In Steve Awodey's book on category theory, he claims the latter is a branch of abstract algebra. I've never seen such a classification before. Is this really correct?
Category Theory – Is Category Theory a Branch of Abstract Algebra?
abstract-algebracategory-theory
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Universal algebra discusses algebraic systems such as groups,rings,etc. independent of elements or specific examples of such systems-it discusses algebraic systems in general in terms of the operators and relations between those elements only. A algebraic system is defined as a nonempty set with at least one n-ary operation on it. We discuss then a specific kind of algebraic system and it's operations.For example, in universal algebra, we discuss the collection of all groups as a set with an binary associative operation and 2 UNARY operations corresponding to the general identity and the inverse of each element. No specifics about the elements are allowed to be discussed, only general principles unique to groups. Equational relations are added as axioms. In short, it is strictly a "big picture" approach to algebra.But note it's different from the "big picture" approach of category theory since it only discusses one kind of object at a time and does not consider the relations between collections of different kinds of objects.
Category theory takes this one step further by discussing the operations and relations between different kinds of collections of objects-note the objects do not necessarily have to be algebraic systems- codified by functors and commutative diagrams.
In many ways. category theory can be seen as a direct generalization of universal algebra the same way point set topology can be seen as a generalization of ordinary calculus,real and complex analysis. As point set topology strips away the specific algebraic and ordering properties of the real and complex Euclidean spaces to lay bare the common structures that makes continuity and convergence possible on such systems, category theory allows one to discuss the relations between collections of "the same" objects while universal algebra discusses the internal operations of single categories of a single kind-namely, algebraic systems.
At least,that's how I understand it. That help?
It seems that the idea of growth of groups is extremely relevant here. In the example in the question the operation may be viewed as concatenation of words, and so is very similar to the standard way in which we view the free group on $m$ generators $F(\mathbf{x}_m)$. The operation $V\circ W$ takes $O(|V|+|W|)$-time* to evaluate in $F(\mathbf{x}_m)$.
Every finitely-generated group can be constructed by taking such a (finite rank) free group $F(\mathbf{x}_m)$ and adding relators. These relators correspond to "short cuts" in our computation. For example, consider the free group $F(a)$ and add the relator $a^3=\epsilon$ (here, $\epsilon$ denotes the empty word) to obtain the group with presentation $\langle a\mid a^3\rangle$. (This group is actually cyclic of order $3$.) Now, I can evaluate $a^{100}\circ a^{34}$ to get $a^{134}$, but why would I? As I am in my group I know that $a^{100}=a$ and $a^{34}=a$ so $a^{100}\circ a^{34}=a\circ a=a^2$. The idea of growth of groups is to make sense of these "shortcuts" computationally. Roughtly, the "growth" of a group corresponds to how many group elements there are at distance $n$ from the identity element. Crucially, growth rate does not depend on the choice of generating set. Please see the above Wikipedia link for more details. However, I will leave you with some examples of groups and their growth rate (taken and amended from Wikipedia, of course!):
- A free group with a finite rank $k > 1$ has an exponential growth rate.
- A finite group has constant growth – polynomial growth of order $0$.
- A group has polynomial growth if and only if it has a nilpotent subgroup of finite index. This is a massive result due to Gromov see wiki. It was one of the first results to "link" analytic behaviour of a group with its algebraic structure.
- If M is a closed negatively curved Riemannian manifold then its fundamental group $\pi _{1}(M)$ has exponential growth rate. Milnor proved this using the fact that the word metric on $\pi_{1}(M)$ is quasi-isometric to the universal cover of $M$.
- $\mathbb{Z}^d$ has a polynomial growth rate of order $d$.
- The discrete Heisenberg group $H^3$ has a polynomial growth rate of order $4$. This fact is a special case of the general theorem of Bass and Guivarch that is discussed in the article on Gromov's theorem on groups of polynomial growth.
- The lamplighter group has an exponential growth.
- The existence of groups with intermediate growth, i.e. subexponential but not polynomial was open for many years. It was asked by Milnor in 1968 and was finally answered in the positive by Grigorchuk in 1984 (see Girgorchuk's group). This is still an area of active research.
- The class of triangle groups include infinitely many groups of constant growth, three groups of quadratic growth, and infinitely many groups of exponential growth.
*This is an easily-computed upper bound. You can probably do something clever to reduce this though.
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Best Answer
One could likewise argue that General Algebra (which encompasses groups, rings, lattices, and many other structures) is the study of a special class of categories, and so argue that General Algebra is a branch of Category Theory... (For example, George Bergman's General Algebra book is heavily category-theoretically flavored).
I would say rather that Category Theory has some very large areas of intersection with General/Abstract Algebra; I remember George Bergman saying once that in the 80s (?) there was a big conference at Berkeley/MSRI that invited both Universal Algebraists and Category Theorists, and that many times over the course of the conference they discovered that there were results that each "camp" had proven independently and did not realize the other "side" knew about them, or that there were questions that had been raised on the periphery of one whose answer was well-known by the other. (I hope I'm not misremembering and/or misreporting this!).
My particular (heavily algebra-biased) experience leads me to think that Category Theory is closer to abstract algebra than to other branches of pure mathematics (e.g., topology, analysis, etc). Don't know if I would go so far as to call it a "branch" of abstract algebra, though, any more than I would call it a branch of set theory (or set theory a branch of category theory).