remarks on category theory
Here are some remarks that I have noted down while studying basic category theory under the supervision of George Janelidze. Some are quite technical, but one way or another they all have philosophical significance.
I do not claim any originality. In fact, most of the observations belong either to folklore or to Prof. Janelidze.
Both Set Theory and Category Theory can provide foundations for all of mathematics as currently practised. Hence, from a logical point of view, there is not much difference between the two alternatives. However, from a practical point of view, there is a huge difference. Each approach leads to different insights. Although it may be possible to derive a statement p in each framework, it may not be humanly possible to see p in one of the frameworks. For instance, monadicity of the power set functor could not be recognized before the advent of category theory, because consideration of the power set of the power set of the power set of the power set of a set was too unnatural from set theoretical point of view.
How hard is it to generate entirely novel and practically useful mathematical structures? To what extent does the development of mathematics rest on the recycling of old ideas? For example, vector bundle was thought to be a new structure. Then Category Theorists came to the scene and demonstrated that a vector bundle over a topological space X is just a vector space object in the slice category (Top↓X).
In Category Theory, all the important concepts can be stated in terms of each other. (Existence of an initial object can be seen as the existence of a left-adjoint. Existence of a left-adjoint can be seen as the existence of a family of initial objects. Existence of a colimit can be seen as the existence of a left-Kan extension. Existence of a left-Kan extension can be seen as the existence of a family of colimits. So on...) Is this a manifestation of the poverty of human imagination? Everything we do seems to be variations on the same theme.
It may be possible to reformulate a structure in a philosophically less troublesome manner. For instance, there are several equivalent ways of presenting adjuction data. The equational characterization circumvents the controversies involving the use of existential quantifiers over huge collections. How many of our philosophical problems have such syntactic origins? How many would disappear if only we knew the right way of reformulating them? (An example from physics: Lagrangian reformulation of classical mechanics introduces teleological aspects that render it philosophically more problematic than the usual formalism.)
Introduction of a property in a specific setting may have to rely on the same property being true in a greater setting. For instance, defining associativity of a tensor product ⊗ in a specific small category ε requires the use of associativity of the tensor product × in the category of all small categories. More specifically, associativity of ⊗ requires the existence of a natural isomorphism α, called the associator, with components α_(A,B,C) between (A⊗B)⊗C and A⊗(B⊗C). Here A,B,C are objects in ε, and α is a natural transformation from the composite functor ⊗o(1×⊗)oθ to the composite functor ⊗o(⊗×1) where θ is the (ε,ε,ε) component of the associator of ×.
Category Theory can guide us to the "right" way of defining certain objects. Here is an example from topology... Andrei Nikolayevich Tychonoff introduced a new product topology in his proof of the assertion that arbitrary products of compact topological spaces is again compact. The first reactions to the result were mixed. People had the impression that Tychonoff had chosen the product topology in a way that would make his theorem work. Therefore the proof had a air of arbitrariness to it. Years later, however, Category Theorists showed that Tychonoff had indeed imposed the "right" topology on the set theoretical product. He was now truly vindicated.
Category Theory can uncover the systematic biases of human mathematicians. When teaching arithmetic to kids, we first introduce sums. On the other hand, when teaching category theory to university students, we first introduce products. (Arithmetic sum of numbers 2 and 3 is the size of the coproduct (i.e. the disjoint union) of the sets {1,2} and {1,2,3}.) We do this because modern mathematics uses products more often than coproducts. Also, the actual constructions of coproducts tend to be more complicated and category-specific than those of products. The discrepancy arises from the choices made by human mathematicians: Having found it easier to work with products, they defined algebraic structures in a way that is more friendly to product formations. (This highlights one of the reasons why discovering a duality between two well-known categories is so important: It allows one to compute a colimit by passing to the dual category and computing the corresponding more tractable limit.)
Yoneda's embedding reveals one of the most important insights of Category Theory: One can safely replace an object with the network of its relationships. This is the mathematical analogue of what sociologist George Herbert Mead, one of the founders of Symbolic Interactionism, once wrote: "The individual mind can exist only in relation to other minds with shared meanings." This view is actually prevalent in Eastern cultures: "Many have noted that the 'meaning' of an individual in Japan is not intrinsic, implanted in a single person, as in the West. There is no unique soul or substance. The meaning is in relation to another. We can see this in the very word for 'human being'. It is composed of two Chinese characters, one meaning 'human' (nin) and the other meaning 'between' (gen). One way of interpreting this is that a human being is by definition a relationship, not a self-sufficient atom. Thus the very idea of the separate, autonomous 'person', the basic premise of Western thought and Western individualism, is missing in Japan." (Macfarlane - Japan Through the Looking Glass, p.76-77) Similarly, Henri Poincaré claimed the following: "The aim of science is not things themselves, as the dogmatists in their simplicity imagine, but the relations among things; outside these relations there is no reality knowable."
