Fundamental Concepts of the Theory

Category theory unifies mathematical structures in two different ways. First, as we have seen, almost every set theoretically defined mathematical structure with the appropriate notion of homomorphism yields a category. This is a unification provided within a set theoretical environment. Second, and perhaps even more important, once a type of structure has been defined, it is imperative to determine how new structures can be constructed out of the given one. For instance, given two sets A and B, set theory allows us to construct their Cartesian product A × B. It is also imperative to determine how given structures can be decomposed into more elementary substructures. For example, given a finite Abelian group, how can it be decomposed into a product of certain of its subgroups? In both cases, it is necessary to know how structures of a certain kind may combine. The nature of these combinations might appear to be considerably different when looked at from a purely set theoretical perspective.

Category theory reveals that many of these constructions are in fact certain objects in a category having a “universal property”. Indeed, from a categorical point of view, a Cartesian product in set theory, a direct product of groups (Abelian or otherwise), a product of topological spaces, and a conjunction of propositions in a deductive system are all instances of a categorical product characterized by a universal property. Formally, a product of two objects X and Y in a category C is an object Z of C together with two morphisms, called the projections, p : Z → X and q : Z → Y such that—and this is the universal property—for all objects W with morphisms f : W → X and g : W → Y, there is a unique morphism h : W → Z such that p ○ h = f and q ○ h = g.

Note that we have defined a product for X and Y and not the product for X and Y. Indeed, products and other objects with a universal property are defined only up to a (unique) isomorphism. Thus in category theory, the nature of the elements constituting a certain construction is irrelevant. What matters is the way an object is related to the other objects of the category, that is, the morphisms going in and the morphisms going out, or, put differently, how certain structures can be mapped into a given object and how a given object can map its structure into other structures of the same kind.

Category theory reveals how different kinds of structures are related to one another. For instance, in algebraic topology, topological spaces are related to groups (and modules, rings, etc.) in various ways (such as homology, cohomology, homotopy, K-theory). As noted above, groups with group homomorphisms constitute a category. Eilenberg & Mac Lane invented category theory precisely in order to clarify and compare these connections. What matters are the morphisms between categories, given by functors. Informally, functors are structure-preserving maps between categories. Given two categories C and D, a functor F from C to D sends objects of C to objects of D, and morphisms of C to morphisms of D, in such a way that composition of morphisms in C is preserved, i.e., F(g○ f) = F(g) ○ F(f), and identity morphisms are preserved, i.e., F(id_X) = id_FX. It immediately follows that a functor preserves commutativity of diagrams between categories. Homology, cohomology, homotopy, K-theory are all example of functors.

A more direct example is provided by the power set operation, which yields two functors on the category of sets, depending on how one defines its action on functions. Thus given a set X, ℘(X) is the usual set of subsets of X, and given a function f : X → Y, ℘(f) : ℘(X) → ℘(Y) takes a subset A of X and maps it to B = f(A), the image of f restricted to A in X. It is easily verified that this defines a functor from the category of sets into itself.

In general, there are many functors between two given categories, and the question of how they are connected suggests itself. For instance, given a category C, there is always the identity functor from C to C which sends every object/morphism of C to itself. In particular, there is the identity functor over the category of sets.

Now, the identity functor is related in a natural manner to the power set functor described above. Indeed, given a set X and its power set ℘(X), there is a function h_X which takes an element x of X and sends it to the singleton set {x}, a subset of X, i.e., an element of ℘(X). This function in fact belongs to a family of functions indexed by the objects of the category of sets {h_Y : Y → ℘(X) | Y in Ob(Set)}. Moreover, it satisfies the following commutativity condition: given any function f : X → Y, the identity functor yields the same function Id(f) :Id(X) → Id(Y). The commutativity condition thus becomes: h_Y ○ Id(f) = ℘(f) ○ h_X. Thus the family of functions h(-) relates the two functors in a natural manner. Such families of morphisms are called natural transformations between functors. Similarly, natural transformations between models of a theory yield the usual homomorphisms of structures in the traditional set theoretical framework.

