After seeing the answer to the question Category theory, a branch of abstract algebra, I would like to ask
Are there literature discussing the difference/indifference/comparison between category theory and universal algebra?
2 Answers
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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?
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1$\begingroup$ Thanks. Correct me if I am wrong, my interpretation of your answer is, universal algebra is intra-structure while category theory is inter-structure. Is there any example that universal algebra also put its hands on non-algebraic objects. $\endgroup$– NobodyNov 6, 2011 at 4:34
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$\begingroup$ Although this is a very nice answer, I hesitate to accept it yet because I am still waiting for links to books/papers for more info. Thanks. $\endgroup$– NobodyNov 6, 2011 at 5:39
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$\begingroup$ There are two classes, one is the class of all groups, the other is the class of all semigroups. The group class is a subclass of the semigroup class because all groups are semigroups. Does the study of the inter-relationships between these two classes belong to category theory or universal algebra, or both? $\endgroup$– NobodyNov 6, 2011 at 8:35
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$\begingroup$ @Mathemagician1234 Good answer, by the way I've one question for you: you said that universal algebra deals with operations defined on just one object, so how modules come in the context of universal algebra? $\endgroup$ Nov 23, 2011 at 10:57
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2$\begingroup$ @ineff (continued) Defining an R-module rigorously in terms of set theory is a little tricky when you stop to think about it.I distinctly remember asking Kenneth Kramer this question in my honors algebra class.He made a terrific observation:One can think of an R-module as a group action where the scalar ring elements act on the Abelian group of "vectors". The main set here in this case is the Cartesian product of the scalar ring R and the nonempty set which is the Abelian group and all corresponding operations are defined on the product set. $\endgroup$ Nov 24, 2011 at 19:49
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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