. I refrain from making such obvious remarks.)

(PPS: Ignore those examples below which you don’t follow.)

  • PSet, (meaning, pointed sets) with nonempty sets with a fixed element (special point) of every set as objects, and morphisms as set maps with the property that special points are mapped to special points,
  • Grp, or the category with groups as objects and group-homomorphisms as morphisms,
  • R-Mod, or the category of R-modules with R-module-homomorphisms as morphisms,
  • FDkVec, or finite-dimensional vector spaces over a field k with k-linear homomorphisms as morphisms,
  • PPTop, or Pointed path-connected topological spaces with a special point and with continuous functions (respecting special-points) serving as morphisms,
  • \mathcal C^0(\mathbb R) where the objects are open sets in \mathbb R and morphisms are continuous functions on these open sets.
  • Top\mathbb X with objects as open sets of a topological space \mathbb X and inclusion maps as morphisms. Note that here, Mor(U, V) is empty if U \not\subseteq V and consists of one object, the inclusion map from U to V otherwise.
  • Hol, where objects are regions (open & path-connected subsets) of \mathbb C and morphisms are holomorphic functions on them,
  • kVar Let k be an algebraically closed field. (Affine or Projective) algebraic varieties constitute objects in this category and (variety) morphisms are morphisms here. (PS: If you know this category, then you know most of the categories above :P)

We also have something known as Subcategories; a subcategory is a category whose objects and morphisms are the objects and morphisms of another (bigger) category. A few examples to illustrate this —

  • PSet is a subcategory of Set.
  • Ab, the category of abelian groups  is a subcategory of Grp,
  • Rng = Category of rings (possibly without identity), Ring = Category of rings with identity and CRing = Category of commutative rings, Field = Category of fields.  Then Field \subset CRing \subset Ring \subset Rng (PS: This fancy notation is by Jacobson.)

Why study Category Theory?

Categories were introduced to make rigorous the meaning of the word natural used in many contexts. As an example, consider this. Given a k-vector space V and a basis of V, we have an explicit map from V to its dual, V^*, but there is no such explicit map without a basis. However, we have a natural map from V to its bidual, V^{**}, namely, v \mapsto (F_v: V^* \to k); F_v(f) = f(v).

A category, say of groups, has group-homomorphisms that take a group to another retaining the juice or information of the original group. We can then study a tough group by looking at its image in a relatively easier group and can deduce information about the tough group. In a somewhat similar fashion, we have maps (called as functors) that take objects and morphisms of one category to another. Thus, study of a tougher category can be simplified by looking at corresponding objects (and morphisms) in the other category.

For instance, a geometric assertion on affine varieties can be transferred to a statement in commutative algebra with the use of a functor that associates to each affine variety, its coordinate ring. Similarly, one observes that homeomorphism of path-connected topological spaces implies  group-isomorphism of their fundamental groups. Thus, if the fundamental groups are different, then the two spaces cannot be homeomorphic.

More on Functors in a future post!

Future posts:

  • Functors
  • Products & coproducts
  • Limits & colimits
  • Yoneda Lemma