The above notions, while important, are not fundamental to category theory. The latter heading arguably include the notions of limit/colimit; in turn, these are special cases of what is certainly the cornerstone of category theory, the concept of adjoint functors, first defined by Daniel Kan in 1956 and published in 1958.

Adjoint functors can be thought of as being conceptual inverses. This is probably best illustrated by an example. Let U : Grp → Set be the forgetful functor, that is, the functor that sends to each group G its underlying set of elements U(G), and to a group homomorphism f : G → H the underlying set function U(f) : U(G) → U(H). In other words, U forgets about the group structure and forgets the fact that morphisms are group homomorphisms. The categories Grp and Set are certainly not isomorphic, as categories, to one another. (A simple argument runs as follows: the category Grp has a zero object, whereas Set does not.) Thus, we certainly cannot find an inverse, in the usual algebraic sense, to the functor U. But there are many non-isomorphic ways to define a group structure on a given set X, and one might hope that among these constructions at least one is functorial and systematically related to the functor U. What is the conceptual inverse to the operation of forgetting all the group theoretical structure and obtaining a set? It is to construct a group from a set solely on the basis of the concept of group and nothing else, i.e., with no extraneous relation or data. Such a group is constructed “freely”; that is, with no restriction whatsoever except those imposed by the axioms of the theory. In other words, all that is remembered in the process of constructing a group from a given set is the fact that the resulting construction has to be a group. Such a construction exists; it is functorial and it yields what are called free groups. In other words, there is a functor F : Set → Grp, which to any set X assigns the free group F(X) on X, and to each function f : X → Y, the group homomorphism F(f) : F(X) → F(Y), defined in the obvious manner. The situation can be described thusly: we have two conceptual contexts, a group theoretical context and a set theoretical context, and two functors moving systematically from one context to the other in opposite directions. One of these functors is elementary, namely the forgetful functor U. It is apparently trivial and uninformative. The other functor is mathematically significant and important. The surprising fact is that F is related to U by a simple rule and, in some sense, it arises from U. One of the striking features of adjoint situations is precisely the fact that fundamental mathematical and logical constructions arise out of given and often elementary functors.

The fact that U and F are conceptual inverses expresses itself formally as follows: applying F first and then U does not yield the original set X, but there is a fundamental relationship between X and UF(X). Indeed, there is a function η : X → UF(X), called the unit of the adjunction, that simply sends each element of X to itself in UF(X) and this function satisfies the following universal property: given any function g : X → U(G), there is a unique group homomorphism h : F(X) → G such that U(h) ○ η = g. In other words, UF(X) is the best possible solution to the problem of inserting elements of X into a group (what is called “insertion of generators” in the mathematical jargon). Composing U and F in the opposite order, we get a morphism ξ : FU(G) → G, called the counit of the adjunction, satisfying the following universal property: for any group homomorphism g : F(X) → G, there is a unique function h : X → U(G) such that ξ ○ F(h) = g ○ FU(G) constitutes the best possible solution to the problem of finding a representation of G as a quotient of a free group. If U and F were simple algebraic inverses to one another, we would have the following identity: UF = I_Set and FU = I_Grp, where I_Set denotes the identity functor on Setand I_Grp the identity functor on Grp. As we have indicated, these identities certainly do not hold in this case. However, some identities do hold: they are best expressed with the help of the commutative diagrams:

U	η ○ U →	UFU		F	F ○ η →	FUF
	↘	↓U ○ η			↘	↓ξ ○ F
		U				F

where the diagonal arrows denote the appropriate identity natural transformations.

This is but one case of a very common situation: every free construction can be described as arising from an appropriate forgetful functor between two adequately chosen categories. The number of mathematical constructions that can be described as adjoints is simply stunning. Although the details of each one of these constructions vary considerably, the fact that they can all be described using the same language illustrates the profound unity of mathematical concepts and mathematical thinking. Before we give more examples, a formal and abstract definition of adjoint functors is in order.

