After seeing the answer to the question Category theory, a branch of abstract algebra,
I would like to ask
I do not know of any books that make such comparisons their main theme, but
texts on the categorical approach to universal algebra will often also discuss
how their approach relates to “traditional” universal algebra.
I think that Lawvere’s Thesis from 1963, available as reprint with commentary
is probably the best way to start. Beside the obvious advantage of being freely
available, it is from the main inventor of this type of categorical algebra,
includes comments from 2004 on subsequent developement and also has additional
references. Textbook treatments are the article of Pedicchio and Rovatti, the
book of Pareigis and the small book of Wraith (all cited in the references of
the above on page 20f).
Another good textbook treatment along with discussion is provided in Chapter 3
of the book by Borceux “Categorical Algebra II”.
Francis Borceux, Handbook of categorical algebra. 2,
Encyclopedia of Mathematics and its Applications 51,
Cambridge University Press, 1994
Slightly older textbooks are e.g.
Ernest G. Manes, Algebraic theories,
Graduate Texts in Mathematics, No. 26,
Springer-Verlag, New York 1976
Günther Richter, Kategorielle Algebra,
Studien zur Algebra und ihre Anwendungen 3,
Akademie-Verlag, Berlin 1979
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?