As opposed to Set Theorists, a Category Theorist does not define his structures directly. Instead he writes down a certain behaviour. This behaviour is specific enough so that any two objects exhibiting it have to be isomorphic to each other. The resulting lack of uniqueness is not big loss for the Category Theorist since he does not like to distinguish isomorphic objects which interact in exactly the same way with all other objects.
In Set Theory, one makes a canonical choice for all direct products right from the start: The ordered pair (a,b) is defined as the unordered pair {{a,b},{a}}, and AxB is defined as the set of all ordered pairs (a,b) such that a∈A and b∈B. But is there any good reason for choosing {{a,b},{a}} over the alternative {{a,b},{b}}? Instead of making such an arbitrary choice, is it not better to make no choice at all? A Category Theorist defines the product as an isomorphism class of sets which satisfy a certain universal property. He makes a choice from this class only when he is forced to manipulate AxB directly.
Although Category Theorists favour "isomorphism" over "equality", this conceptual hierarchy is not really genuine. One can relax the equational constraints, but one can never obliterate the presence of equalities. For instance, in the case of bicategories, equalities creep back into the picture in the form of coherence axioms.
The proposition that every surjective function has a right inverse is equivalent to the Axiom of Choice (AC). This categorical characterization of AC lends itself to an easy generalization: We say that AC holds in an arbitrary category ε if each epimorphisms in ε has a right inverse. (Note that epimorphisms in the category Set are the surjection functions.) AC is false in many categories. Consider, for instance, the category of Groups. Here all epimorphisms are surjective homomorphisms. Hence, given an epimorphism f, one can invoke the AC of Set to get a right inverse to f. But this will not help much, because the set theoretical right inverse may not be a group homomorphism. (Consider the homomorphism from Z to the factor group Z/2Z reducing integers modulo 2. It does not have a right inverse, because the only homomorphism from Z/2Z to Z is the zero homomorphism.) The categorical characterization pinpoints the source of AC's philosophical content: As objects assume greater structure AC is less likely to hold. (Recall that sets have no structure at all.)
Power of Topos Theory does not lie in its ability to replicate Set Theory. It lies in its ability to place Set Theory in the greater context of all mathematical theories. It allows us to see how exactly sets differ from other mathematical structures. It enables us to investigate intermediary structures and see exactly what properties lead to set-like, space-like or algebra-like behaviours.
Existence of exponents in Cat implies that functors know natural transformations. In other words, higher dimensional information is encoded inside the initial two dimensional data.
The notion of associativity can be seen as a special case of a monoid homomorphism. Let C be a category with products. If products are specified beforehand, we can view C as a monoid where the multiplication operator is just the product. (We assume that Ax1 is chosen as A for each object A of C.) We can define a set theoretical map from this monoid to the monoid of endomorphisms of C by sending each object A to the functor Ax(-). This map is a monoid homomorphism if and only if the product operator is associative.
As opposed to Set Theory, in Category Theory, topological groups is considered as a special case of groups rather than a generalisation. (Topological groups are simply the internal category of group objects in Top.)
Category of internal categories in modules is equivalent to the category of complexes. This equivalence takes natural transformations to chain homotopies. (Previously, chain homotopies could only be motivated via geometry. Now we know that they also spring from purely categorical considerations.) So, in a sense, abelian homological algebra is just category theory internal to modules. Hence, non-abelian homological algebra should just be category theory! (In reality, this becomes too vast a generalisation. For instance, instead of considering category theory internal to Set, Ronald Brown considers category theory internal to Grp which lies in between Mod and Set.)
Say your indexing set is I. For each i∈I, you make a choice of a set A_{i}. In other words, this family is just a function from I to the category Set. Viewing I as a category itself, the I-indexed families organise themselves into a presheaf category Fun(I,Set). Note that to define I-indexed families, one is naturally led to introduce the category Set whose objects constitute a collection that is larger than a set. (We call such collections as classes.) In other words, you are forced to venture beyond the most usual set theory even to define something as basic as families. Moreover, there is an arbitrariness in the definition of families. Should one require A_{i} to be disjoint? The answer to this question should not matter. But from a set theoretical point of view it does. If one defines an I-indexed family as a functor F from I to Set, then the disjointness assumption is not made. However, if one defines it as a function f from some set A to I, then the assumption is forced since the inverse of each i∈I under f is disjoint. Note that, from a categorical point of view, these two approaches are actually the same since the slice category Set/I is equivalent to Fun(I,Set). Notice how Category Theory is an expert at resolving arbitrarinesses. Before the notion of categorical equivalence, people knew instinctively that the above two approaches were the same but they had no mathematical means of articulating this thought. (Another well-known similarity that can not be stated without Category Theory is "Linear maps are like matrices.")