Definition: Let F : C → D and G : D → C be functors going in opposite directions. Fis a left adjoint to G (G a right adjoint to F), denoted by F ⊣ G, if there exists natural transformations η : I_C → GF and ξ : FG → I_D, such that the composites

G

η ○ G →

GFG

G ○ ξ →

G

and

F

F ○ η →

FGF

ξ ○ F →

F

are the identity natural transformations. (For different but equivalent definitions, see Mac Lane 1971 or 1998, chap. IV.)

Here are some of the important facts regarding adjoint functors. Firstly, adjoints are unique up to isomorphism; that is any two left adjoints F and F' of a functor G are naturally isomorphic. Secondly, the notion of adjointness is formally equivalent to the notion of a universal morphism (or construction) and to that of representable functor. (See, for instance Mac Lane 1998, chap. IV.) Each and every one of these notions exhibit an aspect of a given situation. Thirdly, a left adjoint preserves all the colimits which exist in its domain, and, dually, a right adjoint preserves all the limits which exist in its domain.

We now give some examples of adjoint situations to illustrate the pervasiveness of the notion.

1. Instead of having a forgetful functor going into the category of sets, in some cases only a part of the structure is forgotten. Here are two standard examples:
   • There is an obvious forgetful functor U : AbGrp → AbMon from the category of abelian groups to the category of abelian monoids: U forgets about the inverse operation. The functor U has a left adjoint F: AbMon → AbGrpwhich, given an abelian monoid M, assigns to it the best possible abelian group F(M) such that M can be embedded in F(M) as a submonoid. For instance, if Mis ℕ, then F(ℕ) “is” ℤ, that is, it is isomorphic to ℤ.
   • Similarly, there is an obvious forgetful functor U : Haus → Top from the category of Hausdorff topological spaces to the category of topological spaces which forgets the Hausdorff condition. Again, there is a functor F : Top → Haus such that F ⊣ U. Given a topological space X, F(X) yields the best Hausdorff space constructed from X: it is the quotient of X by the closure of the diagonal Δ_X ⊆ X × X, which is an equivalence relation. In contrast with the previous example where we had an embedding, this time we get a quotient of the original structure.
2. Consider now the category of compact Hausdorff spaces kHaus and the forgetful functor U : kHaus → Top, which forgets the compactness property and the separation property. The left adjoint to this U is the Stone-Cech compactification.
3. There is a forgetful functor U : Mod_R → AbGrp from a category of R-modules to the category of abelian groups, where R is a commutative ring with unit. The functor Uforgets the action of R on a group G. The functor U has both a left and a right adjoint. The left adjoint is R ⊗ − : AbGrp → Mod_R which sends an abelian group G to the tensor product R ⊗ G and the right adjoint is given by the functor Hom(R, −) : AbGrp → Mod_R which assigns to any group G the modules of linear mappingsHom(R, G).
4. The case where the categories C and D are posets deserves special attention here. Adjoint functors in this context are usually called Galois connections. Let C be a poset. Consider the diagonal functor Δ : C → C × C, with Δ(X) = ⟨X, X⟩ and for f : X→ Y, Δ(f) = ⟨f, f⟩ : ⟨X, X⟩ → ⟨Y, Y⟩. In this case, the left-adjoint to Δ is the coproduct, or the sup, and the right-adjoint to Δ is the product, or the inf. The adjoint situation can be described in the following special form:

   X ∨ Y ≤ Z ⇕ X ≤ Z, Y ≤ Z

   Z ≤ X ∧ Y ⇕ Z ≤ Y, Z ≤ X

   where the vertical double arrow can be interpreted as rules of inference going in both directions.

