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This post follows the previous two posts on Categories. (First and second). It explains how one uses the universal properties to prove nontrivial results. Since \LaTeX rendering in WordPress is not so efficient, I decided to upload the notes in my Notes page, and can also be accessed via this link.

This post is a continuation of the previous post – A mild introduction to Categories. Before I jump into the technical details of Categorical stuff like Functors, I would like to introduce the notion of a universal mapping property (henceforth referred to as UMP).

I won’t bog you down with the details of jargon that one has to absorb before getting the definition. Those interested in the definition of a UMP may refer the Wikipedia article here. Loosely speaking, a UMP of an object in a category is a property that characterizes it completely. In this post, I shall only present the universal properties of some familiar objects. We start with the property of a kernel of a group homomorphism.

Kernel of a group homomorphism: Let Grp be the category of groups. (One may similarly define the UMP of the kernel of a ring, module, field homomorphism). Let A and B be groups and f: A \to B be a group homomorphism. To familiarize ourselves with the categorical notation, we continue this example in categorical language- Let A, B \in \text{Obj} (Grp) and f \in \text{Mor} (A, B). There exists an object K\in \text{Obj} (Grp) and a morphism \iota \in \text{Mor} (K, A) such that the pair (K, \iota) satisfies the following properties:

  • f \circ \iota = 0 as a morphism (group homomorphism) from K to B.
  • Whenever $latex (K’, \iota’ \in \text{Mor}(K’, A))$ is such that f \circ \iota' = 0, then there is a unique morphism \Phi \in \text{Mor}(K', K) such that \iota \circ \Phi = \iota'.

In group-theoretic parlance, kernel (of a homomorphism) is a group which has an embedding into the preimage and whose image under the homomorphism is trivial. The group is unique in that any other group that has these properties must factor uniquely via the kernel. This property of the kernel, in fact, characterizes it.

We begin with examples that create new rings from old ones:

Direct sum of R-Modules: Given R-modules M_1, M_2, the direct product of M_1 and M_2 is an R-module M_1\oplus M_2 along with (inclusion) maps \iota_i: M_i \to M_1\oplus M_2\; (i=1,2) such that given any R-module N and any R-module homomorphisms \phi_i : M_i \to N, there is a unique homomorphism \Phi: M_1 \oplus M_2 \to N such that \Phi \circ \iota_i = \phi_i.

Note: This can be generalized to an arbitrary family of submodules of an R-module. The existence follows from by taking all but finitely many nonzero elements of each submodule.

Polynomial ring in one variable over a ring: Given rings R, S, a ring-homomorphism \phi: R \to S and an element a\in S, there exists a unique ring-homomorphism \Phi: R[X] \to S extending \phi such that \Phi(X)=a.

Quotient Ring: Given a ring R and its ideal I (possibly trivial or the whole ring), the quotient ring is a ring Q along with ring homomorphism \pi: R \to Q such that

  • \pi(I)=0,
  • For every ring S and homomorphism \phi:R\to S with \phi(I)=0, there is a unique ring homomorphism \Phi: Q\to S such that \phi = \pi \circ \Phi.

The UMP of a free module (or free group) is in some sense, more fundamental.

Free R-module: Let S be a set and R be a ring. A free R-module over S is an R-module F(S) along with an (injective) set map \iota: S \to F(S) such that given any R-module M and a set map \phi: S \to M, there is a unique R-module homomorphism \Phi: F(S) \to M such that \Phi \circ \iota = \phi.

The UMP of a ring of fractions of a ring is often useful.

Ring of fractions of a ring: Given a domain R and a multiplicatively closed subset S of R, (that contains 1 but no zerodivisors of R), the ring of fractions of R is a ring (denoted by) S^{-1}R along with a map \iota: R \to S^{-1}R such that given a ring Q and ring homomorphism \phi: R \to Q such that \phi(S) consists of units (of Q), then there is a unique homomorphism \Phi: S^{-1}R \to Q such that \Phi\circ\iota=\phi.

Note: If we take R to be a domain and S=R\backslash \{0\} then we get the field of fractions of R.

We give two examples to show that UMPs exist for non-algebraic objects too. (whatever that means!)

Quotient of a set: Let X be a set with an equivalence relation \tilde{} on it. The quotient of X with respect to \tilde{} is a set \overline{X} with a map \pi: X \to \overline{X} such that given any set Z and a map \phi: X\to Z with \phi(x) = \phi(x') whenever x \tilde{} x', then there is a unique map \Phi: \overline{X}\to Z such that \Phi\circ\pi=\phi.

Product of sets: Let X_i\; (i\in I) be an arbitrary family of sets. We define the product set of X_i‘sto be a set \Pi_{i\in I} X_i alongwith (projection) maps \displaystyle\pi_i : \Pi_{i\in I} X_i \to X_i such that for any __ (complete this!)

The above two properties – quotient and product of sets generalize immediately and can be used to define the UMP for Quotient topology and Box topology in the category Top of topological spaces. To state the UMP of the Product topology, one uses an analog of Direct Sum of Modules stated above.

Exercise: Deduce the UMP for cokernel of a module homomorphism.

Uniqueness of a UMP: The universal property of an object need not be unique. A trivial example to see this is as follows: The identity element of a group (denote it by e) satisfies the two following properties:

  1. Given any group G, there is a unique homomorphism f:\{e\} \to G.
  2. Given any group G, there is a unique homomorphism f: G \to \{e\}.

Future post:

  • A small application of UMP.

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

About me

Abhishek Parab

I? An Indian. A mathematics student. A former engineer. A rubik's cube addict. A nature photographer. A Pink Floyd fan. An ardent lover of Chess & Counter-Strike.

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