New Levels of Abstraction: Homological Algebra and Category Theory
New Levels of Abstraction: Homological Algebra and Category Theory
Two of the more abstract branches of mathematics are homological algebra and category theory. Important progress was made in both during the first half of the twentieth century. Indeed, since the fields both arose in the latter part of the nineteenth century, virtually all work in them took place in the twentieth. While their practical effects may not be as great as their mathematical importance, research is still worth pursuing because the field of mathematics provides such an accurate description of the universe in which we live. This leads to the assumption that, even if these fields are seemingly of little import, the future may hold something more.
According to the website Eric Weisstein's "World of Mathematics," category theory is "the branch of mathematics which formalizes a number of algebraic properties of collections of transformations between mathematical objects (such as binary relations, groups, sets, topological spaces, etc.) of the same type, subject to the constraint that the collections contain the identity mapping and are closed with respect to compositions of mappings. The objects studied in category theory are called categories." This definition includes a link to "category," about which it is said, "A category consists of two things: a collection of objects and, for each pair of objects, a collection of morphisms (sometimes called 'arrows') from one to another." While these definitions sound slightly abstruse, they can be understood without much effort.
To start, an object is simply any mathematical construct, be it a group, a space, a manifold, or anything else that can be defined mathematically. While we tend to think of an object as a tangible "thing," mathematical objects are no less real, even if they are less tangible. For example, a ball is a sphere. An equation can be written that, when plotted in three-dimensional space, will produce a sphere. Although the ball can be touched and the mathematical construct is intangible, is the graphed sphere any less real than the ball? And, by extension, aren't all surfaces, lines, curves, shapes, and other objects described by mathematical equations equally real?
If two objects can be related to one another through some sort of consistent mathematical relationship, this relationship is called a "morphism," or a map. Thinking about it, the term "map" is not unreasonable so much as it is unexpected. The typical image that comes to mind when hearing the word "map" is of a street map, not a mathematical relationship. However, a street map is precisely that—the "real world" constructed as (or related to) an image on a sheet of paper by means of a mathematical expression, in this case, simply a factor by which distances and sizes are divided. In a 1:250,000 map, each real world distance is divided by a factor of 250,000, giving a map distance. Of course, although we know by experience that the real world is mapped onto our street map, mathematically it makes as much sense to say that the map served as a template for the real world; in other words, we could speculate that our home city was created by simply scaling up the features shown on our map. The mathematical relationships used to map mathematical objects onto one another are much more complex than simple multiplication or division, but this added complexity does not make the result any less a map in the mathematical sense.
Category theory understands and explicitly states the rules that link these mathematical objects together by mapping one onto the other. By formalizing relationships in this manner, these collections of related mathematical objects can be categorized.
Homological algebra, a somewhat related topic, is defined by Weisstein as "[a]n abstract algebra concerned with results valid for many different kinds of spaces. Modules (another sort of mathematical object in which members can be added together in certain ways) are the basic tools used in homological algebra." It's also worth noting that algebra is a type of mathematics that studies systems of numbers and the mathematical operations that can be performed between the members of these systems. Incidentally, when mathematicians speak of an algebra, they are describing a set of mathematical rules and operations much more complex than what is taught in secondary schools (which is usually referred to as arithmetic); used mathematically, algebra means something much more complex.
The last term that needs defining is "homology," which was first used by Henri Poincaré (1854-1912) to define a relation between manifolds that was mapped onto a manifold of higher dimension in a particular way. Although this, too, sounds abstruse, the three-dimensional world is more complex than a two-dimensional piece of paper. So building a house, for example, from a blueprint would be one way of mapping a manifold (of sorts) onto a manifold of higher dimension in a particular way described in the instructions to the blueprint. In recent years, however, homology has come to mean what used to be called a homology group, which is a slightly looser definition of the term, extending it to a space rather than simply a manifold (which is a kind of surface).
Taking all of this together, we can see that homological algebra is a way to describe the mathematical rules by which objects can be related to (or mapped onto) each other, just as a blueprint is mapped into a house by following the general rules of construction.
One of the more interesting impacts of category theory lies in its application to one of the more philosophical problems facing mathematicians: whether mathematics and the physical world are composed of sets and set theory operations on these sets (i.e., unions of sets, intersections of sets, etc.). Under set theory and much of formal mathematics, this is the case. However, category theory is an alternative to this way of thinking. In category theory, there is no distinction between objects and the operations on them (i.e., transformations, morphisms, etc.). Whether category theory is a more complete or more accurate way of mathematically describing the world remains to be seen, but it is an alternate way that, along with other alternatives, has forced mathematicians to look more closely at their work and, by so doing, to understand it better.
Homological algebra is perhaps more abstract than category theory, although both disciplines are seemingly far removed from everyday life. Homological algebra, like category theory, primarily affects the manner in which mathematicians view their work and the effect their work has on the rest of mathematics. In this sense, both category theory and homological algebra, while not of earth-shattering importance, are of more than passing interest to the field of mathematics.
Much of the interest in homological algebra lies in the fact that multi-dimensional spaces are very important to both mathematics and the real world. We live in a multidimensional space—a space with three linear dimensions and a time dimension. Some physics theories require 10 dimensions to work correctly, positing that six of them are "compactified" in the same manner a box can be smashed into a flat sheet, seeming to lose a dimension in the process.
From this perspective, it is important to be able to map features from 10 dimensional space onto three or four, just as having the mathematics to map 10-dimensional space might be useful to physicists exploring this realm. This is one of the ways that seemingly abstract mathematics can connect with the real world: the world is described with mathematics and, no matter how complex the mathematics seem to be, the world is still more complicated.
In general, too, we must remind ourselves that the laws of the universe seem written in mathematics. There have been a great many instances in which arcane mathematics have been found to be precisely what was needed to permit a more appropriate mathematical description of real physical phenomena. In fact, while we think of Albert Einstein (1879-1955) as a great physicist, his contemporaries thought of him as a mathematician because his breakthroughs were made possible by the application of new mathematical techniques that allowed him to formalize and describe his insights in a manner that was convincing to the scientific world. From that perspective, it might just be that homological algebra and category theory, while currently without much direct impact outside the world of mathematics, may end up being as useful to future generations as were the mathematics that gave us relativity theory.
Even without a direct physical impact, however, it is still important to research even the most abstract and arcane mathematics. One of the things that sets us apart from other animals is our ability to consciously explore beyond our immediate surroundings, even when we are fed and comfortable. By pushing against the boundaries, be they physical, geographic, or mathematical, we are engaging in the activities that make us human. And we tend to learn lessons that help us in our everyday life, even if this process takes awhile.
Studies into homological algebra or category theory may never lead to world peace or feed the hungry. But, then, on the other hand, maybe they will. Thus far in our history as an intelligent species, we have had much success at turning seemingly abstract theory into practical reality. But, even if these fields never affect the average person, they are still worth pursuing because, someday, we or our descendants will be able to take some measure of pride in having understood our universe even a little better.
P. ANDREW KARAM
Weibel, C.A. An Introduction to Homological Algebra. Cambridge University Press, 1994.
Eric Weisstein's World of Mathematics. http://mathworld.wolfram.com