5. Implication can also be introduced. Consider a functor with a parameter: (− ∧ X) : C→ C. It can easily be verified that when C is a poset, the function (− ∧ X) is order preserving and therefore a functor. A right adjoint to (− ∧ X) is a functor that yields the largest element of C such that its infimum with X is smaller than Z. This element is sometimes called the relative pseudocomplement of X or, more commonly, the implication. It is denoted by X ⇒ Z or by X ⊃ Z. The adjunction can be presented as follows:

   Y ∧ X ≤ Z	⇕
   Y ≤ X ⇒ Z

6. The negation operator ¬X can be introduced from the last adjunction. Indeed, let Z be the bottom element ⊥ of the lattice. Then, since Y ∧ X ≤ ⊥ is always true, it follows that Y ≤ X ⇒ ⊥ is also always true. But since X ⇒ ⊥ ≤ X is always the case, we get at the numerator that X ⇒ ⊥ ∧ X = ⊥. Hence, X ⇒ ⊥ is the largest element disjoint from X. We can therefore put ¬X =_def X ⇒ ⊥.
7. Limits, colimits, and all the fundamental constructions of category theory can be described as adjoints. Thus, products and coproducts are adjoints, as are equalizers, coequalizers, pullbacks and pushouts, etc. This is one of the reasons adjointness is central to category theory itself: because all the fundamental operations of category theory arise from adjoint situations.
8. An equivalence of categories is a special case of adjointness. Indeed, if in the above triangular identities the arrows η : I_C → GF and ξ : FG → I_D are naturalisomorphisms, then the functors F and G constitute an equivalence of categories. In practice, it is the notion of equivalence of categories that matters and not the notion of isomorphism of categories.

It is easy to prove certain facts about these operations directly from the adjunctions. Consider, for instance, implication. Let Z = X. Then we get at the numerator that Y ∧ X ≤ X, which is always true in a poset (as is easily verified). Hence, Y ≤ X ⇒ X is also true for all Yand this is only possible if X ⇒ X = ⊤, the top element of the lattice. Not only can logical operations be described as adjoints, but they naturally arise as adjoints to basic operations. In fact, adjoints can be used to define various structures, distributive lattices, Heyting algebras, Boolean algebras, etc. (See Wood, 2004.) It should be clear from the simple foregoing example how the formalism of adjointness can be used to give syntactic presentations of various logical theories. Furthermore, and this is a key element, the standard universal and existential quantifiers can be shown to be arising as adjoints to the operation of substitution. Thus, quantifiers are on a par with the other logical operations, in sharp contrast with the other algebraic approaches to logic. (See, for instance Awodey 1996 or Mac Lane & Moerdijk 1992.) More generally, Lawvere showed how syntax and semantics are related by adjoint functors. (See Lawvere 1969b.)

Dualities play an important role in mathematics and they can be described with the help of equivalences between categories. In other words, many important mathematical theorems can be translated as statements about the existence of adjoint functors, sometimes satisfying additional properties. This is sometimes taken as expressing the conceptualcontent of the theorem. Consider the following fundamental case: let C be the category whose objects are the locally compact abelian groups and the morphisms are the continuous group homomorphisms. Then, the Pontryagin duality theorem amounts to the claim that the category C is equivalent to the category C°, that is, to the opposite category. Of course, the precise statement requires that we describe the functors F : C → C° and G : C° → C and prove that they constitute an equivalence of categories.

Another well known and important duality was discovered by Stone in the thirties and now bears his name. In one direction, an arbitrary Boolean algebra yields a topological space, and in the other direction, from a (compact Hausdorff and totally disconnected) topological space, one obtains a Boolean algebra. Moreover, this correspondence is functorial: any Boolean homomorphism is sent to a continuous map of topological spaces, and, conversely, any continuous map between the spaces is sent to a Boolean homomorphism. In other words, there is an equivalence of categories between the category of Boolean algebras and the dual of the category of Boolean spaces (also called Stone spaces). (See Johnstone 1982 for an excellent introduction and more developments.) The connection between a category of algebraic structures and the opposite of a category of topological structures established by Stone's theorem constitutes but one example of a general phenomenon that did attract and still attracts a great deal of attention from category theorists. Categorical study of duality theorems is still a very active and significant field, and is largely inspired by Stone's result. (For recent applications in logic, see, for instance Makkai 1987, Taylor 2000, 2002a, 2002b, Caramello 2011.)

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